PROBLEM
SOLVING
goal
directed sequence of cognitive operations
CLASSIFYING
PROBLEMS

WellStructured Versus IllStructured
Problems:

Wellstructured, or welldefined
problems have a clear path to their solution.

Illstructured, or illdefined
problems do not.

In reality, problems fall on a
continuum between these two extremes.

Greeno (1978) has suggested a
classification scheme which has three categories.

Not all problems need to be classified
into these three types; however, it is a framework of ideal typeshelps
to determine whether a given problem primarily consists of the need for
a particular skill.
A.) Arrangement
Arrangement problems present some
objects and require the problem solver to arrange them in a way that satisfies
some criterion; i.e., Duncker's 'candle' problem.
The problem requires the rearrangement
of objects to form a new relation among them. Often much trial and error
is The skills needed to solve such problems include
(1) Fluency
generating possibilities and rejecting poor ones.
(2) Retrieval
of previous solution patterns.
(3) Knowledge
of principles that constrain search.

Arrangement problems received
the most initial attentionthey were first studied in great depth by the
Gestalt psychologists (who are best typified by the work of Kohler with
his apes).

Gestalt psychology emphasized
the structure of patterns and analyzed problem solving from this perspective.

It argued that discovering the
correct solution usually occurs as a flash of insight. Insight can
be defined as the sudden discovery of a correct solution following a period
of incorrect attempts based on trial and error.

One factor which sometimes interferes
with finding a correct arrangement functional fixedness which is
a tendency to perceive an object only in terms of its most common use.
B.) Inducing Structure

In problems of inducing structure
the relation is fixed and the problem is to discover what that fixed relation
is.

The psychological processes involve
identifying relations among the components and fitting the relations together
in a pattern.The classic example is an analogy problem.

It involves several steps:
(1) The
encoding process
identifies attributes that could be important in establishing relations.
(2) The
inference process
establishes a valid relation between the 1st and 2nd items.
(3) The
mappingprocess
establishes relations between the 1st and 3rd items.
(4) Finally the application
process attempts to establish a relation between the 3rd and 4th items
that is analogous to the one between the 1st and 2nd items. Ex: Washington
: Lincoln :: 1 : ______ (5/10/20)
C.) Transformational Problems

These consist of an initial state,
a goal state, and a sequence of operations for changing the initial state
into the goal state, i.e., the water jar or cannibals problem.

Solution requires skills in planning
based on means/end analysisidentifying differences that exist
between the current state and the goal state and selecting operators that
will reduce the differences. Since the goal is given, one can compare the
current problem state with the goal state.

One factor which often interferes
with the solution of these problems is called
persistence of set.

Sometimes in the literature this
is called Einstellung, or habituation, and refers to the use of
a previously successful formula to a current problem, although it may not
be the most efficient formula to use.

This is like a mental set functional
fixedness.
NEWELL
& SIMON'S THEORY OBJECTIVES & METHOD

Although Newell & Simon's
methods have not been widely adopted, their theory of problem solving has
been influential in determining how psychologists think about human information
processing in general and problem solving in particular.

The theory provides a framework
for specifying how IP characteristics, the structure of the problem, and
different sources of knowledge interact to influence behavior.

They examined how individual subjects
tried to structure their problemsolving protocols.
A.) Theoretical Assumptions
Performance
on a problemsolving task is influenced by capacity, storage time, and
retrieval time of STM and LTMlimited STM capacity places a constraint
how many sequential operations that can be carried out mentally. The time
required to store new information in LTM can influence the efficiency of
a human problem solver. (Shades of Kintsch & language comprehension!)
B.) The Problem Space

The sequential nature of many
problems raises the question of what options are available at each point
in solving the problem.

Newell & Simon use the term
problem
space to refer to choices that the problem solver evaluates while solving
a problem.

There are several sources of information
that influence how we construct a problem space. These include:
(1)
the task instructions that give a description of the problem and which
may contain helpful information.
(2) previous
experience with the same task or a similar one
(3) previous
experience with analogous tasks.
(4) plans
stored in LTM that generalize over a range of tasks.
(5) information
accumulated while solving a problem.
COMPUTERSIMULATION
DEFINITION OF THINKING

This approach assumes that a problem
solver solves problems by applying operators to problem states.

An
operator is any move
that the problem solver deems to be legal, and which can be applied physically
or mentally.

A
problem state is a description
of the elements in a problem. Applying an operator results in changing
the problem from one state to another.

A
table of connections
– combines operators and problems states, showing all possible combinations
and their results.

Thus, problem solving involves
three major components:

The problem solver  the
information processing system (human or machine).

The problem  which is
also called the task environment.

The problem representation
 which is synonymous with the problem state.
A.) Means/Ends Analysis

An example of computer programmed
problem solving is the heuristic of means/end analysisspecifically General
Problem Solver (GPS).

