Transcript Problem Solving - Carnegie Mellon University

Chinese Ring Puzzle and its
Isomorphs (Kotovsky & Simon)
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Non-conscious problem-solving
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Announcements
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Midterm grades
Quiz tomorrow in recitation (thru today’s lec.)
Exam Thursday Oct. 30
Review session
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Tuesday, Oct. 28, 7pm, DH 2210
A very brief outline of the material will be shown
and you will be given time to ask questions about
the material
Review notes sample exam up soon
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Problem Solving
Definition of a problem
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A problem exists when you want to get from
“here” (a knowledge state) to “there”
(another knowledge state) and the path is
not immediately obvious.
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What do we have so far?
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Basic biology of the nervous system
Motivations
Senses
Learning
Perception
Memory
Thinking and mental representations
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What do we have so far?
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All of these topics give a basic sense of the
structure and operation of our mind
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General architecture of mind
What kinds of tasks does our mind engage
in?
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Language
Problem Solving
Decision Making
Others
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What are problems?
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Everyday experiences
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Domain specific problems
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How to get to the airport?
How to study for a quiz, complete a paper, and
finish a lab before recitation?
Physics or math problems
Puzzles/games
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Crossword, anagrams, chess
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A Problem Typology
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Well-defined vs. ill-defined problems: Problems
where the goal or solution is recognizable--where
there is a right answer (ex. a math or physics
problem) vs. problems where there is no "right"
answer but a range of more or less acceptable
answers.
Knowledge rich vs. knowledge lean problems:
problems whose solution depends on specialized
knowledge.
Insight vs. non-insight problems--those solved "all
of a sudden" vs. those solved more incrementally-in a step by step fashion.
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Some Problem Examples
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Tower of Hanoi
Weighing problem
Traveling salesman (100 cities = 100! or 10200 or
each electron, 109 operations per sec. would take
1011 years!!) but
100,000 cities within 1% in 2 days via heuristic
breakup (reduce search!)
Missionaries & Cannibals
Flashlight: 1, 2, 5, 10 min. walkers to cross bridge
21 link gold necklace/21 day stay
Subway Problem
Vases (or 3-door)
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Early findings
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Zeigarnik effect, 1927
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Participants were given a set of problems to solve
On some problems, they were interrupted before
they could finish the problem
Participants were given a surprise recall test
They remembered many more of the interrupted
problems than the uninterrupted ones
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Early Findings
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Luchins water jug experiment, 1942
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Participants were given a series of water jug
problems
Example: You have three jugs, A holds 21 quarts,
B holds 127, C holds 3. Your job is to obtain
exactly 100 quarts from a well
Solution is B – A – 2C
Participants solved a series of these problems all
having the same solution
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Early Findings
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Luchins water jug experiment, 1942
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New problem: Given 23, 49, and 3 quart jugs.
Goal is to get 20 quarts.
Given 28, 76, and 3 quart jugs, obtain 25 quarts
Some failed to solve, others took a very long time
Mental set
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People who solved series of problems using one method
tended to over apply that method to new similar
appearing problems
Even when other methods were easier or where the
learned method no longer could solve the problem
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Early Findings
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Duncker’s candle problem, 1945
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Problem: Find a way to fix a candle to the wall
and light it without wax dripping on the floor.
Given: Candle, matches, and a bow of thumbtacks
Solution: Empty the box, tack it to the wall, place
candle on box
Have to think of the box as something other than
a container
People found the problem easier to solve if the
box was empty with the tacks given separately
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Early Findings
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Functional Fixedness
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Inability to realize that something known to have
a particular use may also be used for performing
new functions
But is this really a bad thing?
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We learn and generalize from our experience in order to
be more efficient in most cases
Is it really a good idea to sit around trying to figure out
how many potential uses a pair of nail clippers has?
How often do mental sets and functional fixedness save
time and computation?
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General Problem Characteristics
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What characteristics do all problems share?
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Start with an initial situation
Want to end up in some kind of goal situation
There are ways to transform the current situation
into the goal situation
Can we have a general theory of problem
solving?
