Problem Solving. Outline. What is Problem Solving? Kimberley Clow. Learning vs. Thinking Stages in Creative Thinking

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1 Problem Solving Kimberley Clow 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 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

2 Learning vs. Thinking Puzzle Box Learning Curve First Trial Tendency to perform Exploring Sniffing Grooming Reaching Scratching Reaching with paw Lever-Pressing Reward Later Trial Tendency to perform Lever-pressing Exploring Grooming Reaching Scratching Reaching with paw Sniffing

3 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? Insight Stages of Creative Thinking 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

4 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 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? 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?

5 Using Analogies Proportion of People Accessing Analogy Same Theme Different Theme Close Situation Remote Situation Situational Similarity 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 Evidence for Incubation Short Prep Long Prep, Brief Break Long Prep, Long Break

6 Problems in Arrangement Unscramble: xbo 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 Set One verba luppi bagler thrize Try Some Set Two prega rogena pleap viole Set Three broin arancy chifn relbawr Evidence for Illumination Draw through all 9 dots with 4 straight lines, without lifting your pencil.

7 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 Verification Water Jug Problem Problem # Jar A Jar B Jar C Desired Quarts of water Support the candle on the wall so that it doesn t drip on the table below

8 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 Looking at the Process 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 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

9 Methods of Problem Solving Brute Force Search explore entire problem space Hill Climbing always move toward a better state Work Backward start with goal and work back to start Means-Ends Analysis combination of hill climbing and working backward The Tower of Hanoi The Problem Space

10 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! Hill Climbing 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

11 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 Initial state Destination O O H H H Boat 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 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!

12 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 For the Logic Lover Three 5-handed monsters were holding three crystal globes. Because of the quantum-mechanical peculiarities of their neighbourhood, both monsters and globes come in exactly three sizes: small, medium, and large. The medium-sized monster was holding the small globe; the small monster was holding the large globe; and the large monster was holding the medium-sized globe. Because this situation offended their keenly developed sense of symmetry, they proceeded to transfer globes from one monster to another so that each monster would have a globe proportionate to his own size. Monster etiquette complicated the solution of the problem because it requires: that only one globe may be transferred at a time that if a monster is holding two globes, only the larger of the two may be transferred that a globe may not be transferred to a monster who is holding a larger globe. By what sequence of transfers could the monsters have solved this problem?

13 Symbol Problem Solve the following problem: Mary is ten years younger than twice Susan s age. Five years from now, Mary will be eight years older than Susan s age at that time. How old are Mary and Susan? List Person We know that 1. The person with asthma is in Room Ms. Jones has heart disease. 3. Ms Green is in Room Ms. Smith has tuberculosis. 5. The woman with mononucleosis is in Room Ms. Thomas is in Room Ms. Smith is in Room One of the patients, other than Ms. Anderson, has gall bladder disease. What disease does Ms. Anderson have, and in what room is she?