Problem Solving

 

Problem – 4 aspects:

·        goal: state toward which problem solving is directed

·        givens: conditions and constraints present (explicitly or implicitly) in the problem

·        means of transforming conditions

·        obstacles

 

Types of problems:

·        well-defined

·        four aspects are completely specified

·        e.g., maze; math problems

·        ill-defined

·        aspects are not completely specified or easy to infer

·        e.g., maintaining good relationship with roommates

 

Methods for studying problem solving:

·        reaction time, accuracy

·        good global measures of performance, but insensitive to

·        verbal protocols

·        subjects “think aloud” as they solve problems

·        infer strategies from the protocol

·        problem: people can’t always articulate their thoughts

·        computer simulation

·        what processes lead to the thoughts revealed by the protocols?

·        test hypotheses in simulations

·        forces people to be explicit about the processes they hypothesize

 

Problem solving as search through

a problem space

 

·        Newell & Simon

·        Problem solvers are information processing systems

·        Constrained by information processing limitations

·        Serial processing

·        Limited capacity STM

·        Essentially unlimited LTM

 

·        Problem solving requires search through a problem space

·        Problem space: internal representation of the problem

·        Consists of states and operators

·        State: representation of the problem in some degree of solution

·        Initial state: givens and prior knowledge

·        Goal state: desired outcome

·        Intermediate states: situations on the way to the goal state

·        Operators: means of transforming on state into another state

·        permitted moves

·        e.g., 8-tile puzzle; Tower of Hanoi

·        serial processing: consider current state and potential operators

 

Search through the problem space

·        algorithm: systematic procedure; guaranteed to find a solution

·        e.g., maze strategy

·        problem: too time-consuming to be generally useful

·        heuristic: a useful “rule of thumb” that can be used to guide search

·        does not guarantee a solution, but is more efficient

 

Heuristics for Problem Solving

 

·        given some set of potential next states, which one should be chosen?

 

·        Difference-reduction method (simple search)

·        Choose the state that is closest to the goal state

·        “hill climbing”

·        problems:

·        local maximum

·        considers only the next step, not the larger plan

·        some problems require move away from the goal state

·        e.g., hobbits and orcs problem

·        Means-ends analysis

·        Determine difference between current state and goal state

·        Choose operator that removes largest part of difference

·        Apply operator; continue until goal is reached

·        If operator cannot be applied, do not abandon it; find operator that enables it

·        i.e., create subgoals to enable the operator

·        e.g., fly from Champaign, Illinois to Providence, Rhode Island

·        the means can become an end itself

·        General Problem Solver: computer simulation by Newell & Simon (1972)

·        Uses means-ends analysis

·        Powerful problem solver

·        e.g., tower of Hanoi

 

Representation of the Problem

 

·        successful problem solving often depends on how the problem is represented

·        representation of states

·        e.g., mutilated checkerboard; gorge/rope problem

·        it may be useful to transform the representation of the problem

·        e.g., “going to the extremes” (Levine, 1988)

·        flagpole example

·        functional fixedness: inability to use objects in ways other than their typical use

·        e.g., Duncker’s (1945) candle problem

·        e.g., wrench-pendulum example

·        representation of operators

·        e.g., 9-dot problem

·        set effects: problem representation can be affected by prior experience

·        e.g., Luchins’s water jug problems

·        3 jugs of different capacity; need to measure out a specific quantity of water

·        e.g.,

Jug A: 5 cups    B: 40 cups     C:  18 cups

Need 28 cups

Solution: 2A + C

 

Jug A: 21 cups    B: 127 cups     C:  3 cups

Need 100 cups

Solution: B – A – 2C

·        Einstellung effect (mechanization of thought)

·        group with practice on problems of

     type (B-2C-A):

·        80% used that method when either A+C or A-C could have been used

·        64% could not solve problem 8:

 

          A: 28 cups  B: 76 cups  C: 3 cups  Goal: 25

          Cannot be solved by B-2C-A.

          Can be solved by A-C.

 

·        control group:

·        < 1% used B-2C-A when simpler solution could be used

·        5% failed to solve problem 8

 

·        set effect: practiced subjects developed a mental set -- a bias toward a particular solution