A major goal of education is to help students learn in ways that enable them to use what they have learned to solve problems in new situations. In short, problem solving is fundamental to education because educators are interested in improving students' ability to solve problems. This entry defines key terms, types of problems, and processes in problem solving and then examines theories of problem solving, ways of teaching for problem solving transfer, and ways of teaching of problem solving skill.

## DEFINITIONS

**What is a Problem?** A problem exists when a problem solver has a goal but does not know how to accomplish it. Specifically, a problem occurs when a situation is in a given state, a problem solver wants the situation to be in a goal state, and the problem solver is not aware of an obvious way to transform the situation from the given state to the goal state. In his classic monograph, *On Problem Solving*, the Gestalt psychologist Karl Duncker defined a problem as follows:

A problem arises when a living creature has a goal but does not know how this goal is to be reached. Whenever one cannot go from the given situation to the desired situation simply by action, then there has to be recourse to thinking. Such thinking has the task of devising some action, which may mediate between the existing and desired situations. (1945, p. 1)

This definition includes high-level academic tasks for a typical middle school student such as writing a convincing essay, solving an unfamiliar algebra word problem, or figuring out how an electric motor works, but does not include low-level academic tasks such as pronouncing the sound of the printed word “cat,” stating the answer to “2 2 =___,” or changing a word from singular to plural form.

**What is Problem Solving?** According to Mayer and Wittrock, problem solving is “cognitive processing directed at achieving a goal when no solution method is obvious to the problem solver” (2006, p. 287). This definition consists of four parts: (1) problem solving is cognitive, that is, problem solving occurs within the problem solver's cognitive system and can only be inferred from the problem solver's behavior, (2) problem solving is a process, that is, problem solving involves applying cognitive processes to cognitive representations in the problem solver's cognitive system, (3) problem solving is directed, that is, problem solving is guided by the problem solver's goals, and (4) problem solving is personal, that is, problem solving depends on the knowledge and skill of the problem solver. In sum, problem solving is cognitive processing directed at transforming a problem from the given state to the goal state when the problem solver is not immediately aware of a solution method. For example, problem solving occurs when a high school student writes a convincing essay on the causes of the American Civil War, understands how the heart works from reading a biology textbook, or solves a complex arithmetic word problem.

## PROBLEM SOLVING AS A KIND OF THINKING

Problem solving is related to other terms such as thinking, reasoning, decision making, critical thinking, and creative thinking. Thinking refers to a problem solver's cognitive processing, but it includes both directed thinking (which is problem solving) and undirected thinking (such as daydreaming). Thus, thinking is a broader term that includes problem solving as a subset of thinking (i.e., a kind of thinking, i.e., directed thinking).

Reasoning, decision making, critical thinking, and creative thinking are subsets of problem solving, that is, kinds of problem solving. Reasoning refers to problem solving with a specific task in which the goal is to draw a conclusion from premises using logical rules based on deduction or induction. For example, if students know that all four-sided figures are quadrilaterals and that all squares have four sides, then by using deduction they can conclude that all squares are quadrilaterals. If they are given the sequence 2–4–6–8, then by induction they can conclude that the next number should be 10. Decision making refers to problem solving with a specific task in which the goal is to choose one of two or more alternatives based on some criteria. For example, a decision making task is to decide whether someone would rather have $100 for sure or a 1% chance of getting $100,000. Thus, both reasoning and decision-making are kinds of problem solving that are characterized by specific kinds of tasks.

Finally, creative thinking and critical thinking refer to specific aspects of problem solving, respectively. Creative thinking involves generating alternatives that meet some criteria, such as listing all the possible uses for a brick, whereas critical thinking involves evaluating how well various alternatives meet some criteria, such as determining which are the best answers for the brick problem. For example, in scientific problem solving situations, creative thinking is involved in generating hypotheses and critical thinking is involved in testing them. Creative thinking and critical thinking can be involved in reasoning and decision making.

## TYPES OF PROBLEMS

Problems can be well-defined or ill-defined. A well-defined problem has a clearly specified given state, a clearly specified goal state, and a clearly specified set of allowable operations. For example, “Solve for x: 2x + 11 = 33” is a well-defined problem because there is clear given state (i.e., 2 x 11 = 33), a clear goal state (i.e., x = ___) and a clear set of operations (i.e., the rules of algebra and arithmetic). An ill-defined problem lacks a clearly specified given state, goal state, and/or set of allowable operators. For example, “develop a research plan for a senior honors thesis” is an ill-defined problem for most students because the goal state is not clear (e.g., the requirements for the plan) and the allowable operators are not clear (e.g., the places where students may find information). What makes a problem well-defined or ill-defined depends on the characteristics of the problem. Although most important and challenging problems in life are ill-defined, most problem solving in schools involves well-defined problems.

