Problem SolvingProblem SolvingKimberley ClowKimberley Clow

kclow2@uwo.cakclow2@uwo.ca

http://instruct.uwo.ca/psychology/130http://instruct.uwo.ca/psychology/130

OutlineOutline

► Learning vs. ThinkingLearning vs. Thinking► Stages in Creative ThinkingStages in Creative Thinking

PreparationPreparation IncubationIncubation IlluminationIllumination VerificationVerification

► Hindrances to Problem SolvingHindrances to Problem Solving► Process of Problem SolvingProcess of Problem Solving► Methods of Problem SolvingMethods of Problem Solving► The BrainThe Brain

What is Problem Solving?What is Problem Solving?

► ProblemProblem A situation in which one A situation in which one

is trying to reach a goalis trying to reach a goal

► Problem SolvingProblem Solving Finding a means for Finding a means for

arriving at a goalarriving at a goal► Arises when a goal is Arises when a goal is

blocked and the solution blocked and the solution is not obviousis not obvious

Learning vs. ThinkingLearning vs. Thinking

Puzzle Box

Learning Curve

First TrialFirst Trial

Tendencyto perform

Scratching

Exploring

Sniffing

Grooming

Reaching

Lever-Pressing

Reward

Reaching with paw

Later TrialLater Trial

Tendencyto perform

Scratching

Lever-pressing

Exploring

Grooming

Reaching

Sniffing

Reaching with paw

Classic Problem SolvingClassic Problem Solving

►ArchimedesArchimedes Is the king’s golden Is the king’s golden

crown pure?crown pure?

►Köhler’s ChimpsKöhler’s Chimps How do I get at the How do I get at the

banana that is banana that is outside my cage outside my cage and beyond my and beyond my reach?reach?

InsightInsight

Stages of Creative ThinkingStages of Creative Thinking

►PreparationPreparation

► IncubationIncubation

► IlluminationIllumination

►VerificationVerification

Formulating problem & Formulating problem & making initial attempts to making initial attempts to solve itsolve it

Leaving the problem while Leaving the problem while considering other thingsconsidering other things

Achieving insight to the Achieving insight to the problemproblem

Testing and/or carrying out Testing and/or carrying out the solutionthe solution

PreparationPreparation

►Different types of problems need Different types of problems need different skills and approachesdifferent skills and approaches Problems in Inducing StructureProblems in Inducing Structure

►Analogy ProblemsAnalogy Problems Problems in TransformationProblems in Transformation

►Cheap NecklaceCheap Necklace Problems in ArrangementProblems in Arrangement

►AnagramsAnagrams

Problems in Inducing Problems in Inducing StructureStructure

►Suppose you are a doctor faced with a Suppose you are a doctor faced with a patient who has an inoperable patient who has an inoperable stomach tumour. You have at your stomach tumour. You have at your disposal rays that can destroy human disposal rays that can destroy human tissue when directed with sufficient tissue when directed with sufficient intensity.intensity. How can you use these rays to destroy How can you use these rays to destroy

the tumour without destroying the the tumour without destroying the surrounding healthy tissue?surrounding healthy tissue?

AnalogyAnalogy

►A general wishes to capture an enemy A general wishes to capture an enemy fortress. Radiating outward from the fortress. Radiating outward from the fortress are many roads, each mined fortress are many roads, each mined in such a way that the passing of any in such a way that the passing of any large force will cause an explosion. large force will cause an explosion. This precludes a full-scale direct attackThis precludes a full-scale direct attack How can the general attack the fortress?How can the general attack the fortress?

Using AnalogiesUsing Analogies

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Close Situation Remote Situation

Situational Similarity

Pro

port

ion

of P

eopl

e A

cces

sing

A

nalo

gy

Same Theme

Different Theme

Problems in TransformationProblems in Transformation

► Cheap Necklace ProblemCheap Necklace Problem Make a necklace out of 4 chains. It costs 2 cents to open a Make a necklace out of 4 chains. It costs 2 cents to open a

link and 3 cents to close a link. Make the necklace without link and 3 cents to close a link. Make the necklace without spending more than 15 cents.spending more than 15 cents.

