goldstein chp11-1

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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.

Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.

Chapter 11Part 1

Problem Solving


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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.

Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.

Figure 11.1 (p. 389)Flow diagram for this chapter.

Part 1

Part 2


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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.

Problem Solving

Problem solving is defined: a goal toaccomplish, with an initial state and goal

state, with obstacles to overcome which arenot obvious how to get around

Richard Feynman: You write down theproblem. You think very hard. Then you write

down the answer.

Examples: puzzles and real-world problems


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Salient differences between puzzle

problems and real-world problems

Puzzles

unfamiliar

involve no priorknowledge

all necessaryinformation is present

in the problemstatement

requirements areunambiguous

Real-worldproblems

familiar require prior knowledge

necessary informationnot present

what is the goal?


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Greenos problem types

Types of problems (Greeno, 1978):

Inducing structure (e.g., analogies)

Discovery of a pattern relating elements of a problem to

each other. Transformation (e.g., Towers of Hanoi, water jar problem)

Manipulation of objects or symbols while following certainrules.

Arrangement (e.g., anagrams, seating guests)

All the elements are given, and the task is to re-arrangethem.

Any problem could be in more than one category.


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Gestalt Viewpoint Koehler study with primates in 1910s

Problem solving use of mental representation

Problem-solving is both reproductive and productive

Reproductive Problem Solving involves re-use ofprevious experience (can be beneficial or

detrimental) Productive problem-solving is characterized by

restructuring and insight

Insight accompanied by subjective ah-ha

Problem-solving: trial-and-error or otherwise? Problem-solving cycle


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Koehlers primate studies


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The Problem-Solving CycleProblem Solving Cycle


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Problem Examples

Circle problem F 11.2 Triangle F 11.3 Cheap necklace F 11.3 Runckers Candle-wall problem F 11.5

Maiers Two-string problem F 11.7 Luchins Water jug problem F 11.8 Tower of Hanoi F 11.10, F 11.11 Hobbits-and-Orcs F 11.12 Mutilated checkerboard F 11.13 Monk and the mountain F 11.15 Many others handout


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Figure 11.2 (p. 390)The circle problem. Seen end of chapter for the solution.


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Figure 11.3 (p. 391)(a) Triangle problem and (b) chain problem for two insight problems

demonstration. See end of chapter for solutions.


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Insight and Non-insight

Problems

Insight: problem-solvers poor at predicting success; cantmonitor closeness to solution Sudden understanding of what is needed for the solution. Combining information in new ways

Non-insight or Routine: problem-solvers good at predicting

their success; monitor accurately how close they are tosolution Metcalfe studies

Made a distinction between insight and non-insight problems. Innon-insight problems solvers are accurate to predict success to asolution where for insight problem poor prediction of success to a

solution. Used warmth rating every 15 seconds while solving a problem.

The results shown in F 11.4, p. 393 comparing insight and non-insight problems. For non-sight problems when solvers were closerto the solution the warmth rating went up but not for the insightproblems. This was support for the Gestalt view of problem solving.


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Figure 11.4 (p. 393)Results of Metcalfe and Wiebes(1987) experiment showing howparticipants judged how closethey were to solving algebraproblems (left column) andinsight problems (right column).For the algebra problem, warmthratings move slowly towards thehot end of the scale during the

minute before the problem issolved. For the insight problems,the solution is sudden.


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Figure 11.24 (p. 424)Solution to the circle problem.


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Figure 11.25 (p. 424)Solution to the triangle problem. Arrows indicate movement; colored circlesindicate new positions.


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Figure 11.26 (p. 424)Solution to the chain problem. All the links in one chain are cut and separated (3cuts @ 2 cents = 6 cents). Then each separated link is used to connect the otherthree pieces and are then closed (3 closings @ 3 cents = 9 cents). Total = 15

cents.


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Obstacles to Problem Solving

Duncker (1945) candle-wall problem F 11.5, p. 394

A table A candle A box of matches A box of tacks.

Functional fixednessfix a candle to the wall

Participants are better at this task if the matches andthe tacks are on the table, not in the box. Adamson(1952) investigate effects of problem presentation andfound if the box was empty, problem solving wasmore successful F. 11.6, p. 394

It is easier to think of empty boxes as something otherthan containers to overcome functional fixedness.


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Figure 11.5 (p. 394)Objects for Runckers (1945) candle problem.


