Some problems can be successfully solved by following specific, step-by-step instructions—that is, by using an algorithm. We can correctly assemble the pieces of a new bookcase by following the directions for assembly that come with the package. We can calculate the length of a slanted roof by using the Pythagorean theorem. When we follow an algorithm faithfully, we invariably arrive at a correct solution.

However, the world presents many problems for which no algorithms exist. There are no rules we can follow to identify a substitute metal ship, no list of instructions to help us address the destruction of rain forests. In the absence of an algorithm, learners must instead use a heuristic, a general problem-solving strategy that may or may not yield a successful outcome. For example, one heuristic that we might use in solving the deforestation problem is this: Identify a new behavior that adequately replaces the problem behavior (i.e., identify another way that peasant farmers can meet their survival needs). For another example of a heuristic, consider the addition problem in the exercise that follows.

Experiencing Firsthand • Grocery Shopping- Solve this addition problem as quickly as you possibly can:

You are purchasing three items at the store, at these prices:

\$19.95
\$39.98
\$29.97

About how much money are you spending? (Don’t worry about a possible sales tax.)

The fastest way to solve this problem is to round off and approximate. The first item costs about \$20, the second about \$40, and the third about \$30; therefore, you are spending about \$90 on your shopping spree. Rounding is often an excellent heuristic for arriving quickly at approximate answers to mathematical problems.

At school, students typically get far more practice solving well-defined problems than ill-defined ones, and they are taught many more algorithms than heuristics. For example, they are likely to spend more school time learning problem-solving strategies useful in determining the length of planks needed for a treehouse roof than strategies applicable to the problem of deforestation. And they are apt to spend more time using laws of physics to predict when battleships will float than wrestling with ways of preventing the conflicts that require those battleships in the first place. But many real-world problems cannot be solved with cut-and-dried algorithms. Furthermore, few algorithms exist for solving problems outside the domains of mathematics and science.

Problem-solving strategies, algorithms and heuristics alike, are often specific to particular content domains. But here are several general problem-solving heuristics that students may find helpful in a variety of contexts:

• Identify subgoals. Break a large, complex task into two or more specific subtasks that can be more easily addressed.
• Use paper and pencil. Draw a diagram, list a problem’s components, or jot down potential solutions or approaches.
• Draw an analogy. Identify a situation analogous to the problem situation, and derive potential solutions from the analogy.
• Brainstorm. Generate a wide variety of possible approaches or solutions—including some that might initially seem outlandish or absurd—without initially evaluating any of them. Once a lengthy list has been created, evaluate each item for its potential relevance and usefulness.
• “Incubate” the situation. Let a problem remain unresolved for a few hours or days, allowing time for a broad search of long-term memory for potentially productive approaches.(J. R. Anderson, 1990; J. E. Davidson & Sternberg, 1998, 2003; H. C. Ellis & Hunt, 1983; Halpern, 1997a)

#### Teaching Problem-Solving Strategies

Occasionally students develop problem-solving strategies on their own. For instance, many children invent simple addition and subtraction strategies long before they encounter arithmetic at school (Carpenter & Moser, 1984). But without some formal instruction in effective strategies, even the most inventive of students may occasionally resort to unproductive trial and error to solve problems.

To be truly effective problem solvers, students must have a solid grounding in—that is, a conceptual understanding of—the subject matter in question (more about this point shortly). But they also benefit from explicit instruction in the use of both algorithms and heuristics. The following are some strategies we might use:

For teaching algorithms:

• Describe and demonstrate specific procedures and the situations in which each can be used.
• Provide worked-out examples of algorithms being applied, and ask students to explain what is happening in each step.
• Help students understand why particular algorithms are relevant and effective in certain situations.
• When a student’s application of an algorithm yields an incorrect answer, look closely at what the student has done, and locate the trouble spot

For teaching heuristics:

• Give students practice in making ill-defined problems more specific and well defined.
• Teach heuristics that students can use in situations where no specific algorithms apply; for example, encourage rounding, identifying subgoals, and drawinganalogies.

For teaching both algorithms and heuristics:

• Teach problem-solving strategies within the context of specific subject areas (not as a topic separate from academic content) and, ideally, within the context of authentic activities.
• Engage in joint problem-solving activities with students, modeling effective strategies and guiding students’ initial efforts.
• Provide scaffolding for difficult problems (e.g., break them into smaller and simpler problems, give hints about possible strategies, or provide partial solutions).
• Ask students to explain what they are doing as they work through a problem.
• Have students solve problems in small groups, sharing ideas about problem-solving strategies, modeling various approaches for one another, and discussing the merits of each approach. (R. K. Atkinson, Derry, Renkl, & Wortham, 2000; Barron, 2000; Chinn, 2006; Crowley & Siegler, 1999; Gauvain, 2001; Kirschner et al., 2006; Mayer, 1985; Reimann & Schult, 1996; Renkl & Atkinson, 2003; Rogoff, 2003)