Week 11: Problems Flashcards
What is a problem?
An obstacle between a present state and a goal
Not immediately obvious how to get around the obstacle
When you are at a curb state and you want to move to another state
If there is a barrier, you must figure out how to overcome the barrier
Solve
- Problem of representation
History
Behaviorist
* Edward Thorndike (1911) – Law of Effect
* Trial and error
Gestalt
* Wolfgang Kohler (1917/25) – The Mentality of Apes
* Insight
* Information Processing
* Newell & Simon (1972) – Problem Space
* Means-end
Thorndike, proposed that problem solving relates to trail and error
Argued that people and animals will just try things and keep trying and when they are successful, there will be a reward and they will solve it in the future as it is reinforced
When things don’t result in success, it is a negative outcome and you will not do it again in the future
This method shows what works and doesn’t work in situations
Law of effect
Gestalt
Kohler, mentality of apes
Interacted with chimpanzees
One of the best subjects, solton, found that chimpanzees did things that he didn’t expect
Flash of insight, as shown a problem and witnessed that chimps would try to figure it out and got a sudden realization of how to do it
simon , problem space
4 features of problems
- a goal, or description of what constitutes a solution
- a description of objects relevant to achieving a solution
- a set of operations or allowable actions toward solution
- a set of constraints not to be violated
Certain operations that can be carried out
These are the things that we have to keep in mind
Types of problems
Knowledge-Lean Problems: can be solved (though not always skillfully) by use of instructions for the task and general problem solving skills
– e.g., finding a parking space in the mall
Knowledge-Rich Problems: requires specific knowledge or skill to solve the
problem
– e.g., calculus, computer-programming problems
Well-defined: a problem having a clear- cut solution; can be solved by an algorithm
– E.g., crossword puzzle or 3x = 2 (solve for x)
Ill-defined: a problem usually having multiple possible solutions; cannot be solved by an algorithm
– E.g., writing a hit song or building a career
Dont require a lot of knowledge (knowledge lean)
Usually can be solved fairly easily with general problem solving skills and instructions
Knowledge rich, need specific knowledge and skills
Such as solving a calculus problem
Well-defined: has a very specific solution
E.g., answer to crossword puzzle
Ill-defined: a provelm that can have multiple possible solutions
Such as writing a song
Types of Problems (Greeno)
Arrangment problems
Where you have to rearrange in order to solve something
Such as an anagram
Inducing structure problem
Trying to figure outt whats going on in a particular situation
E.g,. sequence of items and guess what the next one will be
Transformation problems
When you have to manipulate the object in the environment in order to achieve success
E.g., a rubix cube, tower of hanoi (7 moves)
General Problem Solving Methods
generate-test method (cf. “trial and error”) * e.g. fruit with a vowel as its fourth letter
* would take too long
* have to work through the whole problem for each of the many candidates generated
representation
* may require restructuring
* means-end analysis (a heuristic) * break problem into subgoals
* using analogies
* try to retrieve solution to a similar problem
GPSM
Trail and error -> generate-test method
This would take too long in order to reach solutation
A lot of things tha you cannot take back
What is the situation that you are dealing with?
This is what representation is
A lot of times we are doing a mean-end analysis
We are taking the problem and saying we have go here to here somehow
We reach the middle as a subgoal
Subgoals help reach main goal
Use analogies
Think if we have similar situations in the past and apply them here
Representation
The price of a notebook is four times that of a pencil. The pencil costs 30 cents less than the notebook. What is the price of each?
Mutilated Checkerboard Problem
Wickelgren (1974)
64 squares on chessboard cut off two corners, leaving 62
31 dominos
Each domino covers 2 squares
Can you cover the checkerboard with the dominoes?
