Thinking and Problem Solving That is THINKING going
Thinking and Problem Solving
• That is THINKING? • • “going beyond the info given” Bruner ‘ 57 “a complex and high level skill” that “fills up gaps in the evidence” Bartlett ‘ 58 The process of searching through a problem space” Newell & Simon ‘ 72 What we do “when we are in doubt about how to act, what to believe, or what to desire” baron ‘ 94
• • What is critical thinking? Nobody said it better than Francis Bacon, back in 1605:
• For myself, I found that I was fitted for nothing so well as for the study of Truth; as having a mind nimble and versatile enough to catch the resemblances of things … and at the same time steady enough to fix and distinguish their subtler differences; as being gifted by nature with desire to seek, patience to doubt, fondness to meditate, slowness to assert, readiness to consider, carefulness to dispose and set in order; and as being a man that neither affects what is new nor admires what is old, and that hates every kind of imposture.
• • A shorter version is the art of being right. Or, more prosaically: critical thinking is the skillful application of a repertoire of validated general techniques for deciding the level of confidence you should have in a proposition in the light of the available evidence.
• What is this? An obscured image of butt with dimples!
• • Well-defined problems Ill-defined problems
• Generate-and-Test • Try this: • • Think of 10 things to eat or drink that start with c. Write down everything that occurs to you even if they don’t meet the criteria. How are you solving this problem? Useful technique when there aren’t a lot of possibilities to keep track of. (examples? )
• Means-Ends Analysis • • Compare the goal to the starting point. Newell and Simon’s GPS (general problem solver) DONALD +GERALD ROBERT • Doesn’t work well when the most direct route isn’t the most efficient ( you live west of Hotlanta and need to get to San Fran. )
• Working Backward • • Very similar to means-end analysis, establishing sub-goals is important Classic Example: The Tower of Hanoi
• Tower of Hanoi • Fascinating Facts • The Tower of Hanoi (sometimes referred to as the Tower of Brahma or the End of the World Puzzle) was invented by the French mathematician, Edouard Lucas, in 1883. He was inspired by a legend that tells of a Hindu temple where the pyramid puzzle might have been used for the mental discipline of young priests. Legend says that at the beginning of time the priests in the temple were given a stack of 64 gold disks, each one a little smaller than the one beneath it. Their assignment was to transfer the 64 disks from one of the three poles to another, with one important proviso–a large disk could never be placed on top of a smaller one. The priests worked very efficiently, day and night. When they finished their work, the myth said, the temple would crumble into dust and the world would vanish.
• Where's the Math in this Game? • The number of separate transfers of single disks the priests must make to transfer the tower is 2 to the 64 th minus 1, or 18, 446, 744, 073, 709, 551, 615 moves! If the priests worked day and night, making one move every second it would take slightly more than 580 billion years to accomplish the job! You have a great deal fewer disks than 64. Can you calculate the number of moves it will take you to move the disks from one of the three poles to another?
• Tower of Hanoi • • The goal here is to move all of the pieces to another position in the least amount of moves possible Rules: § You may move one disk at a time § No big disks may rest atop of a smaller disk http: //www. cut-the-knot. org/recurrence/hanoi. shtml
• Backtracking • The women, dogs, children, and jobs problem • • • There are 5 women: Cathy, Debbie, Judy, Linda, & Sonya There are 5 occupations: clerk, exec, lawyer, teacher, & surgeon Everyone has a different # of Children: 0, 1, 2, 3, or 4 Cathy owns the Irish setter The teacher has no children The owner of the Labrador retriever is a surgeon Linda does not own the Shetland sheepdog Sonya is a lawyer The owner of the Shetland sheepdog does not have 3 children The owner of the Golden retriever has 4 children
Backtracking
• Reasoning by Analogy • Dunker’s “the tumor problem” • Given an individual with an inoperable stomach tumor, and rays that destroy tissue at sufficient intensity, by what procedure can one free him from the tumor by these rays and at the same time avoid destroying the healthy tissue that surrounds it?
• Reasoning by Analogy • Dunker’s “the tumor problem”
The General and Fortress Problem (after Gick & Holyoak 1980, 1983) A small country was ruled from a strong fortress by a dictator. The fortress was situated in the middle of the country, surrounded by farms and villages. Many roads led to the fortress through the countryside. a rebel general vowed to capture the fortress. The general knew that an attack by his entire army would capture the fortress. He gathered his army at the head of one of the roads. The mines were set so that small bodies of men could pass over them safely, since the dictator needed to move his troops and workers to and from the fortress. However, any large force would detonate the mines. Not only would this blow up the road, but it would also destroy many neighboring villages. It therefore seemed impossible to capture the fortress. However, the general devised a simple plan. He divided his army into small groups and dispatched each group to the head of a different road. When all was ready he gave the signal and each group marched down a different road. Each group continued down its road to the fortress so that entire army arrived together at the fortress at the same time. In this way, the general captured the fortress and overthrew the dictator.
• Mental sets (kinda like perceptual sets) • • olk pronounced oak water jar problems
• Mental sets -- water jar problems
• Mental sets -- nine dots
• Draw through all 9 dots with 4 straight lines, without lifting your pencil.
Weisberg & Alba, 1981 • • there is an insight that helps…you can draw outside the square given the insight, is the solution obvious? • 25% of subjects given insight solved the problem
• Mental sets -- I’ll describe a situation from Perkins ‘ 81 and you decide what the situation is: • • There is a man at home. That man is wearing a mask. There is a man coming home. What is happening? I’ll answer yes/no form questions.
• Functional Fixedness • The string problem
• Using Incomplete or Incorrect Representations • If a problem is misunderstood you’re in trouble as a problem solver. • • The mutilated checkerboard problem. The numbers game.
• A checkerboard contains 8 rows and 8 columns, or 64 squares in all. You are given 32 dominoes, and asked to place the dominoes on the checkerboard so that each domino covers two squares. With 64 squares and 32 dominoes, there actually many arrangements of dominoes that will cover the board. • We now take out a knife, and cut away the top-left and bottom-right squares on the checkerboard. We also remove one of the dominoes. Therefore, you now have 31 dominoes which to cover the remaining 62 squares on the checkerboard. Is there an arrangement of the 31 dominoes that will cover the 62 squares? Each domino, as before, must cover two adjacent squares on the checkerboard. • The mutilated checkerboard problem.
Number Scrabble 1. Players alternate choosing numbers. 2. Whoever gets 3 numbers that total 15 wins.
Number Scrabble 6 7 2 1 5 9 8 3 4
• Lack of Problem-Specific-Knowledge or Expertise • • Expert Novice differences • Chess players • Physics instructors How do the experts differ? • • • Experts represent at a deeper and more principled level than novices who represent superficially Experts spend more time qualitatively analyzing on the front end, novices dive right in. Experts are more likely to check for errors in their thinking as they solve the problem.
• • Insight, aha, lightbulbs, etc… Incubation • Is it real or an illusion
• 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 state
• • Gp 1: brief time working, then 30 minute interruption Gp 2: brief time working, then 4 hour interruption Gp 3: longer prep time, then 4 hour interruption Gp 4: longer prep time, then 30 minute interruption
• • • 2 groups with brief preparation time solved problem approximately 55% of time long prep, short interruption: 64% long prep, long interruption: 85%
possible reasons: w recovery from fatigue w selective forgetting of irrelevant information w release from inappropriate set (related to functional fixedness) w work on the problem unconsciously
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