Problem Solving Skills (14021601-3...

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Problem Solving Skills

(14021601-3 )

Lecture 1These slides are a modified version of those prepared by Dr. André Szameitat in cognitive psychology

Outline

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• Problem solving

• Introduction

• Theories of problem solving• Behaviourism

• Gestalt Psychology

• Representational Change Theory

• Information Processing Approach

• Analogical Problem Solving

• Expertise

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Introduction to Problem Solving

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Problem?

Introduction to Problem Solving What is a “Problem”?

You are in a current state or situation. (start state)

Your goal is to be in a different state. (goal state)

It is not obvious to you how to get from the start state to

the goal state.

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Example: finding food when stranded on a desert island.

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What is not a “Problem”?

If it is obvious to you how to reach the goal state.

If you are hungry and at home, just open the fridge…

However, what might be obvious to you, might be a

problem for others (Thus, problems are “subjective”)

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Types of problems

Definition of start state, goal state, and strategies.

Well-defined

Ill-defined

Knowledge required to solve the problem.

Knowledge-lean problems

Knowledge-rich problems

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Introduction to Problem Solving Types of problems

Well-defined problem: All aspects of the problem

are clearly specified

Start state

Goal state

Range of possible moves or strategies to reach the goal

Examples: Finding the way out of a maze; Playing chess

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Types of problems

Ill-defined problem: The problem is underspecified

Start state, goal state, and/or strategies may be unclear

Example: Keys locked in car, and you have an urgent

appointment.

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Wait for help?May take too long.

Try yourself?Potentially fails.

Smash window?Incurs costs.

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Types of problems

Ill-defined problem: The problem is underspecified

Most everyday problems are ill-defined problems

Further examples

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Writing anessay

Getting from A to Bon a tube strike day


Types of problems

Knowledge-rich problems:

Can only be solved if you have a considerable amount

of specific knowledge.

Studies in expertise often use knowledge-rich problems.

Example: Find the fault and repair an electronic device.

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Types of problems

Knowledge-lean problems:

No specific knowledge is required.

Most of necessary information is given in the problem

statement.

Example: Find your way out of a maze.

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Introduction to Problem Solving Types of problems – Summary

Well-defined vs Ill-defined problems

Knowledge-rich vs knowledge-lean problems

Both types are independent of each other

i.e., all potential combinations exist

In Psychology research

Mostly well-defined knowledge-lean problems are used

because they

Can be performed by everybody.

Have an optimal strategy for their solution.

Have an objectively right answer.

Thus, errors and deficiencies in strategies can be

assessed.

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Theories of Problem Solving

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Solution!

Theories of Problem Solving

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Behaviourism

Gestalt Psychology

Representational Change Theory

Information Processing Approach

Analogical Problem Solving

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Behaviourism & Problem Solving

Trial-And-Error Approach

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Thorndike’s Puzzle Box

Behaviourism Thorndike (1898)

Placed hungry cats inside a box.

Cats had to pull a lever to get out.

They showed “random” behaviour until, by

chance, they pulled the lever.

Slowly, they learned solving the problem.

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Thorndike Puzzle Box

Humans might learn this in one single trial.Cat’s learning curve

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Behaviourism Trial-and-Error learning

Is characterised by

repeated and varied attempts.

Is an unsystematic method

no insight, theory, or organised methodology.

Advantages

Does not rely on specific knowledge.

Thus, often used by animals and children.

Disadvantages

Tedious, time-consuming, monotonous.

Potentially risky (e.g. for attempts with very erroneous

outcomes).

Thus, in adults often last resort.

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Gestalt Psychology and Problem

Solving (Insight Approach)

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Insight, “A-ha Experience”

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Gestalt Psychology

Wolfgang Köhler (1921)

Placed food in chimpanzee cages which can be

retrieved only using tools, e.g.

Two sticks, either of them too short, but long enough when

stuck together

Boxes, which need to be stacked.

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Gestalt Psychology Wolfgang Köhler (1921)

Often observed the following behaviour

Monkey would try initially, but fail.

Monkey retreats frustrated into a corner of the cage.

