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					                                                             11/23/2011
• Outline
    » Problem solving
       ◊ The Gestaltist approach
           – Stages of thinking
    » Analogy
       ◊ Multiconstraint theory
    » Problem Solving Basics
       ◊ Characteristics of problem
         solving
       ◊ Well-defined and ill-defined
         problems
       ◊ Problem typologies



Study Question.                                Math Phobic’s Nightmare

       • Describe Wallas‟ stages of thinking during problem solving
       •When do analogies work best in problem solving? Describe the
multicontraint theory.
          Problem Solving
Two flagpoles are standing, each 20 meters tall. A 30 meter
 rope is strung from the top of one of the flagpoles to the
 top of the other and hangs freely between them. The
 lowest point of the rope is 5 meters above the ground.
 How far apart are the two flagpoles?




          20 m




                             5m

                         ?
            Problem Solving
• The Gestaltist tradition
  » The goal of problem solving is the achievement of a
    Gestalt.
     ◊ A form, configuration or whole pattern.
     ◊ Parallels between perception and problem solving
            Problem Solving
• The Gestaltist tradition
  » Wolfgang Kohler (Circa WW I)




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            Problem Solving
• The Gestaltist tradition
  » Wolfgang Kohler (Circa WW I)
     ◊ Studied visual discrimination in animals
     ◊ Applied gestalt principles to animal perception
     ◊ Coined the term „insight‟ to describe the sudden
       perception of proper relations.
     ◊ Observed insightful behaviour in chimps
       solving problems
            Problem Solving
• The Gestaltist tradition
  » Wallas‟ (1926) stages of thinking
     ◊ Preparation
        – Recognize a problem exists
        – Find a representation for the problem
        – Preliminary attempts at solution
     ◊ Incubation
        – After failing to solve the problem it is set aside
        – No longer work on the problem at a conscious level
        – Work proceeds at an unconscious level
            Problem Solving
• The Gestaltist tradition
  » Wallas‟ (1926) stages of thinking
     ◊ Illumination
        – Flash of insight
        – Answer suddenly appears in consciousness
     ◊ Verification
        – Confirm the insight
        – Usually involves simple checking.
            Problem Solving
• The Gestaltist tradition
  » Preparation: The role of representation
     ◊ A game of “15s”.


               2 7   6         X O     X
               9 5   1         O X     O
               4 3   8         O X     X
            Problem Solving
• The Gestaltist tradition
  » Preparation: The role of representation
     ◊ The Buddhist monk problem
   Top