A general procedure for solving
transformation problems is to select operators that result in a problem
state that is closer to the goal state.

Getting close to the goal is accomplished
by reducing the differences between the current problem state and the goal
state.
B.) Memory and Problem Solving

The computer analogy to problem
solving fails to account for the limitations of human problem solving,
specifically capacity limitations of STM and time needed to access LTM.

Thus, in means/ends analysis STM
places a constraint on the number of moves that can be evaluated at any
time. If there are four possible moves but STM only holds three the best
move might not be the one evaluated.

So people look for a GOOD MOVE,
which may not be the BEST MOVE. In fact, Atwood & Polson (1976) found
that STM capacity seems to be about three moves.

Also, an old state might appear
to be a new state if the problem solver does not remember itold states
need to be stored in LTM and then accessed as necessary.
SOME
GENERAL STRATEGIES
A. Heuristics: rules of thumb—
These
are shortcuts for problem solving (and for decision making). They allow
us to reach a solution relatively quickly and effortlessly however do so
at the expense of accuracythey do not guarantee a correct solution. Example:
'i' before 'e' except after 'c', and in long 'a' as in 'neighbor' and 'weigh'
B. Algorithms:
These
are strict protocols, which, if followed exactly will always lead to a
correct solution. The tradeoff is that they are relatively more time and
effort consuming. Example: look up each word in a dictionary.
C. Use of Subgoals:
This
refers to breaking up a problem into component parts. However, this is
often difficult. See below.
D. Analogy
This
is applying previously successful solution strategies to a new problem.
As research shows, below, most people are very poor at this, especially
when the deep structure of a problem remains the same but the surface structure
changes.
MORE
ON THE PROBLEM SPACE
Solving
a problem, can be seen as finding the correct path through a problem space.
Several techniques accomplish this:
A.) Subgoals 

The problem space can be broken
down into several smaller goals; i.e., converted into several smaller problem
spaces.

Subgoals can make solving a problem
easier: knowing an intermediate state is on the solution path helps prevent
searching unlikely paths.

Problem: Helpful intermediate
states are not always obvious, and reaching a subgoal can create confusion
about what to do next.
B.) Working backward 

The number of alternative paths
in a space can be reduced by working backward from the goal state toward
the initial state.
C.) Related problem spaces 

The correct path can be suggested
by remembering how you solved similar or analogous problems in the past.
THE TOWER
HANOI 
A.) Example of a Problem Space

The
initial state indicates
the three disks are on peg 1 with the largest on the bottom and the smallest
on the top.

The
goal state requires
the disks to be on peg 3 with the largest on the bottom and the smallest
on the top.

The
operators consist of
moving the top disk from one peg to another peg that does not contain a
smaller disk.

Each box in the problem space
represents one possible state of the problem, and each arrow represents
a legal action that could be taken from that state. You can move from an
intermediate state back to a previous state in the problem space.
B.) Strategies to Search the Problem Space

There are several methods used
to find a way from the initial state to the goal state:

1.) Random Trial
and Error  you randomly apply legal operators until you have generated
the goal state. This may involve many wasted moves. This is a good procedure,
however, for unfamiliar problems or when under stress.

2.) Hill Climbing
 a more systematic search in which you continually try to move from your
present state to a state that is closer to the goal. The main drawback
is that you may need to move away from the goal "locally" in order to achieve
the final goal.

3.) Means/Ends
Analysis  Again  the problem solver always works on one goal at a time.
If you are in a certain state, you set a goal of creating the goal state.
If that goal cannot be directly achieved, you set a subgoal of removing
any barriers to directly achieving the goal, etc.

Means/Ends analysis involves the
use of several unique subgoals:

The
transform goal involves
comparing the present state to the goal state and listing any differences
between the two.

The input is a description of
the present state A and the goal state B; the output is a description of
the difference D between the two states.

The
reduce goal involves
finding an operation that can be applied to reduce a certain difference.The
input is a description of the difference D and the output is an appropriate
operator Q that would reduce or eliminate that difference.

The
apply goal involves
applying the operator Q to state A to produce a new state. The input for
the apply goal is a description of the operator Q and the state A to which
it is to be applied.

The output is either a new state
a' of the operator Q can be directly applied to A or a description of the
difference D between state A and the required state if Q cannot be directly
applied to A.

Analogy requires that the problem
solver use the solution of a similar problem to solve a current problem.

Evidence, however, shows people
are not better at solving a second problem analogous to the first. Even
when the instructions reveal the exact relation between the two problems
it does not guarantee that it will be easier to solve the second problem.

An explanation for this is that
it is difficult to remember the correct solution when it consists of a
long sequence of moves. This suggests that the use of analogy may be more
effective when the solution is easier to remember.