Sits still for a while, seemingly doing nothing (“incubation”)

Then, suddenly, jumps up, rushes to the sticks and puts

them together

“Insightful behaviour”

“A-ha experience”

“Illumination”

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Gestalt Psychology

Gestalt Psychology

“The whole is different from the sum of its parts”

Not only in perception, but also in problem solving.

Example: Realising that two items initially considered

separate (the two short sticks) have to be integrated into

a single item (by putting them together).

Mental restructuring. Creation of a new representation.

Often connected with an ‘a-ha’ experience.

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Gestalt Psychology

Restructuring for a new representation

Two-string problem (Maier, 1931)

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Task: Tie the two strings

together.

Problem: Strings not long

enough to grasp both.

Solution: Tie one object to a

string and swing it.

(i.e., change the representation of one object from tool to weight.)

Functional fixednessTendency to consider only the

usual function of objects.

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Gestalt Psychology

Functional fixedness (Birch and Rabinowitz, 1951)

Two groups complete an electrical circuit by using (A) a

switch or (B) a relay

Then, two-string problem with a number of objects at

their disposal, incl. switches and relays

The two groups chose different devices to swing

When used switch for repair, they used relay to swing

When used relay for repair, they used switch to swing

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RelaySwitch

Gestalt Psychology

Summary

Gestalt Psychology “discovered”

The importance of the representation of the elements of

the problem.

The existence of “insight.”

in humans and animals

However, Gestalt Psychology did not

Specify the phenomena of restructuring and insight

Provide explanations

in terms of theories, models, mechanisms

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Representational change theory

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Matchstick arithmetics


Stellan Ohlsson (1992)

Based on Gestalt terminology.

Provides an elaborate mechanism which explains the

processes involved in insight.

Two main processes

Internal representation of the problem.

Constraints relaxation.

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Representational change theory Internal representation of the problem.

Example: The mutilated checkerboard problem (Kaplan

& Simon, 1990)

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• Two diagonally opposite squares are removed from a checker board (8x8 = 64 squares).

• Participants receive 31 dominos, each covering two squares (i.e., 31 dominos cover 62 squares).

• Is it possible to cover the remaining 62 squares with the dominos?

• After trying for a while to cover all squares, people realise that they have come to an impasse.

• There are nearly 1 million possible ways.

• A new representation is required.


Different representation for the checkerboard problem

Count number of white and black squares

Dominos

They always cover 1 black and 1 white square

Thus, 31 dominos will cover 31 white and 31 black

squares

Board

Original: Consists of 64 squares (32 white, 32 black)

Then, two white squares are removed, leaving

30 white squares and 32 black squares

A different representation highlights that there cannot be a

solution.

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A further representation

Change size of

checkerboard from 8x8 to 2x2

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Constraints relaxation

Example: Nine-dot problem

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Join all 9 dots by drawing 4

continuous straight lines, without

lifting the pencil

Representational change theory Constraints relaxation

Example: Nine-dot problem

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According to Gestalt laws, the 9 dots form the

shape of a square. Most people automatically

assume (wrongly) that one has to stay within

this square shape.

Relaxing the constraint of staying within the

square allows to solve the problem.

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Representational change theory Constraints relaxation

Example: Matchstick problem

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Produce a correct equation by moving

one matchstick.

Constraints relaxation hint:Consider not only the numbers but also the

operators (plus, minus, equal signs).

Representational change theory Summary


Is based on Gestalt principles.

But offers a much more detailed explanation how insight

can be achieved.

Mechanisms

Changing the problem representation

Constraints relaxation

Limitations

Unclear, when or in what way the representation

changes.

Many more factors affect solution finding in insight

problems (e.g., incubation)

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Tower of Hanoi task


In everyday life, most problems do not involve the

element of insight.

Example: You have a date and the car does not

start

Try to fix it yourself.

Call a friend to drive you.

Use public transport.

These solutions do not involve insight

What strategies are used for these problems?

