# Learning

## Problem Solving

Cognitive processing aimed at figuring out how to achieve a goal is called problem solving. In problem solving, the problem solver seeks to devise a method for transforming a problem from its current state into a desired state when a solution is not immediately obvious to the problem solver. Thus, the hallmark of problem solving is the invention of a new method for addressing a problem. This definition has three parts: (1) problem solving is cognitive–that is, it occurs internally in the mind (or cognitive system) and must be inferred indirectly from behavior; (2) problem solving is a process–it involves the manipulation of knowledge representations (or carrying out mental computations); and (3) problem solving is directed–it is guided by the goals of the problem solver.

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. This definition has three parts: (1) the current state–the problem begins in a given state; (2) the goal state–the problem solver wants the problem to be in a different state, and problem solving is required to transform the problem from the current (or given) state into the goal state, and (3) obstacles–the problem solver does not know the correct solution and an effective solution method is not obvious to the problem solver.

According to this definition a problem is personal, so that a situation that is a problem for one person might not be a problem for another person. For example, "3 + 5 = ___" might be a problem for a six-year-old child who reasons, "Let's see. I can take one from the 5 and give it to the 3. 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.

### Types of Problems

Routine and nonroutine problems. 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. 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. 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, the following problem (reported by Robert Sternberg and Janet Davidson) is nonroutine for most people: "Water lilies double in area every twenty-four hours. At the beginning of the summer, there is one water lily on the lake. It takes sixty days for the lake to be completely covered with water lilies. On what day is the lake half covered?" In this problem, the problem solver must invent a solution method based on working backwards from the last day. Based on this method, the problem solver can ask what the lake would look like on the day before the last day, and conclude that the lake is half covered on the fifty-ninth day.

Well-defined and ill-defined problems. It is also customary to distinguish between well-defined and ill-defined problems. In a well-defined problem, the given state of the problem, the goal state of the problem, and the allowable operators (or moves) are each clearly specified. For example, the following water-jar problem (adapted from Abrahama Luchins) is an example of a well defined problem: "I will give you three empty water jars; you can fill any jar with water and pour water from one jar into another (until the second jar is full or the first one is empty); you can fill and pour as many times as you like. Given water jars of size 21, 127, and 3 units and an unlimited supply of water, how can you obtain exactly 100 units of water?" This is a well-defined problem because the given state is clearly specified (you have empty jars of size 21, 127, and 3), the goal state is clearly specified (you want to get 100 units of water in one of the jars), and the allowable operators are clearly specified (you can fill and pour according to specific procedures). Well-defined problems may be either routine or nonroutine; if you do not have previous experience with water jar problems, then finding the solution (i.e., fill the 127, pour out 21 once, and pour out 3 twice) is a nonroutine problem.

In an ill-defined problem, the given state, goal state, and/or operations are not clearly specified. 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. Ill-defined problems can be routine or nonroutine; if one has extensive experience in writing then writing a short essay like this one is a routine problem.

### Processes in Problem Solving

The process of problem solving can be broken down into two major phases: problem representation, in which the problem solver builds a coherent mental representation of the problem, and problem solution, in which the problem solver devises and carries out a solution plan. Problem representation can be broken down further into problem translation, in which the problem solver translates each sentence (or picture) into an internal mental representation, and problem integration, in which the problem solver integrates the information into a coherent mental representation of the problem (i.e., a mental model of the situation described in the problem). Problem solution can be broken down further into solution planning, in which the problem solver devises a plan for how to solve the problem, and solution execution, in which the problem solver carries out the plan by engaging in solution behaviors. Although the four processes of problem solving are listed sequentially, they may occur in many different orderings and with many iterations in the course of solving a problem.