Given Goal

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45

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60

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75

80

85

90

Short Prep Long Prep, BriefBreak

Long Prep, LongBreak

Evidence for IncubationEvidence for Incubation

Problems in ArrangementProblems in Arrangement

► AlgorithmAlgorithm Generate all Generate all

possible letter possible letter combinations and combinations and find which one(s) find which one(s) is(are) real wordsis(are) real words

► xboxbo► xobxob► oxboxb► obxobx► bxobxo► boxbox

► HeuristicHeuristic If 2 consonants and If 2 consonants and

a vowel, vowel a vowel, vowel probably in the probably in the middlemiddle

Few words start with Few words start with x, so put b firstx, so put b first

► boxbox

Unscramble: xbo

Try SomeTry Some►Set OneSet One

verbaverba luppiluppi baglerbagler thrizethrize

►Set ThreeSet Three broinbroin arancyarancy chifnchifn relbawrrelbawr

►Set TwoSet Two pregaprega rogenarogena pleappleap violeviole

Evidence for IlluminationEvidence for Illumination

Draw through all 9 dots with 4 straight lines, without lifting your pencil.

Individual DifferencesIndividual Differences

► Remote Associations Test (RAT)Remote Associations Test (RAT) What single word is an associate of these three?What single word is an associate of these three?

►   ResultsResults Correlation between RAT performance and the Correlation between RAT performance and the

generation of associatesgeneration of associates RAT improves following training in the RAT improves following training in the

generation of word associationsgeneration of word associations RAT is a better predictor of implicit learning RAT is a better predictor of implicit learning

than IQthan IQ►IQ is a better predictor of intentional learningIQ is a better predictor of intentional learning

High and High and lowlow RAT scorers deploy the RAT scorers deploy theiir attention r attention in different ways in different ways

VerificationVerification

Water Jug Problem

► Support the candle Support the candle on the wall so that it on the wall so that it doesn’t drip on the doesn’t drip on the table belowtable below

Hindrances to Problem Hindrances to Problem SolvingSolving

►Persistence of SetPersistence of Set Old strategies continue to be used -- even if Old strategies continue to be used -- even if

they are less efficient -- if we fail to they are less efficient -- if we fail to perceive that the situation has changedperceive that the situation has changed

►Functional FixednessFunctional Fixedness A tendency to use objects in their A tendency to use objects in their

customary waycustomary way►Negative SetNegative Set

A tendency to solve problems in one A tendency to solve problems in one particular way, even when a different particular way, even when a different approach might be more productiveapproach might be more productive►Why are these numbers arranged in this order?Why are these numbers arranged in this order?

8, 5, 4, 9, 1, 7, 6, 3, 2, 08, 5, 4, 9, 1, 7, 6, 3, 2, 0

Looking at the ProcessLooking at the Process

►Characteristics of Characteristics of Problem SolvingProblem Solving Goal Goal

DirectednessDirectedness Sequence of Sequence of

StepsSteps Cognitive Cognitive

OperationsOperations Subgoal Subgoal

DecompositionDecomposition

►Sequence of Sequence of problem solvingproblem solving Identify the Identify the

problemproblem Represent the Represent the

problemproblem Plan the solutionPlan the solution Execute the planExecute the plan Evaluate the planEvaluate the plan Evaluate the Evaluate the

solutionsolution

DefinitionsDefinitions

►Problem SpaceProblem Space Includes the initial, intermediate and Includes the initial, intermediate and

goal states of the problemgoal states of the problem►Also includes the problem solver’s Also includes the problem solver’s

knowledge at each of these stepsknowledge at each of these steps

►OperatorsOperators The set of legal moves that can be The set of legal moves that can be

performed during problem solvingperformed during problem solving

►GoalGoal Ultimate solution to the problemUltimate solution to the problem

Methods of Problem SolvingMethods of Problem Solving

►Brute Brute ForceForce SearchSearch explore entire problem spaceexplore entire problem space

►Hill Hill ClimbingClimbing always move toward a better statealways move toward a better state

►Work Work BackwardBackward start with goal and work back to startstart with goal and work back to start

►Means-Means-EndsEnds AnalysisAnalysis combination of hill climbing combination of hill climbing and and

workingworking backward backward

The Tower of HanoiThe Tower of Hanoi

The Problem SpaceThe Problem Space

Brute Force SearchBrute Force Search

►Go through all possible states until Go through all possible states until solution is foundsolution is found Guaranteed to work (eventually)Guaranteed to work (eventually) Impractical with large problem spacesImpractical with large problem spaces

►ExampleExample A 4-letter word meaning “not smart”A 4-letter word meaning “not smart” 456,976 possible states!456,976 possible states!