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Problem solving: a search through a problem space

Problem space

All possible states in a problem (can be very large)

It offers an objective measure of optimal solutions

Initial state

Starting point of a problem, includes information given at

the start of a problem

Goal state

Desired end state/solution

Operators

The set of permissible operations that can be performed

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Example: Tower of Hanoi (Newell & Simon, 1972)

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Information Processing Approach Example: Tower of Hanoi, Problem Space (2-disc

version)

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Problem solving: a search through a problem

space

Problem space

All possible states in a problem (can be very large)

Short-term memory & Working Memory is limited

Often it is impossible to

First generate and hold the entire problem space in mind.

And only then search for the optimal solution.

But then, how is a solution found?

Use of heuristics

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Heuristic

A cognitively undemanding strategy which often

produces a solution

But not necessarily the optimal!

“rule of thumb”

Examples

Trial-and-Error

Hill Climbing

These examples do not require a real understanding of the

problem.

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Hill Climbing Heuristic (Newell & Simon, 1972)

If in doubt, choose a move which brings you closer to

your goal

instead of further away

If you aim for the peak: walk uphill, not downhill

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Hill Climbing Heuristic: Limitation

Sometimes, one needs to choose a move which leads

further away from the final goal to solve the problem.

To reach the highest peak (red), one needs to go a bit

downhill as well (green).

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Summary

Information processing approach

Works well for well-defined problems.

Problem space allows for objective measure of

performance.

Resulted in successful computer models (General

Problem Solver, GPS).

Limitations

Most everyday problems are ill-defined.

Does not perform well on insight-problems.

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Analogical problem solving

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Ideally, we should learn something by solving a

problem

We should be able to solve the same problem

We should be able to solve similar problems by transfer


Making analogies is actually very hard

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Fortress-Problem (Gick and Holyoak, 1980)

A general has to capture a fortress. Numerous roads

lead to the fortress but each of them contains mines,

which prevent the whole army from using one single

road. Yet, in order to overpower the enemy, the entire

army must attack the fortress at the same time. How to

attack?

Solution

Spreading the army on all the roads leading to the

fortress avoids losing soldiers to mines and enables to

attack with full power.

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Analogical problem solving Duncker’s (1945) Problem

A surgeon needs to operate on a patient with a

malignant tumour in the stomach. The tumour can be

removed by directing a kind of ray towards it. However,

a ray strong enough to destroy the tumour would also

destroy the healthy tissue around it. A ray that will not

harm the tissue, on the other hand, would be too weak

to destroy the tumour. How would it be possible to

destroy the tumour without damaging the healthy tissue?

Duncker’s (1945) Problem

Solution: The surgeon needs to spread the rays by using

several weak rays so that he does not harm healthy

tissues while still converging on the tumour with full

power.

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Fortress- and Duncker’s Problems are analogical

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Without any help, 10% solve the tumour problem.

When given the fortress story first, 30% solve the

tumour problem.

Stories do not share surface similarities.

They only have the same deep structure.

This makes drawing an analogy very hard.

When participants are told that the fortress story is

relevant to the tumour problem, almost all find the

correct solution.

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Summary

Much research on the factors determining finding

analogies, e.g.

Importance of superficial, structural, and procedural

similarities between past and present problem.

Limitations

Laboratory: Analogy often superficial and in close

temporal proximity. Real life the opposite.

Source for profound differences in individual’s abilities to

use analogies has not been investigated.

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Theories of Problem Solving Summary

Behaviourism

Trial and Error approach. Learning by reinforcement.

Gestalt Psychology

The whole is different from the sum of its parts

Introduced “Insight”

Representational Change Theory

Explains processes involved in “Insight”


For “non-insight” problems. Formal (start, goal, problem

space)

Analogical Problem solving

Transfer previous solutions to new problems

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Expertise Expertise

Skill or knowledge in a particular area.

Expert

A person with extensive knowledge or ability.

Face highly challenging tasks. Are problem solvers.

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Expert Chess Players

• Find the right “path” or solution among millions and billions of

options.

• Can play multiple games in parallel.

Gary Kasparov; probably the best chess player ever.

Expert Tennis Players

• Return balls served at a speed of over 150 miles per hour

(240 kilometres per hour).

• Less than 500 ms before the ball reaches the player.

• The player needs to determine the side, the distance and the

location where the ball will land

Roger Federer; probably the best tennis player ever.

