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