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Figure 11.6 (p. 394)Results of Adamsons (1952) replication of Dunckers candle problem.


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Figure 11.27 (p. 425)Solution to the candle problem.


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Functional fixedness fixed on the use of certain

objects Another example is Maiers (1931) two-string problem

F 11.7, p. 395 Two strings A chair

A pair of scissors or pliers Participants did not do well in solving the problem, 60%fail to solve. If the participants have not solve theproblem after 10 minutes, the strings are set in motion.It has been found that 23/37 participants who have notsolved the problem solve it within 10 second!

Other examples: panty-hose can be used to make a fanbelt water in a car radiator can be drunk


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Figure 11.7 (p. 395)Maiers (1931) two-string problem. As hard as Sebastian tries, he cant grab thesecond string. How can he tie the two strings together?


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Luchins (1942) water jar problem F 11.8, p. 397.Note that the strategy for solution changes afterproblem 6. There are more efficient solutions.

Mental set is the obstacle to problem solving. Set is a preference for certain operators (things you

can do, actions you can take to solve a problem). Participants who began the water jar problem with

problem 7 used the shorter more efficient version F11.9, p. 397.

The Gestaltist called these obstacles Einstellung(mechanization of thought) people kept using astrategy that worked even when a better one wasavailable.

Summary of problems in Table 1.1, p. 396


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Figure 11.8 (p. 397)Luchins (1942) water-jugproblem. Each problem specifiesthe capacities of jugs A, B, and

C, and a final desired quantity.The task is to use the jugs tomeasure out the final quantity.The solution to problem 1 isshown. All of the other problemscan be solved using the same

pattern of pourings, but thereare more efficient solutions toproblems 7 and 8.


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Figure 11.9 (p. 397)All of the participants whobegan the Luchins water-jugproblem with problem 7 used

the shorter solution (right bar),but less than a quarter whohad established a mental setby beginning with problem 1used the shorter solution tosolve problem 7 (left bar).


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Figure 11.28 (p. 425)Solution to problem 7 of the Luchins water-jug problem. Problem 8 can also besolved using just jugs A and C.


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Table 11.1 (p. 396)Gestalt Restructuring Summary


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Modern

Approaches to Problem Solving

Problem-Space Theory Newell and Simon view problem solving as a searchprocess GPS computer program solving a problem involves negotiating alternative paths to a solution

initial state is linked to goal state F 11.10, p. 399

Problem-space refers to the abstract structure of a problem slide 30 Operators are specific knowledge structures that transform data

knowledge states are produced by the application of mental operators

Algorithms vs. Heuristics Algorithms a step-by-step procedure that guarantees a solution to a problem if followed correctly Heuristics a general approach (rule of thumb) to problem solving that does not guarantee a solution.

Heuristics: means-end analysis: calculate difference between current state and goal; create a

subgoal to reduce that difference; select an operator that will solve this subgoal initialand goal states and operators slide 31 and F 11.11, p. 400

Subgoal structure slide 33 limited processing resources Move problems Towers of Hanoi, Hobbits-and-Orcs F 11.12, p. 401, slide 35 Other Heuristics: working forward, working backwards, generate and test, hill

climbing strategy


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Figure 11.10 (p. 399)(a) Initial and goal states for the Tower of Hanoi problem. (b) Operators thatgovern the Tower of Hanoi problem.


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Problem Space

Slide 30


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Means-Ends analysis

Slide 31


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Figure 11.11 (p. 400)Initial steps in solving the Towerof Hanoi problem, showing howthe problem can be broken down

into subgoals.


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Problem spaceTowers of Hanoi

Slide 33


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Figure 11.12 (p. 401)Initial and goal states for the hobbits-and-orcs problem. See end of chapter forthe solution (but try doing it first).

Problem space a version of hobbits-and-


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Problem spacea version of hobbits-and-

orcs: missionaries and cannibals

Slide 35

i


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Figure 11.29(p. 426)Solution to the hobbits-and-orcs problem. Each

trip indication on theleft (trip 1, trip 2, etc.)indicates the number ofhobbits and orcs thatremain after the trip.The hobbits and orcs in

the next row downindicate how manyhobbits and orcs thereare each time the boatcomes back.