Moreover, it is also customary to distinguish between routine and non-routine problems. When a problem solver knows how to go about solving a problem, the problem is routine. For example, two-column multiplication problems, such as 25 x 12 = ___, are routine for most high school students because they know the procedure. When a problem solver does not initially know how to go about solving a problem, the problem is non-routine. For example, the following problem is nonroutine for most high-school students: “If the area covered by water lilies in a lake doubles every 24 hours, and the entire lake is covered in 60 days, how long does it take to cover half the lake?” Robert Sternberg and Janet Davidson (1995) refer to this kind of problem as an *insight problem* because problem solvers need to invent a solution method (e.g., in this case the answer is 59 days). What makes problems either routine or non-routine depends on the knowledge of the problem solver because the same problem can be routine for one person and non-routine for another. Although the goal of education is to prepare students for solving non-routine problems, most of the problems that students are asked to solve in school are routine.

## COGNITIVE PROCESSES IN PROBLEM SOLVING

Mayer and Wittrock (2006) distinguished among four major cognitive processes in problem solving: representing, in which the problem solver constructs a cognitive representation of the problem; planning, in which the problem solver devises a plan for solving the problem; executing, in which the problem solver carries out the plan; and self-regulating, in which the problem solver evaluates the effectiveness of cognitive processing during problem solving and adjusts accordingly. During representing, the problem solver seeks to understand the problem, including the given state, goal state, and allowable operators, and the problem solver may build a situation model—that is, a concrete representation of the situation being described in the problem. Although solution execution is often emphasized in mathematics textbooks and in mathematics classrooms, successful mathematical problem solving also depends on representing, planning, and self-regulating. In a 2001 review, Jeremy Kilpatrick, Jane Swafford, and Bradford Findell concluded that mathematical proficiency depends on intertwining of procedural fluency (for executing) with conceptual understanding (for representing), strategic competence (for planning), adaptive reasoning, and productive disposition (for self-regulating).

According to Mayer and Wittrock (2006), students need to have five kinds of knowledge in order to be successful problem solvers:

facts: knowledge about characteristics of elements or events, such as “there are 100 cents in a dollar”;

concepts: knowledge of a categories, principles, or models, such as knowing what place value means in arithmetic or how hot air rises in science;

strategies: knowledge of general methods, such as how to break a problem into parts or how to find a related problem;

procedures: knowledge of specific procedures, such as how to carry out long division or how to change words from singular to plural form; and

beliefs: cognitions about one's problem-solving competence (such as “I am not good in math”) or about the nature of problem solving (e.g., “If someone can't solve a problem right away, the person never will be able to solve it”).

Facts and concepts are useful for representing a problem, strategies are needed for planning a solution, procedures are needed for carrying out the plan, and beliefs can influence the process of self-regulating.

## THEORIES OF PROBLEM SOLVING

Many current views of problem solving, such as described in Keith Holyoak and Robert Morrison's *Cambridge Handbook of Thinking and Reasoning* (2005) or Marsha Lovett's 2002 review of research on problem solving, have their roots in Gestalt theory or information processing theory.

**Gestalt Theory.** The Gestalt theory of problem solving, described by Karl Duncker (1945) and Max Wertheimer (1959), holds that problem solving occurs with a flash of insight. Richard Mayer (1995) noted that insight occurs when a problem solver moves from a state of not knowing how to solve a problem to knowing how to solve a problem. During insight, problem solvers devise a way of representing the problem that enables solution. Gestalt psychologists offered several ways of conceptualizing what happens during insight: insight involves building a schema in which all the parts fit together, insight involves suddenly reorganizing the visual information so it fits together to solve the problem, insight involves restating a problem's givens or problem goal in a new way that makes the problem easier to solve, insight involves removing mental blocks, and insight involves finding a problem analog (i.e., a similar problem that the problem solver already knows how to solve). Gestalt theory informs educational programs aimed at teaching students how to represent problems.

**Information Processing Theory.** The information processing theory of problem solving, as described by Allen Newell and Herbert Simon (1972), is based on a humancomputer metaphor in which problem solving involves carrying out a series of mental computations on mental representations. The key components in the theory are as follows: the idea that a problem can be represented as a problem space—a representation of the initial state, goal state, and all possible intervening states—and search heu-ristics—a strategy for moving through the problem space from one state of the problem to the next. The problem begins in the given state, the problem solver applies an operator that generates a new state, and so on until the goal state is reached. For example, a common search heuristic is means-ends analysis, in which the problem solver seeks to apply an operator that will satisfy the problem-solver's current goal; if there is a constraint that blocks the application of the operator, then a goal is set to remove the constraint, and so on. Information processing theory informs educational programs aimed at teaching strategies for solving problems.

