1. Problem-Solving and Reasoning
Chapter 12
Problem-Solving and Reasoning

2. 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?

3. What Is a Problem?
Obstacle between a present state and a goal
Not immediately obvious how to get around the obstacle

4. 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

5. Gestalt Approach
Representing a problem in the mind
Restructuring: changes the problem’s representation
Kohler’s “circle” problem

6. Caption: The circle problem.

7. Insight in Problem-Solving
Sudden realization of a problem’s solution

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

9. Insight in Problem-Solving
Metcalfe and Wiebe (1987)
Insight: triangle problem, chain problem
Noninsight: algebra
Warmth judgments every 15 seconds

10. Insight in Problem-Solving
Metcalfe and Wiebe (1987)
Insight problems solved suddenly
Noninsight problems solved gradually

11. 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.

12. 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

13. Obstacles to Problem-Solving
Functional fixedness: the elevator riddle

14. Obstacles to Problem-Solving
Situationally produced mental set
Situation influences approach to problem
Water-jug problem: given mental set inhibited participants from using simpler solution

15. Caption: Luchins’s (1942) water-jug problem
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.

16. Information-Processing Approach
Newell and Simon
Problem space
Initial state
Intermediate state(s)
Goal state

17. Information-Processing Approach
Tower of Hanoi
Operators: rules specify which moves are allowed and which are not

18. Figure 11. 1 Specialties in psychology. Percentages are approximate
Figure Specialties in psychology. Percentages are approximate. The percentages shown here are based on membership in the American Psychological Association. The APA welcomes both scientists and practitioners. However, some psychological researchers belong to other organizations, such as the American Psychological Society. As a result, the percentage of clinical and counseling psychologists working in the United States may be a little lower than shown in chart a. Nevertheless, it is accurate to say that most psychologists specialize in applied areas and work in applied settings.
Caption: (a) Initial and goal states for the Tower of Hanoi problem. (b) The operators for the Tower of Hanoi problem.

19. Information-Processing Approach
Means-end analysis: reduce differences between initial and goal states
Subgoals: create intermediate states closer to goal

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

21. 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

22. The Importance of How a Problem Is Stated
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

23. 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 , Figure 2. Copyright © 1990, with permission from Elsevier.)

24. The Importance of How a Problem Is Stated
Think-aloud protocol
Say aloud what one is thinking
Shift in how one perceives elements of a problem

25. 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)

26. Using Analogies to Solve a Problem
Gick and Holyoak
Noticing relationship
Mapping correspondence between source and target
Applying mapping

27. Using Analogies to Solve a Problem
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

28. Caption: (a) Solution to the radiation problem
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.

29. Using Analogies to Solve a Problem
Lightbulb problem
High surface similarities aid analogical problem solving
Making structural features more obvious aids analogical problem-solving

30. Using Analogies to Solve a Problem
Analogical encoding: comparing two cases that illustrate a principle
Effective way to get participants to pay attention to structure features that aide problem-solving

31. Using Analogies to Solve a Problem
Analogical paradox
Participants in experiments focus on surface features
People in the real world use structural features

32. Using Analogies to Solve a Problem
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

33. How Experts Solve Problems
Experts solve problems in their field faster and with a higher success rate than beginners

34. How Experts Solve Problems
Experts possess more knowledge about their fields

35. How Experts Solve Problems
Knowledge is organized so it can be accessed when needed to work on a problem
Novice: surface features
Expert: deep structure

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

37. How Experts Solve Problems
Experts spend more time analyzing problem
Experts are no better than novices when given problems outside of their field

38. How Experts Solve Problems
Experts less likely to be open to new ways of looking at problems

39. Creative Problem-Solving
Creativity
Innovative thinking
Novel ideas
New connections between existing ideas

40. Creative Problem-Solving
Divergent thinking: open-ended; large number of potential “solutions”
Convergent thinking: one correct answer

41. Creative Problem-Solving
Design fixation
Fixated on what not to do as demonstrated by sample
Fixation can inhibit problem-solving

42. Creative Problem-Solving
Creative cognition: technique to train people to think creatively
Preinventive forms: ideas that precede creation of finished creative product

43. 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 (Reprinted from R. A. Finke, “Creative Insight and Preinventive Forms,” from The Nature of Insight, by R. J. Sternberg & J. E. Davidson, Eds., pp , Figure 8.6. Copyright © 1995 with permission from the MIT Press.