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Representations of the Problem Some problems are more easily understood and solved if they are

represented in concrete terms (e.g. a mental image), others are more

easily solved in abstract terms. Mutilated checkerboard problem F 11.13, p. 401 Finding the right representationof a problem can be crucial for

finding the solution. Kaplan and Simon (1990) used four version ofthe mutilated checkerboard F 11.14, p. 403. The key to problemsolving was to arrive aparity representation take into account

that each square has a neighbor. Most Ps started by thinking aboutthe number of squares: 64-2=62, there is room for 31 dominos. Whenthe various board emphasized the difference between adjoiningsquares the problem was found to be easier. Bread and Butter average on 1 hint Bland board average of 3.18 hints

The color and black/pink boards somewhere between. There are other ways to arrive atparity representation, see p. 405

example

Monk and Mountain problem Use of image representation to solve problem F 11.15, p. 406


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Figure 11.13 (p. 402)Mutilated-checkerboard problem. See demonstration for instructions.


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Figure 11.14 (p. 403)Conditions in Kaplan and Simons (1990) study of the mutilated-checkerboardproblem.


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Monk and Mountain Problem


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Figure 11.15 (p. 406)Progress of Charlie, who starts climbing up the mountain in the morning, andSusan, who starts climbing down at the same time, showing that at some pointin time they are both at the same place on the mountain, just as the monk, inthe monk and the mountain problem is, during his ascent and descent.


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Heuristics for problem solving

Other Heuristics Working forward

Start at the beginning and try to solve the problem from thestart to the finish

Working backward The problem-solver start at the end and tries to work

backward from there

Generate and test The problem-solver generates a list of alternative ways of

action, not necessarily in systematic way, and then notices in

turn whether each course of action will work Hill climbing strategy

For any particular state, carry out the operation that movesyou closest to the final goal state. (often not a goodstrategy)


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Working backwards Heuristic:

Example


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One (painful) way to solve the waterlilies problem Initial number of water lilies = 1

double the initial value 90 times

Record each of these values

Find the value that is 1/2 of the90th day value.

1 1 31 1073741824 61 1152921504606850000

2 2 32 2147483648 62 2305843009213690000

3 4 33 4294967296 63 4611686018427390000

4 8 34 8589934592 64 9223372036854780000

5 16 35 17179869184 65 18446744073709600000

6 32 36 34359738368 66 36893488147419100000

7 64 37 68719476736 67 73786976294838200000

8 128 38 137438953472 68 147573952589676000000

9 256 39 274877906944 69 295147905179353000000

10 512 40 549755813888 70 590295810358706000000

11 1024 41 1099511627776 71 1180591620717410000000

12 2048 42 2199023255552 72 236118324143482000000013 4096 43 4398046511104 73 4722366482869650000000

14 8192 44 8796093022208 74 9444732965739290000000

15 16384 45 17592186044416 75 18889465931478600000000

16 32768 46 35184372088832 76 37778931862957200000000

17 65536 47 70368744177664 77 75557863725914300000000

18 131072 48 140737488355328 78 151115727451829000000000

19 262144 49 281474976710656 79 302231454903657000000000

20 524288 50 562949953421312 80 604462909807315000000000

21 1048576 51 1125899906842620 81 1208925819614630000000000

22 2097152 52 2251799813685250 82 2417851639229260000000000

23 4194304 53 4503599627370500 83 4835703278458520000000000

24 8388608 54 9007199254740990 84 9671406556917030000000000

25 16777216 55 18014398509482000 85 19342813113834100000000000

26 33554432 56 36028797018964000 86 38685626227668100000000000

27 67108864 57 72057594037927900 87 77371252455336300000000000

28 134217728 58 144115188075856000 88 154742504910673000000000000

29 268435456 59 288230376151712000 89 309485009821345000000000000

30 536870912 60 576460752303423000 90 618970019642690000000000000

Working backwards:- value doubling every day is

equivalent to say that the value ishalved each preceding day- the field was full Day 90th- the field was half full on day 89th


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Starting in the square marked by the circle, draw a line through all thesquares without picking up your pencil, without passing through asquare more than once, without diagonal lines and without leaving thecheckerboard.

Possible or Impossible?


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Crytarithmetic

DONALD + GERALD = ROBERT

SEND + MORE = MONEY

EAT + THAT = APPLE

CROSS + ROADS = DANGER

COUNT COIN = SNUB

With each crytarithmetic problems find a number to substitute for a letter.

Only one number can be used for any given letter and once a number isuse for a letter it cannot be used for another letter. If the letter appearsmore than once in a problem the same number can be used again.

Note: Newell & Simons GPS was able to solve these problems

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