11 CHAPTER PROBLEM SOLVING
What is a Problem?
Obstacle between a present state and a goal
Not immediately obvious how to get around the obstacle
The Match Problem and Its Solution
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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 Key:
"circle" problem
create a mental
representation
Insight in Problem Solving
Sudden realization of a problem's solution
Caption: (a) Triangle problem and (b) chain problem for "Two Insight Problems" demonstration.
Insight in Problem Solving
Metcalfe and Wiebe (1987)
Insight:
triangle problem, chain problem algebra judgments every 15 seconds
Noninsight: Warmth
Insight in Problem Solving
Metcalfe and Wiebe (1987)
Insight
problems solved suddenly problems solved gradually
Noninsight
Obstacles to Problem Solving
One of the major obstacles to problem solving is fixation: focus on specific characteristic of problem that keeps us from arriving at solution 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
Obstacles to Problem Solving
Functional fixedness: The candle problem
Functional fixedness: The 2 String Problem
Functional fixedness: The Elevator Riddle
The Elevator Riddle
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The Elevator Riddle Solution
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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
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: Described problem solving as a search that occurs between the posing of the problem and its solution Problem Space
Initial
state
state(s) state
Intermediate Goal
Information Processing Approach
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.
Information Processing Approach
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 & Reverse Acrobat Problem
One Not
small change in wording of problem a problem is stated can affect its difficultly
just analyzing structure of problem space
How
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
Caption: Conditions in Kaplan and Simon's (1990) study of the mutilatedcheckerboard 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.)
The Importance of How a Problem is Stated
Think-aloud Say protocol
Shift
aloud what one is thinking
in how one perceives elements of a problem
Using Analogies to Solve Problem
Using a solution to a similar problem guides solution to new problem
Russian
marriage problem (source problem) mutilated checkerboard problem (target problem)
Using Analogies to Solve Problem
Gick and Holyoak
Noticing
relationship correspondence between source and
Mapping
target
Applying
mapping
Using Analogies to Solve Problem
Duncker's Radiation Problem
Analogies Often
aid problem-solving features get in the way
hints must be given to notice connection
Surface
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.
Using Analogies to Solve Problem
Lightbulb Problem
High
surface similarities aid analogical problem solving
Making
structural features more obvious aids analogical problem solving
Using Analogies to Solve 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
Using Analogies to Solve Problem
Analogical paradox
Participants
in experiments focus on surface
features
People
in the real-world use structural features
Using Analogies to Solve 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
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 (chess players and memory for actual game positions vs. random positions) Knowledge is organized so it can be accessed when needed to work on a problem
Novice: Expert:
surface features deep structure
Caption: The kinds of physics problems that were grouped together by novices (left) and experts (right; Chi et al., 1981).
How Experts Solve Problems
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 Novel New
thinking
ideas
connections between existing ideas
Creative Problem Solving
Divergent thinking: open-ended; large number of potential "solutions"
Convergent thinking: one correct answer
Creative Problem Solving
Design fixation
Fixated
on what not to do as demonstrated by
sample
Fixation
can inhibit problem solving
Creative 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.
Sleep Inspires Insight
Sleep facilitates discovering hidden structure needed to solve problems
Sleep is an effective study break!
Restorative Helps
consolidate
information