## TEACHING FOR PROBLEM SOLVING

Max Wertheimer (1959) made the classic distinction between learning by rote and learning by understanding. For example, in teaching students how to compute the area of a parallelogram by a rote method, students are shown how to measure the height, how to measure the base, and how to multiply height times base using the formula, area = height x base. According to Wertheimer, this rote method of instruction leads to good performance on retention tests (i.e., solving similar problems) and poor performance on transfer tests (i.e., solving new problems). In contrast, learning by understanding involves helping students see that if they can cut off the triangle from one end of the parallelogram and place it on the other side to form a rectangle; then, they can put 1 x 1 squares over the surface of the rectangle to determine how many squares form the area. According to Wertheimer, this meaningful method of instruction leads to good retention and good transfer performance. Wertheimer claimed that rote instruction creates reproductive thinking—applying already learned procedures to a problem—whereas meaningful instruction leads to productive thinking—adapting what was learned to new kinds of problems.

Mayer and Wittrock (2006) identified instructional methods that are intended to promote meaningful learning, such as providing advance organizers that prime appropriate prior knowledge during learning, asking learners to explain aloud a text they are reading, presenting worked out examples along with commentary, or providing hints and guidance as students work on an example problem. A major goal of meaningful methods of instruction is to promote problem-solving transfer, that is, the ability to use what was learned in new situations. Wittrock (1974) referred to meaningful learning as a generative process because it requires active cognitive processing during learning.

## TEACHING OF PROBLEM SOLVING

In the previous section, instructional methods were examined that are intended to promote problem-solving transfer. However, a more direct approach is to teach people the knowledge and skills they need to be better problem solvers. Mayer (2008) identified four issues that are involved in designing a problem-solving course.

**What to Teach.** Should problem-solving courses attempt to teach problem solving as a single, monolithic skill (e.g., a mental muscle that needs to be strengthened) or as a collection of smaller, component skills? Although conventional wisdom is that problem solving involves a single skill, research in cognitive science suggests that problem solving ability is a collection of small component skills.

**How to Teach.** Should problem-solving courses focus on the product of problem solving (i.e., getting the right answer) or the process of problem solving (i.e., figuring out how to solve the problem)? While it makes sense that students need practice in getting the right answer (i.e., the product of problem solving), research in cognitive science suggests that students benefit from training in describing and evaluating the methods used to solve problems (i.e., the process of problem solving). For example, one technique that emphasizes the process of problem solving is modeling, in which teachers and students demonstrate their problem-solving methods.

**Where to Teach.** Should problem solving be taught as a general, stand-alone course or within specific domains (such as problem solving in history, in science, in mathematics, ETC.)? Although conventional wisdom is that students should be taught general skills in stand-alone courses, there is sufficient cognitive science research to propose that it would be effective to teach problem solving within the context of specific subject domains.

**When to Teach.** Should problem solving be taught before or after students have mastered corresponding lower-levels? Although it seems to make sense that higher-order thinking skills should be taught only after lower-level skills have been mastered, there is sufficient cognitive science research to propose that it would be effective to teach higher-order skills before lower-level skills are mastered.

In this section, three classic problem-solving courses are described that meet these four criteria and that have been subjected to rigorous research study: the Productive Thinking Program developed by Martin Covington, Richard Crutchfield, and Lillian Davies (1966), Instrumental Enrichment developed by Reuven Feuerstein (1980), and Odyssey described by Raymond Nickerson (1994). The Productive Thinking Program consisted of 15 cartoon-like booklets intended to teach thinking skills to elementary school children. Each booklet presented a detective-type story—such a story about a bank robbery— and students learned how to generate hypotheses— such as who might have done it—and evaluate hypotheses using information in the booklet. Child characters in the booklet modeled problem-solving methods, and adult characters offered commentary and hints. Overall, Richard Mansfield, Thomas Busse, and Ernest Krepelka (1978) reported that students who learned with the Productive Thinking Program showed greater improvements in their ability to solve similar detective-type problems as compared to students who had not received the training.

In Instrumental Enrichment, students who had been identified as mentally retarded based on a traditional intelligence test were given concentrated classroom instruction in how to solve traditional intelligence test items. In a typical lesson, the teacher introduces the class to an intelligence test item; then, the class breaks down into small groups to devise ways to solve the problem; next, each group reports on its solution method to the whole class; and finally, a teacher-led discussion ensues in which students focus on describing effective methods for solving the problem. Evaluation studies reported by Feuerstein (1980) show that students who received this training on a regular basis over several years showed greater gains in non-verbal intelligence than did non-trained students.

Finally, in Odyssey, middle-school students received training in how to solve intelligence test problems, using a procedure somewhat like Instrumental Enrichment, and with similar results. David Perkins and Tina Grotzer reported that the training “enhanced the magnitude of students' intelligent behavior [on] authentic tasks at least in the short term” (2000, p. 496). Overall, each of these courses met the criteria for what to teach (i.e., a collection of small component skills), how to teach (i.e., using modeling to focus on the process of problem solving), where to teach (i.e., teaching specific skills), and when to teach (i.e., teaching before all lower-level skills were mastered). Although none of these programs is currently popular, courses based on these four criteria are likely to be successful.

See also:Creativity, Critical Thinking, Decision Making, Learning and Teaching Mathematics, Reasoning

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