Expertise

“Problem Solving” versus “Expertise”

Problem Solving

Focuses on heuristics that are flexible and can be applied

in most tasks.

Heuristics are of limited use in complex domains.

Expertise

Investigates knowledge specific to one domain.

How do people solve problems in always the same

recurring tasks.

Experience in this domain is a key element.

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Expertise Cognitive expertise

Chess, Memory masters, …

Perceptual-motor expertise

Tennis, Swimming, football, …

Seems to be rather related regarding the

underlying mechanisms of the expertise

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Cognitive Expertise

How do experts think?

Adriaan de Groot (1964)

Was chess master himself.

How do expert chess players think?

How do they find good moves?

Used experimental technique:

Pose a certain chess problem to an expert player.

Let them think aloud and use verbal protocols.

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Cognitive Expertise

How do super-experts think?

Pretty much like ordinary experts!

Strategies comparable

Both first inspect the chess position.

Both would then classify the position.

Both anticipated same number of moves.

However, Grand Masters still found better

solutions.

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Cognitive Expertise Then, what is the difference?

Time it takes to grasp the essence of the chess

position.

Grand Masters often within seconds.

Ordinary experts often more than 15 minutes.

Both have the same cognitive limitations.

But Grand Master have much more time to devote their

effort and cognitive resources at solution finding.

And, Grand Masters have a better knowledge base.

Shown with recall task of chess positions.

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Cognitive Expertise Recall task of meaningful chess positions

Grand masters clearly outperform normal experts.

Recall task of random chess positions

Grand masters not better than novices.

Explanation by chunking theory

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Initially: A meaningful chess position

Experts have a large knowledge base of chunks in

their long-term memory.

• Chunks are familiar patterns used as units

• Are acquired through extensive experience

Possible ways of dealing with the problem (i.e.,

solutions) are attached to the chunks.

According to template theory, chunks may further

grouped into more complex templates.

In random chess

positions, chunking

does not work!

Cognitive Expertise

Summary

Cognitive Expertise

Can be investigated by thinking-aloud technique.

Is at least partly based on chunking

i.e., a large knowledge base in long-term memory

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Perceptual-motor Expertise

Simple response task

As soon as you see a stimulus, press a button.

Participants almost never faster than 200ms.

Tennis serve

Up to 150 mph, takes less than 500ms to reach player.

How can a player possibly

Determine side, distance, and landing spot

And execute a complex movement (e.g. swinging the

racket)

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Perceptual-motor Expertise

Experts are better in predicting / anticipating

Occlusion paradigm

Watch video with the full service motion until racket hits the

ball.

Experts outperform novices in predicting landing

position.

What happens to performance when

Video is stopped earlier (before racket hits ball)

Parts of video are occluded (arm, upper/lower body,…)

Experts usually suffer more than novices

Experts use more information than novices, which helps

them to better predict upcoming events.

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Deliberate Practice How does one become an expert?

Practice, Practice, Practice, Practice, Practice,…

Practice, Practice, Practice, Practice, Practice,…

However, there are many people

who “practice” (perform) the activity quite a lot.

But nevertheless do not become real experts.

The type of practice matters.

Only “Deliberate Practice” can make you an expert.

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Deliberate Practice

Deliberate Practice

Highly structured activities that aim to eliminate

weaknesses.

At the appropriate level of difficulty.

Closely monitored so that constant feedback is provided.

Not inherently enjoyable.

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Deliberate Practice

Deliberate Practice

Predicts part of the differences between experts and

novices.

All researchers agree that deliberate practice is

necessary and inevitable to become an expert.

But is it sufficient?

Some say Yes: By using enough deliberate practice,

everybody can become an expert (in anything).

Some say No: Besides deliberate practice, talent is needed

as well.

Compare “nature versus nurture” controversy.

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Expertise

Summary

Expertise is skill or knowledge in a particular area.

Cognitive expertise (e.g. chess)

Based on previous knowledge (cf. chunking)

Perceptual-motor expertise (e.g. tennis)

Based on previous knowledge (cf. perception,

anticipation) and automaticity

Deliberate practice

Specific training regimes are required for becoming an

expert.

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Problem Solving Skills (14021601-3...

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