Problem solving

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Problem solving

• Simon J. Davies

daviess_at_hope.ac.uk
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What is problem solving?

• The use of previous knowledge or new skills
  applied to a situation where a definite outcome
  is sought but currently unavailable.
• Approaches will depend on the problem environment
  (knowledge rich or poor) and problem solver (what
  they know).
• Problems can be tackled in a productive (i.e.
  creative) or reproductive manner.

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Why study problem solving?

• Get an understanding of the mental processes
  involved in problem solving.
• Look at how people learn or transfer knowledge
  from one situation to another.
• Understand what makes the difference between
  novices and experts.
• Find out what cognitive limitations facilitate or
  hinder problem solving.

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Problem-solving concepts 1

• Heuristics rules-of-thumb frequently applied in
  the past with success. These do not ensure a
  definite outcome, but increase the likelihood of
  solving the problem or getting closer to a goal.
• Algorithms known step-by-step procedures that
  ensure problem solution.

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Problem-solving concepts 2

• Well-defined problems situations where goal
  components are understood (the goal, legal
  operators, and restrictions).
• Ill-defined problems situations where some
  components are under-specified or unspecified.
• Domain-general problems require general
  understanding and strategies.
• Domain-specific problems require knowledge of a
  particular subject to solve the problem.

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Studying problems

• Researchers typically employ puzzle-world
  problems these are often simple, and represent
  specific aspects of problem solving behaviour.
• The use of analogy and metaphor are also used to
  test the value of prior experience and problem
  reformulation and representation.
• Comparisons between novice and experts also
  reveal differences between solution approaches
  and the process of learning to solve problems.

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Methods

• Experiments observing behaviour change under
  different condition can be exploratory.
• Computer simulation models supports the
  progress of AI attempts to mimic human
  cognition.
• Verbal protocol (protocol analysis) obtains
  individual problem-solving strategies by getting
  participants to think aloud whilst solving a
  problem.

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Gestalt approaches to problems

• Earlier research emphasised trial-and-error and
  reproduction in a behaviourist context.
• Gestalt psychologists proposed that problem
  solving is achieved by gaining some insight
  into problem structure, and then restructuring
  the problem.
• Kohler (1927) showed an ape (Sultan) joining
  together two sticks (creating a new tool) as
  evidence for insight and restructuring.

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Past experience and problem solving the
nine-dot problem
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Past knowledge can hinder problem solving

• The candle problem
• Candles, nails and matches placed on desk.
• Attach the candle to the wall.
• Subjects frequently try and nail the candle to
  the wall, rather then nail the nail box to the
  wall and place the candle in the box.

• The nine-dot problem
• Nine-dots organised in a square pattern of three
  rows of three dots.
• Join the dots with four straight lines without
  lifting the pen from the page.
• Subjects cant avoid keeping their lines within
  the square shape.

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The nine dot problem Scheerer (1963)
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Reproduction errors

• Subjects become fixed on the function of objects
  from their prior experiences (called functional
  fixedness).
• The result is a failure to think creatively
  beyond an objects perceived function or Gestalt.
• This effect is also found when we become set in
  our ways. We continuously re-apply a way of
  solving a problem even when there is a more
  parsimonious way of achieving our goal (e.g.
  Luchins Luchins, 1959).

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Maiers (1931) pendulum problem
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Restructuring problems

• Gestalt views also saw problem solving as a
  process of restructuring.
• In Maiers (1931) pendulum problem subjects
  typically brush against one of the strings,
  leading them to reformulate the problem to
  include a pendulum weight.
• The solution is thus arrived at through
  production (i.e. a clue and insight) rather than
  reproduction of knowledge.

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Gestalts legacy

• Showed mere reproduction of learning
  insufficient.
• Problems need to be restructured and also can
  involve insight.
• x Poorly specified concepts and theories.

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Information processing approach

• Newell Simon employed an IPT approach to
  problem solving (creating the General Problem
  Solver).
• They conceive the solver being in a problem space
  in which there are a range of possible options.
• Problem solution involves the application of a
  set of operators which incrementally arrive at a
  solution.

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The Tower of Hanoi
A typical well-defined puzzle problem goals,
operators, and operator restrictions are evident.
Start
Goal
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Tower of Hanoi Problem space (aka state
space)
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Problem space theory

• Operators (possible manipulations) are known and
  applied to current state.
• We pass through a series of knowledge states
  (what we know about the problem).
• These states are changed by applying operators.
• By reapplying operators we minimise the
  difference between goal and current states
  (means-ends analysis).
• Problem solving is also limited by the cognitive
  system as a whole.

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Means-ends analysis (heuristic)

• Means-ends analysis simply involves comparing
  what means you have at your disposal and what
  ends these means will achieve.
• Solvers typically create sub-goals to minimise
  the distance to the goal. Operators then apply to
  close this difference.
• Strategies evolve from domain-general to
  domain-specific through a process of learning.