   Base
           6 AM                               6 PM
            Problem Solving
• The Gestaltist tradition
  » Incubation
     ◊ Fulgosi & Guilford (1968)
        – Imagine that all power stations shut down, then list all
          possible consequences.
        – Waiting 20 min before listing improved retrieval of obvious
          (but not remote) consequences
        – No effect for 10 min waiting interval
            Problem Solving
• The Gestaltist tradition
  » Incubation
            Problem Solving
• The Gestaltist tradition
  » Incubation
     ◊ Silveira (1971)
        – The chain necklace problem
        – Control Group: Works on problem for 30 min
        – Four Experimental Groups
             Brief or long preparation
            30 min or 4 hr interuption
            Everyone works on the problem for a total of 30 min
            Problem Solving
• The Gestaltist tradition
  » Incubation
     ◊ Silveira (1971)
             Problem Solving
• The Gestaltist tradition
  » Incubation
     ◊ Is it unconscious problem solving?
        – Silveria‟s verbal protocol
        – Subjects tended to pick up the problem where they left off
     ◊ Another explanation
        – Incubation allows us to change inappropriate sets
             Improves performance
        – Incubation can impair performance as well
             If the initial set had been appropriate
             Problem Solving
• The Gestaltist tradition
  » Illumination and insight
     ◊TTFFSSEN…
     ◊ Archimedes
     ◊ Insight problems
         – The fourth tree
         – Metcalfe & Wiebe (1987)
            » Had subjects complete either algebra or insight
               problems
            » Recorded “warmth” ratings
            Problem Solving
• The Gestaltist tradition
  » Metcalfe & Wiebe (1987)
     ◊ Results
            Problem Solving
• The Gestaltist tradition
  » Difficulties in problem solving
     ◊ Functional Fixedness
        – Duncker‟s candle problem
            Problem Solving
• The Gestaltist tradition
  » Difficulties in problem solving
     ◊ Negative set (Einstellung)
     ◊ Luchin‟s water jug‟s problem
        – Prepare subjects with a set
            B-2C-A
            Then switch to a simpler A - C
            (1% vs. 80%)
            Problem Solving
• The Gestaltist tradition
  » Representational change theory
     ◊ Insight involves first establishing the correct
       representation
  » The checkerboard - domino problem
            Problem Solving
• The Gestaltist tradition
  » The checkboard-domino problem
     ◊ Two ways to solve the problem:
        – Brute force
            » Try each of the 758,148 ways the first 30 dominos can be placed.
            » In each case, note that the 31st domino cannot be placed anywhere.
        – Insight answer
            » Note that each domino must cover one red & one black square
            » There are 30 red squares and 32 black squares left …
                Problem Solving
•   The Gestaltist tradition
    »   Three ways the representation can be changed
        1. Elaboration - Add new information
           – E.g.s, Letters can be used to symbolically represent something else
           – There are 30 red and 32 black squares
        2. Constraint relaxation - removing assumed constraints
           – The Nine dots problem
           – The landscaping problem
           – E.g., Lines drawn outside of the box, trees not planted on a flat surface
        3. Re-encoding
           – E.g., A box can be used as candle holder, pliers can be a pendulum
           – Dominoes cover one red & one black square
The biggest problem in the world
Could have been solved when it was small
           - Lao Tsu
              Problem Solving
• Analogy
  » Gick and Holyoak‟s convergence schema
     ◊ The parade problem
       A small country was controlled by a dictator. The dictator ruled the country
       from a strong fortress. The fortress was situated in the middle of the country,
       surrounded by farms and villages. Many roads radiated outward from the
       fortress, like spokes on a wheel. To celebrate the anniversary of his rise to
       power, the dictator ordered his general to conduct a full-scale military parade.
       On the morning of the anniversary, the generalユs troops were gathered at the
       head of one of the roads leading to the fortress, ready to march. However, a
       lieutenant brought the general a disturbing report. The dictator was
       demanding that his parade had to be more impressive than any previous
       parade. He wanted his army to be seen and heard at the same time in every
       region of the country. Furthermore, the dictator was threatening that if the
       parade was not sufficiently impressive he was going to strip the general of
       his medals and reduce him to the rank of private. But it seemed impossible
       to have a parade that could be seen throughout the country.
              Problem Solving
• Analogy
  » Gick and Holyoak‟s convergence schema
     ◊ Attack-Despersion problem
      A small country was controlled by a dictator. The dictator ruled the country
      from a strong fortress. The fortress was situated in the middle of the country,
      surrounded by farms and villages. Many roads radiated outward from the
      fortress like spokes on a wheel. A general arose who raised a large army
      and vowed to capture the fortress and free the country of the dictator. The
      general knew that if his entire army could attack the fortress at once it could
      be captured. The generalユs troops were gathered at the head of one of the
      roads leading to the fortress, ready to attack. However, a spy brought the
      general a disturbing report. The ruthless dictator had planted mines on each
      of the roads. The mines were set so that small bodies of men could pass
      over them safelybecause the dictator needed to be able to move troops and
      workers to andfrom the fortress. However, any large force would detonate
      the mines. Not only would this blow up the road and render it impassible, but
      the dictator would then destroy many villages in retaliation. It therefore
      seemed impossible to mount a full-scale direct attack on the fortress.
            Problem Solving
• Analogy
  » Gick and Holyoak‟s convergence schema
     ◊ The parade problem
     ◊ Attack-dispersion problem
     ◊ The radiation problem
     ◊ Three manipulations
        – X-ray problem alone or with an analogous problem
        – Similar or dissimilar story
        – Given hints or not given hints
             Problem Solving
• Analogy
  » Gick and Holyoak‟s (1980) results
     Group              Order                Use of analogy
      1 (hint)      Parade, radiation             49%
      2 (hint)      Attack dis., radiation        76 %
      3 (no hint)   no story, radiation            8%