Gut reaction is to say yes as there are 62 squares and 31 dominoes so it should be yes but that is wrong
Problem of representation
Since theres 32 black squares and 30 white squares
Kaplan & Simon (1990)
Conditions differed in how much information provided about the squares
Easier to solve when information is provided that points toward the correct representation of the problem
Looked at conditions that may lead to better or worse actions
Changed conditions such as the type of board given
Found that people were much better at solving problem with last condition compared to the blank condition
Required only one hint and people with blank condition required more than 3
Bread and butter are semantically related so it makes connection obvious
Easier to imagine that there needs to be an equal number
Herb Simon
trained in political science; also worked in economics and psychology
studied protocols of subjects talking during problem solving
won Nobel Prize in Economics (1978)
We don’t necessarily make optimal decisions
We make a solution
As long as we reach a minimal goal, we are good, no need to max out
Way of studying problems that was necessary for data
Rt from imaging are pretty helpful, hard to compare 30 mins and 29 mins and 5 seconds and why there is a difference and how they solved in differently
Verbal protocol, talk about experiences and talking aloud, articulating what you are doing every step
Problem Space
Problem space: the domain of the problem and the choices the solver evaluates during solution
Means-end analysis: a way of solving the problem by reducing the difference between the initial state and the goal state
- Subgoals: create intermediate states closer to goal
While there is problem space, we have to navigate this space that we don’t actually know in our heads
Apply the means-end analysis in this situation
What you are trying to do is reduce the distance the goal state
You are trying to get some distance towards it
Achieved by setting a subgoal
Setting a goal that is more achievable at the current state, you can then achieve the next one and gradually reach the final solution
Simon (1972)
Give equation and each letter in equation represents a number
Protocol Analysis
Newell & Simon (1972)
* thinking aloud during problem solving
* Hint R = 8
Analogy
Drawing a comparison in order to show a similarity in some respect
puppy : kitten :: dog : puppy : dog :: kitten :
gy
There are these other things that we have concepts for and attention acts like these things
Figure out the relationship in one situation and map it onto another
Use whatever information you have
Have you encountered a similar problem in the past that can be applied now
One group was given russian marriage problem
Russian village, 32 males and 32 females, they are going to get married
However, 2 of the males die, can you make 31 hetrosexual marriages?
This cannot be done
Given the mutilated-checkerboard after, question was that if it helped doing the russian thing before
If they could, they have done “analogical problem solving” and if they transferred it, it is called “analogical transfer”
source problem to target problem
Using Analogies to solve problems
Gick and Holyoak (1980)
* Noticing relationship
* Mapping correspondence between source and target
* Applying mapping
You have to notice that there is a relationship (critcal)
What exactly maps on from one situation to the other
The mapping can then be applied
Presented a problem by dunker
Radiation problem
Doctor, patient who has tumor in stomach, not responding to therapy
Could use radiation to destroy tumor, intensity would also damage nearby organs that the beam passes
If organs are destroyed, patient will die
Reducing intensity would not be strong enough to destroy tumor
How would you help patient?
Solution is to shoot the rays towards tumor in multiple directions
Low intensity multiple times and organs won’t be damaged but beams will multiply to be strong enough to destroy tumor
Found that no one came up with this solution
They gave some people another story
Learned about a general who was capturing a castle, but the king got word that they may be attacked and put mines leading to castle
If weight was sufficient, mine would explode
General split his army so they came across without triggering mine
If no story is given, 8% was able to solve
20% that heard story were correct
If hint is given, 76% made the connection
Strong hint, 92% were correct
Seems that more different the situations are, the harder it is to make connection
We really focus on circus features, real relationships between the 2
Cont.
we tend not to use analogies when the relation between problems is not obvious
* that is, when the problems differ markedly in surface features
* therefore, the difficult part of applying past knowledge is finding the relevant knowledge
A problem
Duncker (1945)
* instructions: using only the objects shown in the picture, mount the candle to the wall
Functional Fixedness: treating an object as having only one function; not thinking creatively
people do not think to use the box (the origin of “thinking outside the box”?)
Duncker (1945)
Participants were told that only the things in the image are allowed to be used so that the candle can be attached to the wall without it dripping
How do you do it?
