Problem Solving

1. Problem Solving Kimberley Clow kclow2@uwo.ca http://instruct.uwo.ca/psychology/130

2. Outline • Learning vs. Thinking • Stages in Creative Thinking • Preparation • Incubation • Illumination • Verification • Hindrances to Problem Solving • Process of Problem Solving • Methods of Problem Solving • The Brain

3. What is Problem Solving? • Problem • A situation in which one is trying to reach a goal • Problem Solving • Finding a means for arriving at a goal • Arises when a goal is blocked and the solution is not obvious

4. Learning vs. Thinking Learning Curve Puzzle Box

5. First Trial Exploring Sniffing Grooming Tendency to perform Reaching Scratching Reaching with paw Lever-Pressing Reward

6. Later Trial Lever-pressing Exploring Grooming Tendency to perform Reaching Scratching Reaching with paw Sniffing

7. Classic Problem Solving • Archimedes • Is the king’s golden crown pure? • Köhler’s Chimps • How do I get at the banana that is outside my cage and beyond my reach?

8. Insight

9. Preparation Incubation Illumination Verification Formulating problem & making initial attempts to solve it Leaving the problem while considering other things Achieving insight to the problem Testing and/or carrying out the solution Stages of Creative Thinking

10. Preparation • Different types of problems need different skills and approaches • Problems in Inducing Structure • Analogy Problems • Problems in Transformation • Cheap Necklace • Problems in Arrangement • Anagrams

11. Problems in Inducing Structure • Suppose you are a doctor faced with a patient who has an inoperable stomach tumour. You have at your disposal rays that can destroy human tissue when directed with sufficient intensity. • How can you use these rays to destroy the tumour without destroying the surrounding healthy tissue?

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

13. Using Analogies

14. Problems in Transformation • Cheap Necklace Problem • 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 spending more than 15 cents. Given Goal

15. Evidence for Incubation

16. Algorithm Generate all possible letter combinations and find which one(s) is(are) real words xbo xob oxb obx bxo box Heuristic If 2 consonants and a vowel, vowel probably in the middle Few words start with x, so put b first box Problems in Arrangement Unscramble: xbo

17. Set One verba luppi bagler thrize Try Some • Set Two • prega • rogena • pleap • viole • Set Three • broin • arancy • chifn • relbawr

18. Evidence for Illumination • Draw through all 9 dots with 4 straight lines, without lifting your pencil.

19. Individual Differences • Remote Associations Test (RAT) • What single word is an associate of these three? • Results • Correlation between RAT performance and the generation of associates • RAT improves following training in the generation of word associations • RAT is a better predictor of implicit learning than IQ • IQ is a better predictor of intentional learning • High and low RAT scorers deploy their attention in different ways

20. Verification Water Jug Problem

21. Support the candle on the wall so that it doesn’t drip on the table below

22. Hindrances to Problem Solving • Persistence of Set • Old strategies continue to be used -- even if they are less efficient -- if we fail to perceive that the situation has changed • Functional Fixedness • A tendency to use objects in their customary way • Negative Set • A tendency to solve problems in one particular way, even when a different approach might be more productive • Why are these numbers arranged in this order? 8, 5, 4, 9, 1, 7, 6, 3, 2, 0

23. Characteristics of Problem Solving Goal Directedness Sequence of Steps Cognitive Operations Subgoal Decomposition Sequence of problem solving Identify the problem Represent the problem Plan the solution Execute the plan Evaluate the plan Evaluate the solution Looking at the Process

24. Definitions • Problem Space • Includes the initial, intermediate and goal states of the problem • Also includes the problem solver’s knowledge at each of these steps • Operators • The set of legal moves that can be performed during problem solving • Goal • Ultimate solution to the problem

25. Methods of Problem Solving • Brute ForceSearch • explore entire problem space • Hill Climbing • always move toward a better state • Work Backward • start with goal and work back to start • Means-EndsAnalysis • combination of hill climbing and working backward

26. The Tower of Hanoi

27. The Problem Space

28. Brute Force Search • Go through all possible states until solution is found • Guaranteed to work (eventually) • Impractical with large problem spaces • Example • A 4-letter word meaning “not smart” • 456,976 possible states!

29. Hill Climbing

30. Extreme Hill Climbing • Dog cannot stop hill climbing toward bone • needs to back up • by going away from the goal for a bit, it will eventually solve the problem • This is a problem for people too…

31. Get the Orcs and Hobbits to the other side of the river BUT Can’t have more Orcs than Hobbits anywhere Boat will only take 2 passengers at a time Someone must steer the boat O O O H H H Destination Initial state Boat

32. Means-End Analysis • Five Steps • Set up a goal or subgoal • Look at the difference between the current state and the goal/subgoal state • Look for an operator that will reduce or eliminate the difference • Apply the operator • Apply steps 2 to 4 repeatedly until all goals have been achieved

33. Improving Your Problem Solving • Increase Domain Knowledge • Automate Components • Have a systematic Plan • Draw Inferences • Develop Subgoals • Work Backwards • Search for Contradictions and Relations Among Problems • Use Different Problem Representations • Practice!

34. Back to the Brain • Identified brain regions associated with the mapping process in analogical reasoning • Medial Frontal Cortex • Left Prefrontal Cortex • Left Inferior Parietal Cortex • Solving insight problems involves the right hemisphere

35. For the Logic Lover • Three 5-handed monsters were holding three crystal globes. • Because of the quantum-mechanical peculiarities of their neighbourhood, both monsters andglobes come in exactly three sizes: • small, medium, and large. • The medium-sized monster was holding the small globe; the small monster was holding thelarge globe; and the large monster was holding the medium-sized globe.

36. Because this situation offended their keenly developed sense of symmetry, they proceeded totransfer globes from one monster to another so that each monster would have a globeproportionate to his own size. • Monster etiquette complicated the solution of the problem because it requires: • that only oneglobe may be transferred at a time • that if a monster is holding two globes, only the larger ofthe two may be transferred • that a globe may not be transferred to a monster who isholding a larger globe. • By what sequence of transfers could the monsters have solved this problem?

37. Symbol Problem • Solve the following problem: • Mary is ten years younger than twice Susan’s age. • Five years fromnow, Mary will be eight years older than Susan’s age at that time. • How old are Mary andSusan?

38. List Person • We know that • 1. The person with asthma is in Room 101. • 2. Ms. Jones has heart disease. • 3. Ms Green is in Room 105. • 4. Ms. Smith has tuberculosis. • 5. The woman with mononucleosis is in Room 104. • 6. Ms. Thomas is in Room 101. • 7. Ms. Smith is in Room 102. • 8. One of the patients, other than Ms. Anderson, has gall bladder disease. • What disease does Ms. Anderson have, and in what room is she?