     Effect of providing a hint
      1 (hint)      Attack dis., radiation        92 %
      2 (no hint) Attack dis., radiation          20%
              Problem Solving
• Analogy
  » Keane (1987)
     ◊ presented with either
        – very similar analogy (another story about a surgeon using X-rays)
        – moderately similar analogy (the story about the general‟s attack).
     ◊ Given the x-ray problem a week later.
  » Results
     ◊ 88% vs. 12%
     ◊ people tend to use relevant analogies only if they strongly
       resemble the problem (or are told the analogy exists).
            Problem Solving
• Analogy
  » Multiconstraint Theory (Holyoak & Thagard, 1997)
    ◊ Three factors constraining the use of analogy
       – Similarity
           Between the source and the target domain
           Similarity can be used as a red herring
       – Structure
           Establish parallel structures between the source and target problems
           Mapping
             e.g, Attack -------> Radiation
                    Fortress -----> Tumor
                    Etc.
            Problem Solving
• Analogy
  » Multiconstraint Theory (Holyoak & Thagard, 1997)
    ◊ Three factors constraining the use of analogy
       – Purpose of the analogy
           Problem solver‟s goals vs. goal of the problem
           e.g., Parading „out‟ vs. marching „in‟
             Problem Solving
• Problem Solving Basics
  » Characteristics of problem solving
     ◊ Three parts to a problem (Newell & Simon)
        – Initial state
        – Goal state
        – Operations to move through intermediate states
            Rules to get you from the initial state to the goal state
     ◊ E.g., The Tower of Hanoi
           Problem Solving
• Problem Solving Basics
  » E.g., The Tower of Hanoi


                      Initial State
           Problem Solving
• Problem Solving Basics
  » E.g., The Tower of Hanoi


                      Goal State
             Problem Solving
• Problem Solving Basics
  » E.g., The Tower of Hanoi
     ◊ Operations through intermediate states
        – Disks may be moved one at a time to any post
        – A larger disk may NOT rest on top of a smaller disk
     ◊ The problem space
        – The initial state, the goal state, and the intermediate steps to reach the
          goal. Also includes the problem-solver‟s knowledge at each step
            E.g., Problem graphs
Problem graph for the
Tower of Hanoi Puzzle
            Problem Solving
• Problem Solving Basics
  » Well-defined and ill-defined problems
     ◊ Well defined problems
        – Clearly defined start state and goal state
        – Operations for moving from one state to the next are clearly
          defined
        – E.g.s.
            The Tower of Hanoi
            Hobbits and Orcs
            Most of the problems we look at in this chapter
            Problem Solving
• Problem Solving Basics
  » Well-defined and ill-defined problems
     ◊ Ill-defined problems
        – One or more of the above three are not clearly defined
        – E.g.s.
            Making a purse from a sows ear
            Many of the day to day problems that we attempt to solve
            Problem Solving
• Problem Solving Basics
  » Typologies of problems (Greeno, 1978)
     ◊ Transformation
        – Find the sequence of operations necessary to transform an
          initial state into a goal state.
        – E.g.s
            The Tower of Hanoi
            Hobbits and Orcs
             Problem Solving
• Problem Solving Basics
  » Typologies of problems (Greeno, 1978)
     ◊ Inducing Structure
        – Determine the relationship among several given elements
        – E.g., Analogies
            If all the gems in the world were made of foam rubber:
            Wood: Hard :: Diamond: ?
            A. Valuable, B. Soft, C. Brittle, D. Hardest

                          X
              X   :       X        ::   O   :

                      O                              O        X
             A.               B.        O       C.       D.
                      O                              O        O
             Problem Solving
• Problem Solving Basics
  » Typologies of problems (Greeno, 1978)
     ◊ Inducing Structure
        – Factors affecting completion time (Mulholland et al.)

                                      8                                4 Transformation
                Response Time (sec)




                                      7
                                      6                                3 Transformation

                                      5
                                      4                                2 Transformation

                                      3
                                      2                                1 Transformation

                                      1

                                          1           2            3
                                              Number of elements
             Problem Solving
• Problem Solving Basics
  » Typologies of problems (Greeno, 1978)
     ◊ Arrangement
        – Some element must be rearranged according to some criteria
        – E.g., Anagrams
            YBO, AIFMA, PSAKKASCN
        – Global - local distinction
            Difficulty is affected by the number of ways the component letters
               can be (legitimately) combined

				
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