STAGES OF PROBLEM SOLVING
• PREPARATION
– encode the “problem space” and
allowable “operators”
– retrieve relevant knowledge from LTM
• PRODUCTION
– devise strategy for searching the space
– implement the search
• EVALUATION
– has the goal been reached?
– are you rpgressing toward the goal?
UNDERSTANDING THE
PROBLEM (Greeno, 1977)
• Our internal representation (or model)
of the problem should have
– accurate CORRESPONDENCE between
relevant elements in the world and model
– good COHERENCE between elements in
the model
– appropriate links to PRIOR KNOWLEDGE
that can aid problem solving
links to prior
knowledge
coherence
correspondence
environment
model of
problem
SOLVING SCHEMATICALLY
“ODD” PROBLEMS
“odd” versions of problems are harder
to solve:
• Monsters & Globes (Simon & Hayes ‘76)
“exchange” versus “grow/shrink” rules
• “surreal” problems (Hinkley & Hayes, 1977)
• A boat takes two hours to steam 20 miles up a river
but only one and a third hours for the return trip. Find
the rate of the boat in still water. (original)
• A wine takes seven years to mature properly from
the time the grapes are harvested but only four years
for the return trip. How fast would the wine mature in
still time? (surreal)
FUNCTIONAL FIXEDNESS
Duncker’s (1940) CANDLE
problem: attach the candle to the
wall so it burns properly.
fewer than half the
subjects solve it
in 15 minutes.
Why?
s o lv in g
Adams (1952): put tacks outside of box
to reduce fixedness on “container”
function:
100
80
60
%
40
20
0
in box
out of box
Placement of Tacks
STRATEGIES FOR SEARCHING
THE PROBLEM SPACE
Tower of Hanoi
• RANDOM SEARCH STRATEGIES
– unsystematic search
– systematic random search
• HEURISTIC SEARCH STRATEGIES
– constraining the space
• ruling out search paths
– finding analogies (transfer)
• modelling and planning
• role of surface similarity
• role of “structural” mapping
Solving Problems by Analogy
• Duncker’s (1945) “radiation” problem:
(Holyoak et al., 1980, 83, 87)
no prior problem:
10%
prior “fortress” problem:
30%
“fortress” plus hint:
light bulb problem:
fragile-glass version:
insufficient intensity:
75%
69%
33%
MENTAL SET (“einstellung”)
Luchins (1942)
task: given three containers of known
quantity, produce target quantity:
A
B
C
goal
2
#1:
7
17
4
5
#2:
19
30
3
4
#3:
7
17
3
prior set
use of B-A-2C on #3:
83%
control
___%
2
SETTING THE PROBLEM ASIDE
task: make a necklace out of the
chains for $15:
$2 to open a link
$3 to close a link
1st block
15 min
15 min
15 min
Rest 2nd block %solving
-----15 min
55%
64
15 min 15 min
__%
85
4 hr
15 min
__%
(Silviera, 1971) mental set dissipates?
incubation?
STRATEGIES FOR SEARCHING
THE PROBLEM SPACE
• RANDOM SEARCH STRATEGIES
– unsystematic search
– systematic random search
• HEURISTIC SEARCH STRATEGIES
– constraining the space
– finding analogies (transfer)
– means-end analysis
• reducing the distance between
current state and goal state
• creating subgoals
CREATIVE PROBLEM SOLVING
• the ability to (solve) (ill-defined)
problems in a novel and appropriate
way
• typically results from intensive work
and extensive knowledge
– Edison: 1% inspiration, 99%
perspiration
• individuals vary “normally” by
ability, knowledge and style
• cognitive, social, and motivational
factors influence creative potential
and achievement
MEASURING CREATIVE
PERFORMANCE
• Guilford & others develop tests for
creative aptitude stressing “divergent
production”
– Remote Associations Test
– Uses Test
– Consequences Test
• Correlations of scores with peer-rated
professional creativity are modest (+.2
to +.4)
– reliance on sheer productivity, rather than
rarity or appropriateness
– wrong mental set?
– job constraints on opportunities for
creative work
– lack of “domain-specific” aspects testing
knowledge in field
• Constraining the search space
DONALD
+ GERALD
= ROBERT
BOVAR
= ?
YUNTT
= ?
REGEME = ?
GRUHNY = ?
The “radiation” problem:
A doctor has to treat a patient with a
malignant, inoperable tumor, buried deep
inside the body. There exists a special kind of
ray, which is perfectly harmless at a low
intensity, but at the sufficient high intensity is
able to destroy the tumor - as well as the
healthy tissue on his way to it. What can be
done to avoid the latter?
(Duncker, 1945)
The “Fortress” problem:
A General wanted to capture his enemy's fortress.
He gathered a large army to launch a full-scale
direct attack, but then learned, that all the roads
leading directly towards the fortress were blocked
by mines. These roadblocks were designed in such
a way, that it was possible for small groups of the
fortress-owner's men to pass them safely, but every
large group of men would initially set them off.
(Duncker, 1945)
The solution to the Fortress problem:
Now the General figured out the following
plan: He divided his troops into several smaller
groups and made each of them march down a
different road, timed in such a way, that the
entire army would reunite exactly when
reaching the fortress and could hit with full
strength.
The lightbulb problem/solution:
A very expensive lightbulb in a physics lab
had a broken filament. The surrounding glass
bulb was completely sealed, with no way to
open it. It could be repaired with a highintensity laser beam, fusing the two halves of
the broken filament.
[fragile glass version:] However, the highintensity laser would also break the fragile
glass; at a lower intensity, it wouldn’t break the
glass, but it wouldn’t fuse the filament either.
[insufficient intensity version:] However, the
laser generated only low-intensity beams that
were not strong enough to fuse the filament; a
much more intense laser beam was needed.
[Solution:] Placing several laser beams in a
circle around the lightbulb, low-intensity beams
are administered all at once, converging on the
filament, where their combined effect was
sufficient to fuse it.