Problem Solving

A problem to think about (Nine dot problem):


Try and connect the nine dots by drawing four continuous straight lines without lifting your pen from paper?

Historical Roots

1. Edward Thorndike

He studied cats. More specifically, he studied the escape behavior of cats from puzzle boxes. He found fairly random behavior with success coming by accident. But learning was taking place: when repeated over trials, the time to escape became shorter. The learning can be explained in terms of operant conditioning.

Explained behavior this way:

Law of Effect - “Good” behaviors tend to be repeated; “bad” behaviors drop away

Characterizing behavior: Cat’s behavior was unsophisticated and nonstrategic. This type of learning is referred to as trial and error learning. (It is the lowest form of learning)

2. Wolfgang Kohler - Gestalt psychologist

A problem must be perceived as a whole (a gestalt) that can be reorganized into a new whole for a solution.

Criticized Thorndyke: Cat’s don’t normally find themselves in a box with a pulley. It was a contrived task that didn’t allow the animals to show intelligent behavior.

Kohler worked with chimpanzees on Tenerife during WWI. He designed problems for chimpanzees to solve. The two stick problem.

Insight learning
- the sudden understanding of the relationships between the elements of a problem

But Thorndike’s view became dominant in American psychology.

3. Wallas, a Gestalt psychologist (1926)

Characterized problem solving as a four-stage process (stage approach)

1. Preparation - gathering information; making initial attempts at a solution

2. Incubation - putting the problem aside for a while; Does work on the problem continue outside of conscious awareness? Anecdotal evidence from artists, scientists, and mathematicians. May also occur during sleep. Anecdotes about dreams.

Emperical evidence:

Silveira (1971) – gave subjects a difficult task called the necklace problem. They would work on it and he would give them a break in a different room with refreshments, He varied the length of breaks participants were given. Longer breaks did produce higher success rates.

3. Illumination - experiencing insight into how to solve the problem. Silveira’s subjects didn’t report illumination during their breaks. They just picked up where they had left off. Relationship between incubation and illumination is unclear.

4. Verification - checking the solution to be sure it is correct. Trivially simple for many problems. Laborious process in scientific research.

Problems with stage approach:

Key terms are rather vague.  (preparation, illumination)

Assumes that all problems are the same

Examples of problems:

Analogy

Series Completion

For children: the missionaries and cannibals (AKA Hobbits and Orcs) – 3 missionaries, 3 cannibals on one side of the river. Boat that holds two people at one time. Job is to get all across the river. If the cannibals ever outnumber the missionaries on either side of the river, they’ll eat them.

An animated solution of the Tower of Hanoi puz...

Tower of Hanoi is another example.

String problem – two strings hanging from ceiling to waste level in room. Task is to tie two strings together. You can use anything in the room (including a book of matches, a pair of pliers, and a few pieces of cotton.) (use the pliers as a pendulum to swing one string toward you. Could also use your shoelaces or a belt to lengthen one string. An example of how assumptions can interfere with problem solving.

Nine dot problem can be solved if you go outside of the grid but instruction never specify that. If you assume that drawing outside the lines is “out of bounds”, you can’t solve the problem.

Image via Wikipedia

Candle problem (Duncker) – given situation in a room and told to do something involving some materials. On the table is a book of matches, tacks, and a candle. Tack box on wall. Put candle in box. The crucial variable is how linked the tacks are to the box they are in. people find problem easier if the box is full of all the tacks. Much harder when it is labeled for tacks. Easier when it is completely blank. When some tacks are on table instead of box, people have easier time.

The Candle Problem: a classic experiment of in...

Image via Wikipedia

Duncker called this functional fixedness.

Functional fixedness – the inability to see new uses for familiar objects. Also true for pliers in the string problem.

People vary in functional fixedness.

How does one go about studying problem solving?

Think Aloud Protocol

Subjects “think out loud” as they solve a problem. It is helpful in identifying certain aspects of problem solving. In fact, it helped discover functional fixedness.

It does have its drawbacks:

  1. It’s a form of introspection – can only report what one is consciously aware of.
  2. It’s an unnatural task – may require practice, self-editing
  3. It’s hard with young children – requires linguistic sophistication

Problems may differ in fundamental ways

A taxonomy of problem types might be useful.

Greeno, 1978 classified problem types based on psychological skills knowledge needed for solving.

Three types of problems (although some may be a hybrid)

(1) Arrangement problems

Set of objects that must be arranged to satisfy some criterion. Typically many different arrangements possible but only one is the solution.

Example: Anagrams (“Word Jumble“) NAIRB –> Brain How many possible combinations? 5*4*3*2*1 = 120 But most subjects only consider a few.

Skills needed for such problems:

(a) Fluency in generating possibilities – solver needs to generate many partial solutions and discard those that aren’t working,

(b) Retrieval of solution patterns — relevant patterns must be assessed in LTM, can’t solve an anagram in a language you don’t know;

(c) Knowledge of principles that constrain search — knowledge of the language’s orthography [spelling rules] [more likely word starts with consonant than a vowel]

(2) Inducing structure problems:

a fixed relationship exists, and the solver must figure out what that relationship is.

Example: analogy problems

A:B::C:__

Series completion problems A,B,C,__

(3) Transformation problems

Initial state, goal state, information about operations to get from one state to the other. The problem: which transformation and what order?

Transformation

Initial state → Goal State

Examples: Missionaries and cannibals, tower of Hanoi, String problem, candle problem (many games, like chess)

Problem solving methods:

(1) Algorithms

Guarantee that a solution will be found (examples: long division, escaping from a certain mazes)

Advantages: guaranteed solution

Drawbacks: computationally complex, domain specific, time specific

(2) Heuristics

Mental shortcuts that provide a good chance of finding a solution. No guarantees this time.

Examples: Lost in the woods (moss grows on the north side of the tree)

Advantages: computationally simple, not time consuming, domain general Drawbacks: no guarantees

Problem solving strategies:

(1) means-end analysis

Solving a problem by setting up subgoals. Solution to each subgoal serves as a means toward the final end. Useful for large, complex problems (Heuristic)

(2) Working backwards

Moving from the goal state to the initial state.

Heuristics: judgment and decision making algorithms: artificial intelligence.

Intelligence

Complex and controversial topic

How to define it? Intelligence is what an intelligence test measures. That’s an example of an operational definition: defining something by specifying how it’s measured.

Charles Spearman (1904)

One general factor (g) underlies all human ability. People differ in intelligence because they differ in how much g they possess. But no agreed-upon way of measuring g has been found.

Maybe intelligence has two distinct components:

(1) fluid intelligence - the ability to solve novel problems creatively

(2) crystalized  intelligence - remembering and utilizing facts about the world.

Seems to correspond to people’s beliefs about intelligence: books smarts vs. street smarts; school of hard knocksFormal education emphasizes crystalized intelligence….perhaps at the expense of fluid intelligence.

Theory of Multiple Intelligences

Howard Gardner (1983) – Book Frames of Mind

How do we account for the sparing of cog abilities in brain injuries (amnesia, aphasia)? How do we account for child prodigies (math, music)? How about savants?

Proposes (at least) six relatively independent intelligences.

(1) linguistic intelligence

(2) Logical-Mathematical Intelligence

(3) Musical Intelligence

(4) Spatial Intelligence – representation of environment, gender differences

(5) Bodily Kinesthetic intelligence – athletic ability, acting

(6) personal intelligences – knowledge of the social world, includes emotional intelligence

Critiques: How do these Intelligences differ from skills?

 

Related articles