outline »problem solving ◊the gestaltist approach –stages of thinking »analogy...

• 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.• Describe Wallas’ stages of thinking during problem solving•When do analogies work best in problem solving? Describe the

multicontraint theory.

04/20/23

Math Phobic’s Nightmare

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?

?

5 m

20 m

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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are needed to see this picture.

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.

2 7 69 5 14 3 8

X O XO X OO X X

Problem Solving

• The Gestaltist tradition» Preparation: The role of representation

◊ A game of “15s”.

Top

Base

6 PM6 AM

Problem Solving

• The Gestaltist tradition» Preparation: The role of representation

◊ The Buddhist monk problem

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

◊ T T F F S S E N …

◊ 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 worldCould 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 analogy1 (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 theTower 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. HardestX

XX: :: O :

A.OO B. O C.

OO D.

XO

Problem Solving

• Problem Solving Basics» Typologies of problems (Greeno, 1978)

◊ Inducing Structure– Factors affecting completion time (Mulholland et al.)

Number of elements

6

4

Res

pons

e T

ime

(sec

)

5

3

1 32

8

1

7

2 1 Transformation

2 Transformation

3 Transformation

4 Transformation

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