Psychology
247 Cognitive Psychology
Problem Solving
Erwin Segal
return to syllabus
I. What are problems?
Problems are situations in which one has a goal, but
the way to achieve that goal is not obvious. Anderson wrote that "human
cognition is always purposeful, directed to achieving goals and to removing
obstacles to those goals." (p. 240). Whenever we have a goal which we wish
to reach and the 'path' to that goal is not obvious we are faced with a
problem.
Cognitive psychologists and cognitive scientists try
to understand what experiences, knowledge, and kinds of processes and mechanisms
are involved in solving problems.
II. Examples of Problems
-
Memorizing a list (no transformations, goal known)
-
Barrier avoidance, problem box (one stage?)
-
Memory search: Answering questions, Taking a test, recalling
a phone number,
-
Tower of Hanoi, water jug problems, tile problems
-
anagrams, (transformation)
-
Selection and Categorization problems
-
cryptarithmetic, number sequence
-
Playing strategy games well (Chess, poker, bridge, (consider
particular moves in a game))
-
Problems in Physics and other sciences.
-
Geometry and other math problems, Logic problems
-
Skilled performance problems (Playing golf (correcting a
slice), playing a sonata, painting a picture, carving
a statue, removing a brain tumor, shooting a basketball. Strategy plus
skill problems.
-
Perceptual problems (Where's Waldo, Find words in letter
matrix, Find animals in picture, reading an
x-ray)
-
Creating novel objects: Writing an essay, composing a sonata,
painting a picture, carving a statue, inventing a
device, proposing a new scientific theory
-
Evaluating a claim: scientific theories, charges of a crime,
designing experiments,
-
Planning: Passing a test, Getting an A in a course, getting
a job.
-
Find historical causes (Solving crimes, explaining historical
events)
-
Social or political persuasion (Getting someone to do a favor,
convincing someone of something, getting elected to
public office, getting a job)
III. Structural analysis of problems: A critical aspect
of understanding problem solving is problem analysis. The most famous approach
is that of Newell and Simon and a "state-space"
analysis. The idea is that a problem solver searches
through a problem space until she quits or finds a solution.
Components of problems:
a. Problem space: The problem situation to be
comprehended by the problem solver
b. initial state: What the solver has at her disposal
when the problem commences
c. intermediate states: The state of the situation at
times after the task has begun
d. goal state: The state of the situation when the problem
has been solved.
e. moves, transformations, or operations: Procedures
which transform the situation from one state to another.
IV. Well-defined and ill-defined problems. Well-defined
problems are those when a-e in III are clear and unambiguous. All the other
problems are ill-defined in one way or another.
Hill-Climbing (Difference-Reduction
Method) Hobbits and Orcs.
Means-ends analysis (Setting
up subgoals, working backwards) Tower of Hanoi.
Computational theory of problem solving: This is the implicit
theory accepted in one form or another by most cognitive psychologists.
It claims that the may that problems are solved is by applying a set of
procedures to states to tranfer them into other states until the problem
is solved.
Anderson's method for doing this is by production
systems. This depends on identifying a
set of conditions which lead to particular actions. Act* is his
theory for this. Some underlying concepts which led to this development
follow:
The concept of 'effective procedure' is one of the more
important concepts in the symbolic sciences, and one which is needed at
least on an informal basis in order to work within any cognitive science.
"An effective procedure is a finite,
unambiguous description of a finite set of operations. The operations must
be effective in the sense that there is a strictly mechanical procedure
for completing them"
Algorithm: An effective
procedure which is guaranteed to solve a problem.
Universal Turing Machine:
An particular abstract information processing system consisting of a linear
tape, a read head, and a finite set of states. It can read the tape for
either of two symbols, it can write either of the two symbols, it can move
one unit to either the left or right, and it can switch from one state
to another. That is all. Correctly programmed a Turing machine can solve
any problem for which one can specify an algorithm. There are, however,
unsolvable problems (Godel proved that)
There are other systems of effective
procedures, Church's Lambda calculus, and Emil Post's production
systems to name two, which using a different architecture, can
solve any solvable problem.
Two critical ideas that are necessary
to have computational systems serve as the basis of complex problem solving
are rules of choice and recursive rules.
Rules of choice are
rules in which the data are evaluated to decide what to do next.
