Problem Solving
• Well-defined versus ill-defined problems
• Creativity
• Polya’s stages of problem solving
– processes and problems at each stage
• Improving your problem solving skills
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Problem Solving
• Definition of a problem
– goal state differs from initial state
– actions are not immediately obvious
• Well-defined problem
– clear goal state
– clear initial state
– clear set of operations
• Ill-defined problem
– unclear goal
– unclear operations
– unsure about what is relevant
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Problem Solving
• Creativity
– synthesizing novel combinations of ideas
– new, more adaptable mental picture of issue
– social utility very important
• The study of genius
– many factors
• divergent thinking
• intrinsic motivation
• devotion
• environment
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Polya’s Four Stages of Problem Solving
Polya’ terms
Information Processing terms
1. Understand the problem
1. Encode problem into working
memory
2. Devise a plan
3. Carry out the plan
2. Search LTM for plan or relevant
information
4. Looking backwards
3. Execute plan
4. Evaluate results
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 1 Understanding The Problem
• Initial representation may be a crucial factor
– some problems are trivial to solve in one representation and impossible to
solve in another
– e.g.
• which word does not fit
– skyscraper, cathedral, temple, prayer
• Buddhist monk problem
• Very hard to change from the initial representation
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 2 Devise a Plan
• Many of the problem reflect two general characteristics
of human cognition
– limited working memory so focus on one view at a time
– tendency to go beyond the given
• implicit inferences
• activation of inappropriate information from LTM
• e.g. the nine dot problem
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 2 Devise a Plan
• Three types of rigidity interfere with stage 2
– response set
• continue to use method that worked in the past
• Luchin’s problem solving set
– perceptual set
• encode only one perspective (representation)
– functional fixedness
• established use
• memory interferes with creative use of an object
• e.g. matchbox problem
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 2 Devise a Plan
• Need search strategies
– overall plans and sub goals
– overall plan may be too large for working memory
• Search space
– representation of the search
– all possible paths to all possible solutions
– often represented as a tree diagram
• Example: Water jug problem
– Given 2 jugs
• one holds 4 gallons the other holds 3 gallons
• there are no marks on the jugs
– Problem: how to get exactly 2 gallons in the 4 gallon jug
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Water Jug Search Space
Initial state
0,0
4,0
0,0
4,3
0,3
4,0
4,0
3,0
0,3
0,0
3,0
4,3
3,3
4,2
0,2
2,0
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 2 Devise a Plan
• Search strategies
– algorithms = step by step procedures that guarantee a solution
– heuristics = short cuts, no guarantee, but quicker and more efficient
• Common types of strategies
– generate and test - trial and error
– means-ends analysis
• set up sub goals
• difference reduction - each step changes only one small differences
– hill-climbing - each step move closer
• not always a good plan e.g. missionaries and cannibals problem
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 2 Devise a Plan
• Protocol analysis
– have subject “think out loud” while problem solving
– gives information about their initial representation and search strategy
– shows where errors turn up
• One frequent difficulty is in redirecting the search
– need to change representation
– try new mental operations
– Reorganize
• Incubation helps redirection
– very hard to study
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 2 Devise a Plan
• Incubation: three general findings
– ONLY after much time and effort are already invested
– burst of insight (the “aha!” feeling)
– not in final form
• Five hypotheses about why incubation works
– decrease fatigue
– forget inappropriate sets and directions
– added practice - divided attention with part of your mind still working on the
problem
– change occurrence of an external event that triggers a new association
• e.g. waffle iron and running shoes
– unconscious processing - run automatized search strategies
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 2 Devise a Plan
• Incubation results in
– innovation - synthesis of ideas
– insight - pieces come together
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Stage 3 and Stage 4
• Stage 3: Carry out the plan
– get a solution
• Stage 4: Evaluate the solution
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“satisficing” (Simon’s Nobel prize)
humans generally do not maximize or optimize they satisfice
accept solutions that are “good enough”
too many possible solutions to check them all
may be problematic because of differing criteria for “good enough”
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Improving Your Problem Solving
• Try
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novel or atypical associations
working backwards
solving a similar but easier problem
diagramming the problem
visualization
brainstorming (withhold evaluation)
analogies
evaluating other people’s solution to other problems
• read biographies
– breaking the problem into sub goals
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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Improving Your Problem Solving
• Techniques for systematizing your search
– inferences - what else applies
• consciously draw as many inferences as possible from the goals, given,
and actions
– classification of action sequences
• record all actions tried
• organize equivalence classes I.e. inverse actions
– state evaluation
• define a “quantitative” evaluation for all momentary outcomes
• choose actions that have evaluations closest to goals
– contradictions
• derive inferences from the given that are inconsistent with the goal
• avoid actions that lead to inconsistent subgoals
Cognitive - problem_solving.ppt © 2001 Laura Snodgrass, Ph.D.
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