Hill ClimbingHill Climbing

Extreme Hill ClimbingExtreme Hill Climbing

► Dog cannot stop hill Dog cannot stop hill climbing toward boneclimbing toward bone needs to back upneeds to back up by going away from the by going away from the

goal for a bit, it will goal for a bit, it will eventually solve the eventually solve the problemproblem

► This is a problem for This is a problem for people too…people too…

O

H

Boat

Initial state Destination

OO

H

H

Get the Orcs and Hobbits to the other side of the riverBUT

Can’t have more Orcs than Hobbits anywhereBoat will only take 2 passengers at a time

Someone must steer the boat

Means-End AnalysisMeans-End Analysis

►Five StepsFive Steps Set up a goal or subgoalSet up a goal or subgoal Look at the difference between the Look at the difference between the

current state and the goal/subgoal current state and the goal/subgoal statestate

Look for an operator that will reduce or Look for an operator that will reduce or eliminate the differenceeliminate the difference

Apply the operatorApply the operator Apply steps 2 to 4 repeatedly until all Apply steps 2 to 4 repeatedly until all

goals have been achievedgoals have been achieved

Improving Your Problem Improving Your Problem SolvingSolving

► Increase Domain KnowledgeIncrease Domain Knowledge► Automate ComponentsAutomate Components► Have a systematic PlanHave a systematic Plan► Draw InferencesDraw Inferences► Develop SubgoalsDevelop Subgoals► Work BackwardsWork Backwards► Search for Contradictions and Relations Search for Contradictions and Relations

Among ProblemsAmong Problems► Use Different Problem RepresentationsUse Different Problem Representations► Practice!Practice!

Back to the BrainBack to the Brain

► Identified brain regions associated Identified brain regions associated with the mapping process in analogical with the mapping process in analogical reasoningreasoning Medial Frontal CortexMedial Frontal Cortex Left Prefrontal CortexLeft Prefrontal Cortex Left Inferior Parietal CortexLeft Inferior Parietal Cortex

►Solving insight problems involves the Solving insight problems involves the right hemisphereright hemisphere

For the Logic LoverFor the Logic Lover

► Three 5-handed monsters were holding Three 5-handed monsters were holding three crystal globes.three crystal globes.

► Because of the quantum-mechanical Because of the quantum-mechanical peculiarities of their neighbourhood, both peculiarities of their neighbourhood, both monsters andmonsters and globes come in exactly three globes come in exactly three sizes:sizes: small, medium, and largesmall, medium, and large..

► The medium-sized monster was holding the The medium-sized monster was holding the small globe; the small monster was holding small globe; the small monster was holding thethe large globe; and the large monster was large globe; and the large monster was holding the medium-sized globe.holding the medium-sized globe.

► Because this situation offended their Because this situation offended their keenly developed sense of symmetry, keenly developed sense of symmetry, they proceeded tothey proceeded to transfer globes from transfer globes from one monster to another so that each one monster to another so that each monster would have a globemonster would have a globe proportionate to his own size.proportionate to his own size.

► Monster etiquette complicated the solution Monster etiquette complicated the solution of the problem because it requires:of the problem because it requires: that only onethat only one globe may be transferred at a timeglobe may be transferred at a time that if a monster is holding two globes, only the that if a monster is holding two globes, only the

larger oflarger of the two may be transferredthe two may be transferred that a globe may not be transferred to a monster that a globe may not be transferred to a monster

who iswho is holding a larger globe.holding a larger globe.► By what sequence of transfers could the By what sequence of transfers could the

monsters have solved this problem?monsters have solved this problem?

Symbol ProblemSymbol Problem

►Solve the following problem:Solve the following problem: Mary is ten years younger than twice Mary is ten years younger than twice

Susan’s age.Susan’s age. Five years fromFive years from now, Mary will be eight now, Mary will be eight

years older than Susan’s age at that time.years older than Susan’s age at that time. How old are Mary andHow old are Mary and Susan?Susan?

List PersonList Person

► We know thatWe know that 1. The person with asthma is in Room 101.1. The person with asthma is in Room 101. 2. Ms. Jones has heart disease.2. Ms. Jones has heart disease. 3. Ms Green is in Room 105.3. Ms Green is in Room 105. 4. Ms. Smith has tuberculosis.4. Ms. Smith has tuberculosis. 5. The woman with mononucleosis is in Room 5. The woman with mononucleosis is in Room

104.104. 6. Ms. Thomas is in Room 101.6. Ms. Thomas is in Room 101. 7. Ms. Smith is in Room 102.7. Ms. Smith is in Room 102. 8. One of the patients, other than Ms. Anderson, 8. One of the patients, other than Ms. Anderson,

has gall bladder disease.has gall bladder disease.► What disease does Ms. Anderson have, and What disease does Ms. Anderson have, and

in what room is she?in what room is she?