Chapter 12 Problem-Solving and Reasoning. Some Questions to Consider What makes a problem hard? Is there anything special about problems that seem to

Chapter 12

Problem-Solving and Reasoning

Some Questions to Consider

• What makes a problem hard?

• Is there anything special about problems that seem to be solved in a flash of “insight”?

• How can analogies be used to help solve problems?

• How do experts in a field solve problems differently than nonexperts?

What Is a Problem?

• Obstacle between a present state and a goal

• Not immediately obvious how to get around the obstacle

What Is a Problem?

• Well-defined: correct answer, certain procedures will lead to solution

• Ill-defined: path to solution is unclear, no one “correct” answer

Gestalt Approach

• Representing a problem in the mind

• Restructuring: changes the problem’s representation

– Kohler’s “circle” problem

Caption: The circle problem.

Insight in Problem-Solving

• Sudden realization of a problem’s solution

Caption: (a) Triangle problem and (b) chain problem for “Two Insight Problems” demonstration.


• Metcalfe and Wiebe (1987)

– Insight: triangle problem, chain problem

– Noninsight: algebra

– Warmth judgments every 15 seconds


• Metcalfe and Wiebe (1987)

– Insight problems solved suddenly

– Noninsight problems solved gradually

Caption: Results of Metcalfe and Wiebe’s (1987) experiment showing how participants judged how close they were to solving insight problems and algebra problems for the minute just before solving the problems.

Obstacles to Problem-Solving

• Functional fixedness: restricting use of an object to its familiar functions

–Candle problem: seeing boxes as containers inhibited using them as supports

–Two-string problem: function of pliers gets in the way of seeing them as a weight


• Functional fixedness: the elevator riddle


• Situationally produced mental set

– Situation influences approach to problem

– Water-jug problem: given mental set inhibited participants from using simpler solution

Caption: Luchins’s (1942) water-jug problem. Each problem specifies the capacities of jugs A, B, and C, and a final desired quantity. The task is to use the jugs to measure out the final quantity. The solution to problem 1 is shown. All of the other problems can be solved using the same pattern of pourings, indicated by the equation, but there are more efficient ways to solve 7 and 8.

Information-Processing Approach

• Newell and Simon

• Problem space

– Initial state

– Intermediate state(s)

– Goal state


• Tower of Hanoi

• Operators: rules specify which moves are allowed and which are not

Caption: (a) Initial and goal states for the Tower of Hanoi problem. (b) The operators for the Tower of Hanoi problem.


• Means-end analysis: reduce differences between initial and goal states

– Subgoals: create intermediate states closer to goal

Caption: Initial steps in solving the Tower of Hanoi problem, showing how the problem can be broken down into subgoals.

The Importance of How a Problem Is Stated

• Acrobat and reverse acrobat problem

– One small change in wording of problem

– Not just analyzing structure of problem space

– How a problem is stated can affect its difficultly


• Mutilated-checkerboard problem

– Conditions differed in how much information provided about the squares

– Easier to solve when information is provided that points toward the correct representation of the problem

Caption: Conditions in Kaplan and Simon’s (1990) study of the mutilated-checkerboard problem. (Reprinted from Cognitive Psychology, Volume 22, C.A. Kaplan & H.A. Simon, “In Search of Insight,” pp. 374-419, Figure 2. Copyright © 1990, with permission from Elsevier.)


• Think-aloud protocol

– Say aloud what one is thinking

– Shift in how one perceives elements of a problem

Using Analogies to Solve a Problem

• Using a solution to a similar problem guides solution to new problem

– Russian marriage problem (source problem) mutilated-checkerboard problem (target problem)


• Gick and Holyoak

– Noticing relationship

– Mapping correspondence between source and target

– Applying mapping


• Duncker’s Radiation Problem

– Analogies aid problem-solving

– Often hints must be given to notice connection

• Surface features get in the way

• Structural features must be used

Caption: (a) Solution to the radiation problem. Bombarding the tumor, in the center, with a number of low-intensity rays from different directions destroys the tumor without damaging the tissue it passes through. (b) Radiosurgery, a modern medical technique for irradiating brain tumors with a number of beams of gamma rays, uses the same principle. The actual technique uses 201 gamma ray beams. (c) How the general solved the fortress problem.


• Lightbulb problem

– High surface similarities aid analogical problem solving

– Making structural features more obvious aids analogical problem-solving


• Analogical encoding: comparing two cases that illustrate a principle

– Effective way to get participants to pay attention to structure features that aide problem-solving


• Analogical paradox

– Participants in experiments focus on surface features

– People in the real world use structural features


• In vivo problem-solving research

– People are observed to determine how they solve problems in the real world

• Advantage: naturalistic setting

• Disadvantages: time-consuming, cannot isolate and control variables

How Experts Solve Problems

• Experts solve problems in their field faster and with a higher success rate than beginners


• Experts possess more knowledge about their fields


• Knowledge is organized so it can be accessed when needed to work on a problem

– Novice: surface features

– Expert: deep structure

Caption: The kinds of physics problems that were grouped together by novices (left) and experts (right; Chi et al., 1981).


• Experts spend more time analyzing problem

• Experts are no better than novices when given problems outside of their field


• Experts less likely to be open to new ways of looking at problems

Creative Problem-Solving

• Creativity

– Innovative thinking

– Novel ideas

– New connections between existing ideas


• Divergent thinking: open-ended; large number of potential “solutions”

• Convergent thinking: one correct answer


• Design fixation

– Fixated on what not to do as demonstrated by sample

– Fixation can inhibit problem-solving


• Creative cognition: technique to train people to think creatively

– Preinventive forms: ideas that precede creation of finished creative product

Caption: How a preinventive form that was constructed from the half-sphere, wire, and handle can be interpreted in terms of each of the eight categories in Table 11.1. (Reprinted from R. A. Finke, “Creative Insight and Preinventive Forms,” from The Nature of Insight, by R. J. Sternberg & J. E. Davidson, Eds., pp. 255-280, Figure 8.6. Copyright © 1995 with permission from the MIT Press.

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Chapter 12 Problem-Solving and Reasoning. Some Questions to Consider What makes a problem hard? Is there anything special about problems that seem to