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.
Insight in Problem-Solving
• Metcalfe and Wiebe (1987)
– Insight: triangle problem, chain problem
– Noninsight: algebra
– Warmth judgments every 15 seconds
Insight in Problem-Solving
• 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
Obstacles to Problem-Solving
• Functional fixedness: the elevator riddle
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
• Problem space
– Initial state
– Intermediate state(s)
– Goal state
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 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
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
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.)
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
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)
Using Analogies to Solve a Problem
• Gick and Holyoak
– Noticing relationship
– Mapping correspondence between source
and target
– Applying mapping
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
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 a Problem
• Lightbulb problem
– High surface similarities aid analogical
problem solving
– Making structural features more obvious
aids analogical problem-solving
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
Using Analogies to Solve a Problem
• Analogical paradox
– Participants in experiments focus on
surface features
– People in the real world use structural
features
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
How Experts Solve Problems
• Experts solve problems in their field faster
and with a higher success rate than
beginners
How Experts Solve Problems
• Experts possess more knowledge about their
fields
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
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
How Experts Solve Problems
• 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
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.