Somewhere between openended, creative thinking and the focused learning of content lies problem solving, the analysis and solution of tasks and situations that are somewhat complex or ambiguous and that pose difficulties, inconsistencies, or obstacles of some kind (Mayer & Wittrock, 2006)^{[1]}. Problem solving is needed, for example, when a physician analyzes a chest Xray: the photograph of the chest is far from clear and requires skill, experience, and resourcefulness to decide which foggylooking blobs on the Xray to ignore, and which to interpret as real physical structures (and real medical concerns). Problem solving is also needed when a grocery store manager has to decide how to improve the sales of a product: should she put it on sale at a lower price, or increase publicity for it, or both? And will these actions actually increase sales enough to pay for their costs?
Problem solving happens in classrooms, too, when teachers present tasks or challenges that are in some way complex or for which the path to a solution is not straightforward or obvious. The responses of students to such problems, as well as the strategies that teachers use to assist them, show some of the key features of problem solving when it happens in school. Consider this example—and especially students’ responses to it. We have numbered and named the paragraphs to make it easier to comment on them afterwards:
 Scene #1: A Problem To Be Solved
 The teacher gave these instructions: “Can you connect all of the dots below using only four straight lines?” She drew the following display on the chalkboard:


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 The problem itself and the procedure for solving it seemed very clear: simply experiment with different arrangements of four lines. But two volunteers who tried doing it at the board were unsuccessful. Several others worked at it at their seats, but also without success.
 Scene #2: Coaxing Students To Reframe the Problem
 When no one seemed to be getting it, the teacher asked, “Think about how you’ve set up the problem in your mind—about what you believe the problem is about. For instance, have you made any assumptions about how long the lines ought to be? Don’t stay stuck on one approach if it’s not working!”
 Scene #3: Alicia Abandons a Fixed Response
 After the teacher said this, Alicia did indeed think about how she saw the problem. “The lines need to be no longer than the distance across the square,” she said to herself. So she tried several more solutions, but none of them worked either.
 The teacher walked by Alicia’s desk and saw what Alicia was doing. She repeated her earlier comment: “Have you assumed anything about how long the lines ought to be?”
 Alicia stared at the teacher blankly, but then smiled and said, “Hmm! You didn’t actually say that the lines could be no longer than the matrix! Why not make them longer?” So she experimented again using oversized lines and soon discovered a solution:


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 [Note: Displaying this solution will require a graphic here using lines—not sure yet how to do this on a Wiki.]
 Scene #4: Willem’s General Strategy: Assume It Is a Trick Question
 Meanwhile, Willem worked on the problem. As it happened, Willem loved puzzles of all kinds, and had had ample experience with them. He had not, however, ever seen this particular problem. “It must be a trick,” he said to himself, because he knew from experience that problems posed in this way often were not what they first appeared to be. He mused to himself: “Think outside the box, they always tell you…” And that was just the hint he needed: he drew lines outside the box by making them longer than the matrix and soon came up with this solution:


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 [Note: Use same solution as Alicia’s but rotated 90 degrees clockwise.]
 Scene #5: Rachel Recalls a Specific Strategy Learned Previously
 When Willem and Alicia began work, Rachel took one look at the problem and knew the answer: she had seen this problem before, though not for a long time and she could not at first remember what the solution was. She had also seen other drawingrelated like it, and knew that their solution always lay with making the lines longer, shorter, or differently angled than first expected. After staring at the dots briefly, she wrote down a solution even before Alicia or Willem had begun serious work on theirs:


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 [Note: Use same solution as Alicia’s but rotated 180 degrees (i.e. rotated 90 degrees from Willem’s as well).]
This story illustrates several common features of problem solving (though not all of them). Consider these four features in particular: the structure or constraints set by the problem, the effect of structure on how to solve a problem, mental obstacles to solving problems, and common techniques for solving problems.
Wellstructured versus Illstructured ProblemsEdit
Problems vary in how well they provide information needed to solve the problem, and in how many rules or procedures might contribute to a solution. A wellstructured problem provides much or all of the information needed and can in principle be solved using relatively few clearly understood rules. Classic examples are the word problems often taught in math lessons or classes: everything you need to know is contained within the stated problem and the solution procedures are relatively clear and precise. An illstructured problem has the converse qualities: the information is not necessarily within the problem, solution procedures are potentially quite numerous, and a multiple solutions are likely (Voss, 2006)^{[2]}. Extreme examples are problems like “How can the world achieve lasting peace?” or “How can teachers insure that students learn?”
By these definitions, the ninedot problem is relatively wellstructured—though not completely. Most of the information needed for a solution is provided in Scene #1: there are nine dots shown and instructions given to draw four lines. But not all necessary information was in fact given: it turned out (Scene #2: “A Hint”) that students needed to consider lines that were longer than implied in the teacher’s original statement of the problem. Students had to “think outside the box,” in this case literally.
Heuristics and AlgorithmsEdit
Not only do problems vary in how completely they are structured, but so do solution procedures. An algorithm is a set procedure for solving a particular kind of problem in a particular way, like the procedures for multiplying or dividing two numbers or the instructions for using a computer (Leiserson, et al., 2001)^{[3]}. Algorithms work best when a problem is very wellstructured and there is no question about whether the algorithm is appropriate for the problem. In that situation it pretty much guarantees a correct solution. They do not work well, however, with illstructured problems, where they are ambiguities and questions about how to proceed or even about precisely what the problem actually is. In those cases it is more effective to use heuristics, which are general strategies—rules of thumb—that often work or that may provide partial solutions. When beginning research for a term paper, for example, a useful heuristic is to scan the library catalogue for titles that look relevant. There is no guarantee that that this strategy will yield the books you most need for the paper, but the strategy works enough of the time to make it worth trying.
In the ninedot problem, most students began in Scene #1 with a simple algorithm that can be phrased like this: “Draw one, then draw another, and another, and another.” Unfortunately this simple procedure did not produce a solution, so they had to find other ways to find a solution. Two of the alternatives are described in Scenes #3 (for Alicia) and 4 (for Willem). Of these, Willem’s response resembled a heuristic the most: he knew from experience that a good general strategy that often worked for such problems was to suspect a deception or trick in how the problem was originally stated. So he set out to question what the teacher had meant by the word line, and came up with a solution as a result.
Common Obstacles to Solving ProblemsEdit
The example also illustrates two common problems that sometimes happen during problem solving. One of these is functional fixedness: a tendency to regard the functions of objects or ideas as fixed (German & Barrett, 2005)^{[4]}. Over time, we might get so used to one particular purpose for an object that we overlook other possible uses. We may think of a dictionary, for example, as always “just” something to verify spellings and definitions, but it also can function as a gift, a doorstop or a footstool. For students working on the ninedot matrix problem, the notion of a line was also initially fixed; they assume it something that connected dots but could not extend beyond the dots.
Functional fixedness is also a form of response set, the tendency for a person to frame or think about each subsequent problem in a series in the same way as the previous problem, even when doing so is not appropriate. In the example of the ninedot matrix described above, students often tried one solution after another, but each solution was constrained by a set response not to extend any line beyond the matrix. Similarly, an even more complex example of response set is illustrated in Figure 81, which is about a classic problem of filling water jars to specified levels (Luchins & Luchins, 1994)^{[5]}.
Both functional fixedness and response set are pitfalls or obstacles in problem representation, the way that a person understands and organizes information provided in a problem. If information is misunderstood or used inappropriately, then mistaken solutions are likely—if indeed the problem can be solved at all. With the ninedot matrix problem, for example, understanding the instruction to draw four lines as meaning “draw four lines entirely within the matrix” means that the problem simply could not be solved.
But this is not the only kind of misunderstanding or misrepresentation possible. Consider this problem: “The number of water lilies on a lake doubles each day. Each water lily covers exactly one square foot. If it takes 100 days for the lilies to cover the lake exactly, how many days does it take for the lilies to cover exactly half of the lake?” If you think that the size of the lilies affects the solution to this problem, you have not represented the problem correctly. The information about lily size is not relevant to the solution, and only serves to distract from the truly crucial information, that the lilies double their coverage each day. (The answer, incidentally, is that the lake is half covered in 99 days; can you think why?)
Strategies To Assist Problem SolvingEdit
Just as there are common obstacles that often happen in all sorts of problem solving, there are also strategies that often help or support the process, regardless of the content specific in the problem (Thagard, 2005)^{[6]}. One helpful strategy is problem analysis—identifying parts of the problem and working on the parts separately. Analysis is especially useful when a problem is complex. Consider, for example, this illstructured problem: “Devise a plan to improve bicycle transportation in the city.” With a problem like this, try identifying its parts or component subproblems, such as 1) installing bicycle lanes on busy streets, 2) educating cyclists and motorists to ride safely, 3) fixing potholes on streets used by cyclists, and 4) revising traffic laws that interfere with cycling. Each component is itself a separate problem, and the solution of each contributes, but is not equivalent to, a solution to the original, larger problem.
Another helpful strategy is working backward, from a final solution to the originally stated problem. This approach is especially helpful when a problem is wellstructured but also has misleading elements when approached in a forward or normal direction. The water lily problem described above is a good example: starting with when all the lake is covered (Day 100), ask what day would it therefore be half covered (answer: by the terms of the problem, it would be the day before, or Day 99). Working backward in this case encourages reframing the extra information in the problem (i.e. the size of each water lily) as merely distracting, not as crucial to a solution.
A third helpful strategy is analogical thinking—using knowledge or experiences with similar features or structures to help solve the problem at hand (Bassok, 2003)^{[7]}. In devising a plan to improve bicycling in the city, for example, an analogy of cars and bicycles is helpful in thinking of solutions: improving car driving requires many of the same measures (improving the roadways, educating drivers) as improving bicycling. Even solving much simpler, more basic problems than this is helped by using analogies. A firstgrade student can partially decode unfamiliar printed words by analogy to words he or she has learned already. If the child cannot yet read the word screen, for example, he can note that part of this word looks similar to words he may already know, such as seen or green, and from this observation derive a clue about reading the word screen. Teachers can assist this process, as you might expect, by suggesting reasonable, helpful analogies for students to consider.
ReferencesEdit
 ↑ Mayer, R. & Wittrock, M. (2006). Problemsolving transfer. In D. Berliner & R. Calfee (Eds.), Handbook of Educational Psychology, pp. 4762. Mahwah, NJ: Erlbaum.
 ↑ Voss, J. (2006). Toulmin’s model and the solving of illstructured problems. Argumentation, 19(3), 321329.
 ↑ Leiserson, C., Rivest, R., Cormen, T., & Stein, C. (2001). Introduction to algorithms. Cambridge, MA: MIT Press.
 ↑ German, T. & Barrett, H. (2005). Functional fixedness in a technologically sparse culture. Psychological Science, 16(1), 15.
 ↑ Luchins, A. & Luchins, E. (1994). The waterjar experiment and Einstellung effects. Gestalt Theory: An International Interdisciplinary Journal, 16(2), 101121.
 ↑ Thagard, R. (2005). Mind: Introduction to Cognitive Science, 2nd edition. Cambridge, MA: MIT Press.
 ↑ Bassok, J. (2003). Analogical transfer in problem solving. In Davidson, J. & Sternberg, R. (Eds.). The psychology of problem solving. New York: Cambridge University Press.