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General Theory of Problem Solving
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Newell & Simon proposed a general theory in
1972 in their book Human Problem Solving
They studied a number of problem solving
tasks
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Proving logic theroems
Chess
Cryptarithmetic
DONALD
+ GERALD
ROBERT
D=5
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General Theory of Problem Solving
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Verbal Protocols
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Record people as they think aloud during a
problem solving task
Computational simulation
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Write computer programs that simulate how
people are doing the task
Yields detailed theories of task performance that
make specific predictions
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General Theory of Problem Solving
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Problem spaces
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Initial state
Goal state(s)
Operators that transform one state into another
o1
Initial
Initial
o2
Initial
………………….
Goal
Goal
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An Example
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Tower of Hanoi
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Given a puzzle with three pegs and three discs
Discs start on Peg 1 as shown below, and your
goal is to move them all to peg 3
You can only move one at a time
You can never place a larger disc on a smaller disc
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An Example
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Tower of Hanoi problem space
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Initial condition: three discs on peg 1
Goal: three discs on peg 3
Operators: Move a disc following the problem
constraints
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Tower of Hanoi
Taken from Zang &
Norman, 1994
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Another example
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Missionaries and cannibals problem
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Six travelers must cross a river in one boat
Only two people can fit in the boat at a time
Three of them are missionaries and three are
cannibals
The number of cannibals on either shore of the
river can not exceed the number of missionaries
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Problem
Space
Taken from Jeffries et al., 1977
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Operators
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How do we choose which operators to apply
given the current state of the problem?
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Algorithm
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Series of steps that guarantee an answer within a certain
amount of time
Heuristic
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General rule of thumb that usually leads to a solution
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Algorithm Examples
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Columnar algorithm for addition
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Add the ones column
Carry if necessary
Add the next column, etc.
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+23
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People don’t have a simple algorithm for
solving most problems
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Heuristics
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Hill climbing
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Just use the operator which moves you closer to
the goal no matter what
What about problems where you have to first
move away from the goal in order to get to it?
Fractionation and Subgoaling
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Break the problem into a series or hierarchy of
smaller problems
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Heuristics
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Working Backwards from the goal
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Works well if there are fewer branchings going
from the goal to the initial state
Only works if you can reverse the operators
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Heuristics
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Means-ends analysis
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Always choose an operator that reduces the difference
between your current state and the goal state
Tests for their applicability of the operator on the current
problem state
Adopts subgoals if there is no move that will take you to the
goal in one step
Must have a difference-operator table or its equivalent
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Tells you what operator(s) to use given the current difference
between the state of the problem and the goal
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Simple Example
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Difference-operator table
Operators
Differences
Subtract
number from
both sides
Add number
to both sides
Multiply both
sides by
number
Divide both
sides by
number
Extra number added
on one side
Extra number
subtracted on both
sides
Extra constant
multiplier for x,
neither of the first
two differences
Extra constant divisor
for x, neither of the
first two differences
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First AI programs
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Newell & Simon
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Logic Theorist (LT)
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LT completed proofs for a number of logic theorems
General Problem Solver (GPS)
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GPS incorporated means-ends analysis, capable of
solving a number of problems
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Planning problems
Cryptarithmetic
Logic proofs
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Centrality of Representation
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Problem space and representation
Problem difficulty and representation
The interaction of representation and
processing limitations (problem isomorphs)
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Representation: Example
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Number scrabble
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1 2 3 4 5 6 7 8 9
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Limitations of GPS
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What about problems where there is no
explicit test for a goal state?
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Well-defined problems have a clearly defined goal
state
Ill-defined problems don’t have a clearly defined
goal state
GPS and other AI programs work only on
well-defined problems
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Examples of ill-defined problems
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Engineering Design
Architecture
Painting
Sculpture
How to run a business?
A number of other creative or difficult tasks
that people engage in
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Limits of AI?
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Can AI programs be applied to ill-defined
problems?
AARON
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Program created by Harold Cohen
Produces paintings using a number of heuristics
and general conceptions of aesthtics
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Art by AARON
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What makes problems hard?