Recursion rules are
rules in which the output of the rule may be used as an input to the same
rule either immediately or after other operations have occurred.
Church-Turing Thesis:
Any problem solving device which uses an effective set of operations to
solve certain arithmetic problems is logically equivalent to any other
system which can solve such problems. Any such device contains all the
procedures necessary to solve any solvable well-specified problem.
(click here for an article on the Church-Turing
Thesis)
Strong AI extension of Church-Turing
Thesis: Humans can solve these arithmetic
problems. Therefore their mechanisms are computationally equivalent to
those of machines that solve them. Thus any problems humans can solve,
are solvable by computational machines. I have a book chapter on line which
discusses some of these issues, you can access it here.
V. Analogy and transfer. Difficulty in finding appropriate
analogies.
VI. Gestalt analyses and influence.
Gestalt psychology
-
Their primary areas of study were form perception and thinking.
-
Their analyses start with whole situations. Gestalt psychologists
argue that one interprets the elements in terms of the situation within
which they are found.
-
The locus of the element within the structure is of paramount
importance.
-
Form, wholes and
structure are terms that underlie
the basic concepts of Gestalt Psychology. These are words that have been
used to try to define Gestalt.
Gestalt Psychologists:
Some problems Gestalt Psychologists have discussed:
-
The nine dots problem: Connect nine dots arranged in a 3
X 3 square with four straight lines without lifting your pencil or retracing
your step.
-
The mutilated checkerboard problem: How many dominoes (rectangles
that can cover exactly two adjacent squares of a checkerboard are needed
to cover all of the remaining squares of a checkerboard that has two diagonal
squares removed?
-
Add a sequence of numbers, e.g., 1+2+3+4+…+99+100.
-
Finding the area of a parallelogram and similar figures.
-
The two string problem (Functional fixedness)
-
The candle problem
-
the water jug problems (rigidity of set)
Problem solving structure according to two Gestalt influenced
researchers.
| Wallas
(1926) |
Polya
(1957) |
| 1. Preparation |
1. Understanding the problem |
| 2. Incubation |
2. Devising a plan |
| 3. Illumination |
3. Carrying out the plan |
| 4. Verification |
4. Looking back |
David Marr's (1981) approach: Using a general Gestalt
backdrop as a framework David Marr (a computational and neuroscientist)
set up a kind of strategy for a problem solver. There are three stages
of analysis involved in solving most problems.
1) Computation.
The problem solver must analyze the task that needs to be done rather carefully.
This requires an analysis of the specific parts. What are the inputs to
the problem? What are the relations between the parts?
2) Algorithm (and
representation). The second task for the problem solver is to specify an
effective procedure that one can carry out in order to achieve the goal
of the task. This requires a specific characterization of the sequence
of operations that operate on a given data base; if the sequence is followed,
it will lead to a solution of the problem.
3) Implementation. This
requires identifying a set of physical objects which can carry out the
algorithm automatically.
VII. Role of Content: There is a great deal of research
that shows that the logical structure of a problem is often not the major
variable in solving the problem. The framing
of the problem, or the schema to which
the problem solver relates it often play an important role in solving the
problem. It is very difficult to integrate the facts of the influence of
content with the standard computational theories.
VIII. Anagrams--A set of
problems on which much research was done, primarily by behaviorists and
other associationist theorists.
a. Examples
i. verba, luppi, bagler, thrize
ii. prega, rogena, pleap, viole,
iii. broin, arancy, chifn, relbawr
b. Variables that affect solution time.
i. Familiarity of goal word
ii. Transition probability of letter sequences in target
iii. Number of moves required to reach target
iv. Transition probability of presented letter string
v. Context within which anagram is presented
c. An explanation from association theory
i. The anagram has responses associated with
it
ii. Habit family hierarchy--Same stimulus is associated
with different responses at different strengths
iii. Strength of responses determine its likelihood of
expression
iv. A response that is not reinforced is weakened
v. Strong habits are harder to override than weaker ones
vi. Higher frequency patterns have stronger responses
associated with them
d. An explanation based on a
task analysis
Structure:
Words, syllables, consonant clusters, frequency, target areas
Strategies
Count vowels and consonants,
Try for high frequency groupings in the right place in a
word
Search for words with same letters as presented string
Search within a category
return to syllabus