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Representation

• How a problem is represented can affect how it is
  solved, even if the two problems appear to be
  identical.
• Learning (i.e. transfer) from one problem to
  another is also not guaranteed if they appear the
  same.
• Solvers are poor at applying past experience or
  analogy to current problems.

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Problem representations

• Isomorphic
• This is where two problems share the exact same
  state space but appear different.
• Useful to examine the effect representation has
  upon problem solving.

• Homomorphic
• Where problems appear the same but have a
  different underlying state space.
• Useful for examining the effects of state space
  upon solving.

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Simon Hayes Monsters and globes problem
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M G move problem

• Monsters and globes is an isomorph of the ToH.
• Monsters must end up holding the same size globe
  as themselves (proportionate).
• Only one globe can be passed at once.
• Only the larger of any globes may be passed.
• A globe cannot be passed to a monster holding a
  bigger globe.

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M G change problem

• The globes now must be imaged to shrink and
  expand rather than move.
• Only a single globe can be changed at once.
• When two globes are of equal size only the globe
  held by the larger monster can be changed.
• A globe cannot be changed to the same size as a
  globe of a larger monster.

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Findings

• The change problem was twice as hard as the move
  problem.
• Hayes Simon (1976) proposed a rule application
  hypothesis the change rule was more difficult
  to articulate and apply mentally.
• Kotovsky, Hayes, and Simon (1987) proposed a
  rule-learning hypothesis, this said that
  difficulty was determined by how hard a rule was
  to learn.

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Homomorphic problems

• The missionaries and cannibals and the jealous
  husbands problems involve similar representation,
  except that the JH problem has an additional
  restriction on moves.
• In the MC problem solvers often fail due to one
  stage in the solution involving an apparent move
  away from the goal state.
• Though these problems are homomorphic JH is more
  difficult than MC.

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Transfer in problems

• Transfer between problems can be negative or
  positive.
• Negative transfer when past experience degrades
  performance on the current problem.
• Positive transfer when past experience improves
  performance on the current problem.

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M C vs. JH (transfer)

• Reed, Ernst and Banerji (1974) compared
  experience with MC (easy) followed by JH (harder)
  and vice versa.
• Found positive transfer only when MC (easy)
  followed the JH.
• Found no transfer when JH followed MC.
• Thus transfer only when easier problem follows
  harder problem.

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Luger and Bauer (1978)

• Compared isomorphic problems ToH with the
  Chinese tea ceremony.
• Found unconditional positive transfer between
  problems.
• Why different to Reed et al.?
• Knowledge appears to be transferred more easily
  between similar problems with a clear sub-goal
  structure.

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Limitations of puzzle problems

• Most puzzle problems are well defined. Most real
  problems are not.
• Any effects of transfer may therefore be
  questionable.
• Research into the use of analogy focuses upon
  ill-defined problems problems with greater
  ecological validity.

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Analogical problem solving

• Gick and Holyoak (1980) used stories to explore
  analogical transfer between ill-defined problems.
• They first told the fortress problem and then the
  radiation problem. The conditions differed in
  respect of how solutions to the fortress problem
  was presented.
• Both problems could only be solved with one type
  of solution (the same).

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Analogy fails without a hint

• Gick and Holyoak found that solvers told one type
  of solution in the first story tended to re-apply
  it in the second, even when it didnt work.
• They next used a similar setup but employed a
  contrast between hints and no hints.
• Without hints only 20 solved the new problem,
  whilst 80 who were given a hint solved it.

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Novices and experts

• What makes the difference between a novice and
  an expert?
• Pattern recognition the knowledge base they
  have developed. This is particularly important in
  semantically rich domains.
• Problem representation.
• Strategies.
• Problem-solving schemas.

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Chess expertise

• Chase and Simon (1973 DeGroot, 1965) looked at
  expert and novice chess players.
• Asked them to remember random vs. meaningful
  chess piece placements.
• Experts performed better only in the meaningful
  condition, suggesting they had chunked some
  pieces together.

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Physics expertise

• Chi, Feltovich and Glaser (1981) asked expert and
  novice physicists to group problems.
• Experts grouped problems together due to deep
  structural similarities, novice by surface
  similarities.
• Experts thus had more developed schemas based
  on solution criteria.
• Experts also work towards a goal, novices work
  from the goal backwards (Bhaskar Simon, 1977)

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Applications of PS

• Problem solving has been used to develop
  computational models of cognition as well as
  cognitive architectures.
• How solvers learn and develop expertise has also
  contributed to pedagogical understanding.
• Understanding what makes some problems hard has
  informed the design of everyday objects to give
  more efficient usage.