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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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Koehlers primate studies
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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.
The Problem-Solving CycleProblem Solving Cycle
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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 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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Figure 11.2 (p. 390)The circle problem. Seen end of chapter for the solution.
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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 11.3 (p. 391)(a) Triangle problem and (b) chain problem for two insight problems
demonstration. See end of chapter for solutions.
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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Figure 11.4 (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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Figure 11.25 (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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Figure 11.5 (p. 394)Objects for Runckers (1945) candle problem.
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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 11.6 (p. 394)Results of Adamsons (1952) replication of Dunckers candle problem.
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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 11.27 (p. 425)Solution to the candle problem.
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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.
Obstacles to Problem Solving
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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Obstacles to Problem Solving
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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Figure 11.8 (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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Figure 11.10 (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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Cognitive Psychology: Mind, Research, and Everyday Experience by Bruce Goldstein.Copyright 2005 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.
Figure 11.11 (p. 400)Initial steps in solving the Towerof Hanoi problem, showing howthe problem can be broken down
into subgoals.
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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 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