Solution: tack the box onto the wall
Issue was that most people thought the box was a container as there were tacks in there instead of a pedestal
Origins of thinking of outside the box
Illustrates functional fixedness
2 string
They are far apart and goal is to tie them together
Solve: take pliers, tie them to one of the ropes and swing it and you can grab the other string and you catch it and tie it together
Professors aren’t immune
When universities hire profs
Bring in candidate and they are interviewed and then decide if they want to hire or not
One job had projector issues
Trying to figure it out and couldn’t get it display properly
Asked others
Someone offered textbook but then it was too high
One of the profs said just open the textbook
In that situation, everyone was fixated on the book being a closed object
Luchins (1942) Water Jug Problems
Gave people scenarios
Each problem is a situation, 3 water jugs, size is indicted by number
Take water and fill up jugs so you have this amount of water
Found that when people were given first 5 things to do, had a lot of difficulty for 8 and 6-7 kept using same solution
If people were given last 3 without first 5, people will then pick the simpler solutions
This is called mental set
Mental Set (Einstellung): a tendency to repeat a solution process that has succeeded previously
Obstacles to solving problems
- Misrepresenting the problem
- Functional Fixedness
- Mental Set
- Failing to notice analogous solution
Incubation and Insight
Why should an interruption help?
* Posner (1973) suggested three reasons:
* recovery from fatigue
* forget inappropriate approaches
* reorganization
Posner (1973) 3 possibilities
As we keep trying to come up with solutions, the longer it takes and more fatigued we become
If we take a break, we can relax and regenerate and come back with new energy
Forget the past approaches as chances are that if they didnt work last time, they wont work this time
May not be the appropriate thing to do
Going away from situation and coming back may allow us to notice something that we didn’t notice before
May find the missing piece
This is called insight
Dates back to BC times
Archimedes using water displacement happened randomly (idea)
Eureka: something triggers the critical missing piece, you can solve the situation quite easily (insight)
Insight in Problem Solving
- Sudden realization of a problem’s solution
- Often requires restructuring the problem
Metcalfe and Wiebe(1987)
* Insight: triangle problem, chain problem *
Non-insight: algebra
* Warmth judgments every 15 seconds
* How close are you to the solution?
Weibe (1987)
Constructed 2 situations
Used some situations as insight problems
Other situations were non-insight problems
While they were completieng these problems, every 15 seconds were told to say how far they are into the problem (hot or cold)
1–7 scale of how close they were
Looked at minute before when they figured out the solution
Algebra, level of 3 so they were not really close
Gradually as they got closer, warmth rating went up
Insight, even at 15 seconds before, they are still at a 3
Last 15 seconds, they get flash of insight and figure out how to do it
How Experts Solve Problems
What is an expert?
* “A person who, by devoting a large amount of time to learning about a field and practicing and applying that learning, have become acknowledged as being extremely knowledgeable or skilled in that field.”
* Experts solve problems in their field faster and with a higher success rate than beginners
* Experts possess more knowledge about their fields
Knowledge is organized so it can be accessed when needed to work on a problem
* Novice: surface features
* Expert: structural features
- Experts spend more time analyzing problem * Slow, but correct start
- Experts are no better than novices when given problems outside of their field
- Experts less likely to be open to new ways of looking at problems
Cont.
Experts do better within their fields
Chess players are better at solving chess players
They can remember legal moves, but if you randomly put them on board, then chess player is no better than the novice
Expert solving problem in different field, doesn’t make them different than novice
Experts have more knowledge and analogies that make it easier for them to solve
Study that asked people to organize problem
When people were given physics problem, novice focused on the surface features
Expert focused on the structural features such as potential energy etc
The way we conceptualize a problem depends on our knowledge on the item
Experts spend more time getting started on solving a particular problem
Slower start, it was a correct start and come up with solution much easier
Expertise can actually also be a negative thing
Sometimes mental set can come into play again
Bias towards examining information in a different way