For example, consider the butter problem described by Mary Hegarty, Richard Mayer, and Christopher Monk: "At Lucky, butter costs 65 cents per stick. This is two cents less per stick than butter at Vons. If you need to buy 4 sticks of butter, how much will you pay at Vons?" In the problem translation phase, the problem solver may mentally represent the first sentence as "Lucky = 0.65," the second sentence as "Lucky = Vons - 0.02," and the third sentence as "4 × Vons = ___." In problem integration, the problem solver may construct a mental number line with Lucky at 0.65 and Vons to the right of Lucky (at 0.67); or the problem solver may mentally integrate the equations as "4 × (Lucky + 0.02) = ____." A key insight in problem integration is to recognize the proper relation between the cost of butter at Lucky and the cost of butter at Vons, namely that butter costs more at Vons (even though the keyword in the problem is "less"). In solution planning, the problem solver may break the problem into parts, such as: "First add 0.02 to 0.65, then multiply the result by 4." In solution executing, the problem solver carries out the plan: 0.02 + 0.65 =0.67, 0.67 × 4 = 2.68. In addition, the problem solver must monitor the problem-solving process and make adjustments as needed.

### Teaching for Problem Solving

A challenge for educators is to teach in ways that foster meaningful learning rather than rote learning. Rote instructional methods promote retention (the ability to solve problems that are identical or highly similar to those presented in instruction), but not problem solving transfer (the ability to apply what was learned to novel problems). For example, in 1929, Alfred Whitehead used the term inert knowledge to refer to learning that cannot be used to solve novel problems. In contrast, meaningful instructional methods promote both retention and transfer.

In a classic example of the distinction between rote and meaningful learning, the psychologist Max Wertheimer (1959) described two ways of teaching students to compute the area of a parallelogram. In the rote method, students learn to measure the base, measure the height, and then multiply base times height. Students taught by the A = b × h method are able to find the area of parallelograms shaped like the ones given in instruction (a retention problem) but not unusual parallelograms or other shapes (a transfer problem). Wertheimer used the term reproductive thinking to refer to problem solving in which one blindly carries out a previously learned procedure. In contrast, in the meaningful method, students learn by cutting the triangle from one end of a cardboard parallelogram and attaching it to the other end to form a rectangle. Once students have the insight that a parallelogram is just a rectangle in disguise, they can compute the area because they already know the procedure for finding the area of a rectangle. Students taught by the insight method perform well on both retention and transfer problems. Wertheimer used the term productive thinking to refer to problem solving in which one invents a new approach to solving a novel problem.

### Educationally Relevant Advances in Problem Solving

Recent advances in educational psychology point to the role of domain-specific knowledge in problem solving–such as knowledge of specific strategies or problem types that apply to a particular field. Three important advances have been: (1) the teaching of problem-solving processes, (2) the nature of expert problem solving, and (3) new conceptions of individual differences in problem-solving ability.

Teaching of problem-solving processes. An important advance in educational psychology is cognitive strategy instruction, which includes the teaching of problem-solving processes. For example, in Project Intelligence, elementary school children successfully learned the cognitive processes needed for solving problems similar to those found on intelligence tests. In Instrumental Enrichment, students who had been classified as mentally retarded learned cognitive processes that allowed them to show substantial improvements on intelligence tests.

Expert problem solving. Another important advance in educational psychology concerns differences between what experts and novices know in given fields, such as medicine, physics, and computer programming. For example, expert physicists tend to store their knowledge in large integrated chunks, whereas novices tend to store their knowledge as isolated fragments; expert physicists tend to focus on the underlying structural characteristics of physics word problems, whereas novices focus on the surface features; and expert physicists tend to work forward from the givens to the goal, whereas novices work backwards from the goal to the givens. Research on expertise has implications for professional education because it pinpoints the kinds of domain-specific knowledge that experts need to learn.

Individual differences in problem-solving ability. This third advance concerns new conceptions of intellectual ability based on differences in the way people process information. For example, people may differ in cognitive style–such as their preferences for visual versus verbal representations, or for impulsive versus reflective approaches to problem solving. Alternatively, people may differ in the speed and efficiency with which they carry out specific cognitive processes, such as making a mental comparison or retrieving a piece of information from memory. Instead of characterizing intellectual ability as a single, monolithic ability, recent conceptions of intellectual ability focus on the role of multiple differences in information processing.

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RICHARD E. MAYER