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Large problem spaces are usually harder to
search than small ones
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Compare playing tic-tac-toe to chess
What factors from our architecture of mind
play a role in determining how hard a
problem is?
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Memory constraints
Memory contents
Types of mental representations we use
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Memory constraints
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Kotovsky, Hayes, & Simon, 1985
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Work on isomorphs of the Tower of Hanoi
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An isomorph of a problem is one in which the structure
of the problem space is the same but the appearance of
the problem is different
Remember the Tower of Hanoi?
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Isomorphs
Taken from Kotovsky, Hayes, & Simon, 1985
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Isomorphs
Taken from Kotovsky, Hayes, & Simon, 1985
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Isomorph Difficulty
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Results of Isomorphs
Adapted from Kotovsky, Hayes, & Simon, 1985
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Memory constraints
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In the original Tower of Hanoi and in the
condition with monster models there was an
external memory aid
Change problems are harder than move
problems
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Takes more processing to assess whether a
change is valid than it does for a move
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Spatial proximity of the information
Working with unchanging discs (stable representation)
vs. changing discs
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Computational Model
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Tested understanding via a computer model
that was:
Goal driven, subgoaling, limited memory capable of perfect
behavior except for limited working memory
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To see if we were in right “ballpark”
To separate actions of various mechanisms to see
which had the most control/influence
To be able to experiment with the separate
postulated mechanisms
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Model-Human Agreement
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Chinese Ring Puzzle and its
Isomorphs (Kotovsky & Simon)
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Non-conscious problem-solving
Strategy acquisition can
be unconscious--
Contents of Memory
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Does the contents of memory influence how
easy a problem is?
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Knowledge rich problems
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Require domain knowledge to answer, physics problems
Knowledge lean problems
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Can use a general problem solving method to solve,
don’t need a lot of domain knowledge
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Expertise
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Hayes on ten year rule
Expertise: What’s being
Learned in the Ten Years?
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DeGroot and Chase & Simon’s work on
chunking and chess
Estimates of knowledge base size
Practice Makes Perfect!
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Power law of practice: Ta = cPb + d
Expertise
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Physics (Simon et al., 1980)
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Physics experts approach physics problems differently than
do novices
Chess (Chase & Simon, 1973)
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Given a mid-game chessboard, grandmasters can
reconstruct it almost perfectly after studying it for only 5
seconds
Novices can only place 3-5 pieces correctly after the same
amount of study
However, if the pieces are randomly placed on the board,
novices and experts perform at the same level
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Knowledge in Chess
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Why do experts and novices perform
differently?
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Experts have more knowledge and experience
But the organization of this knowledge is crucial
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Experts can chunk the chess board into meaningful units
that are already in memory
Novices have no such chunking mechanism
Random placement of pieces eliminates this chunking
from an expert’s performance
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Mental Representations
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Insight problems
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Insight is a seemingly sudden understanding of a
problem or strategy that aids in solving the
problem
Sometimes require a change in mental
representation before the problem can be solved
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Mutilated Checkerboard
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Place dominoes
on the mutilated
checkerboard
until it is entirely
covered
Taken from Kaplan & Simon, 1990
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Mutilated Checkerboard
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Subjects had
difficulty solving
this problem
Average of 38
minutes
Requires parity to
be part of the
representation
Taken from Kaplan & Simon, 1990
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Learning in Problem Solving
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Can knowledge learned on one problem be
transferred to another problem?
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Sometimes, if people notice a similarity between
the source and target problems
How do people map knowledge from a source
problem to a target problem
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Analogy
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Analogy
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Classic example (Gick & Holyoak, 1983)
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Army problem
Cancer problem
Mapping between the two leads to a solution for
the cancer problem
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Conclusions
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Problem solving is an everyday activity
We can use findings from problem solving to
further our understanding of the mind and its
processes
We can use our knowledge of the mind’s
structure and operation to understand
elements of problem solving
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What are some methods of problem solving?
Why are some problems harder than others?
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