Problem Solving
Page 1
Problem Solving
Examples of Problems
• How to lose 10 lbs.
• Solving a crossword puzzle
• Select a good chess move
• Building a spaceship to go to Mars
• Building a successful career
• How to get an "A" in this course
• Writing a good term paper
Parts of a Problem
• Goal
o Where you would like to end up
• Givens
o Where you start from plus the rules
• Means
o The tools you can use to get from the initial state to the
goal
• Obstacles--must have some steps or not a problem.
o not “2+2=?”
Types Of Problems
• Well-defined: Goal and starting point are clear; you know when
it’s been solved
o Researchers mainly study well-defined problems
Problem Solving
Page 2
• Ill-defined problems: Goal and starting point are unclear; hard
to tell when solution is reached
o Many real-life problems are ill-defined
Terms
• Task Environment
o the problem (objectively)
• Problem Space
o how the problem is encoded/represented
• Problem State
o the current state of the problem space
Methods
• Intermediate products: look at work subject does in getting
answer
• Verbal protocol: “think aloud”
• Recognition test: high false alarm to intermediate stages
• Computer simulation: existence proof
Simulation
• Newel and Simons General Problem Solver (GPS)
• Limited capacity STM--fast access; infinite capacity LTM--slow
access
• Searches through the problem space to solve problems
Problem Solving Strategies
• Think of problem space as a real space
• Describe the process of traveling through the space (searching
the problem space)
Problem Solving
Page 3
Terms
• Algorithm
o guarantees a solution if there is one; might not be fast
• Heuristic
o fast; does not guarantee a solution
Algorithm (guaranteed a solution)
• Chess
o Calculate all possible moves and countermoves
o After the sixth move you are up to 225 million possibilities
o Not very efficient for problems that have large problem
spaces
Heuristics (not guaranteed)
• Reduce the problem space by use of a short-cut.
• Guide search in an intelligent way.
Heuristic: Random
• Move in a random direction
Heuristic: Working Backwards
• Start at goal state, move to start state
o Mazes
o Mathematical proofs
Problem Solving
Page 4
Heuristic: Means-End Analysis
• Compare current state with end state
• Move in direction that minimizes distance left/ difference
between current and end
• Guess a number between 1 and 100
o Choose 50
o Eliminate either 1-50 or 50-100
Heuristic: Generate and Test
• Generate a solution.
• Test to see if it works.
• If not, try again.
The importance of representation
• Represent the problem in terms of a list of features describing
the problem.
• Internal representation of the problem
o May include extra information
o May leave out critical information
o May lead to functional fixedness
 focus on just one function for an item
9 Dot Problem
Functional Fixedness
• Reverse “McGyver” problem
• The inability to use objects other than how they are usually
used.
Problem Solving
Page 5
Maier Two-String Problem
Duncker Candle Problem
Functional fixedness and the 9 Dot Problem
Problem Representation
• If crucial information is missing from problem space, problem
may be impossible
• If additional “information” is added to a problem space, problem
may be impossible
Transfer (analogy)
• People don’t spontaneously transfer solutions
• Physics, statistics, maths books:
o if homework problem like example problem, most solve
correctly (~70%)
o if homework problem superficially different from example
problem, most do not solve (~20%)
Similarity
• Surface similarity
o medical versus military
• Structural similarity
o both convergence problems
Mental Models
• Create an internal model of the situation
• Use that model to solve problems
Problem Solving
Page 6
Intuitive physics
Individual differences in decision making capabilities.
• Good decision makers construct more alternative and accurate
models.
Incubation Effect
Not “unconscious processing”
• Subjects did not come back with solution; rather, began working
again
• Explained best by “forgetting previous problem space”
Predicting Success
• Insight problems (e.g., cheap necklace, nine dots)
o Rate how likely solve in next 10 seconds
 Correct: 3.47; Incorrect: 5.25
• Premonition of insight more accurately predicts mistakes than
solution
Expertise
• How are experts different from novices?
o Overcome memory limitations
• Knowledge base
o Experts know more and that information is more easily
accessed
• Representation
o Experts represent the problem in terms of its solution, not
in terms of the problem.
Problem Solving
Page 7
o They are less susceptible to functional fixedness.
o Appreciating structural similarity
• Experts are not distracted by surface features.
• Experts solve problems rapidly and efficiently.
o Many parts of the problem have become automatized.
• Experts are much more aware of how close they are to a
solution. They are able to predict accurately whether they will
solve a problem.
• Experts practice in their area of expertise (at least 4 hours a
day).
o Question: How do you get to Carnegie Hall?
o Answer: Practice, practice, practice.
10 Year Rule
• Become an expert after 10 years of practice
o chess
o musical performance/composition
o mathematics
o tennis
o livestock evaluation
o medical diagnosis
Domain Limited
• Expertise is domain limited
o Does not transfer
• Span for digits = 80
Problem Solving
Page 8
• Span for letters ~7
Creativity
• Divergent production test
Stages of creativity
• Preparation
• Incubation
• Illumination
• Verification