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A Rational Analysis of Alternating Search and
Reflection Strategies in Problem Solving
Niels Taatgen ([email protected])
Cognitive Science and Engineering, University of Groningen
Grote Kruisstraat 2/1, 9712 TS Groningen, the Netherlands
Abstract
In this paper two approaches to problem solving, search and
reflection, are discussed, and combined in two models, both
based on rational analysis (Anderson, 1990). The first model is
a dynamic growth model, which shows that alternating search
and reflection is a rational strategy. The second model is a
model in ACT-R, which can discover and revise strategies to
solve simple problems. Both models exhibit the explore-
insight pattern normally attributed to insight problem solving.
Search vs. Insight
The traditional approach of problem solving is one of prob-
lem space search (see for example Newell & Simon, 1972).
Solving a problem means no more or less than finding an
appropriate sequence of operators, that transform a certain
initial state into a state that satisfies some goal criterium.
Problem solving is difficult if the sequence needed is long, if
there are many possible operators, or if there is no or little
knowledge on how to choose the right operator.
A different approach to problem solving is that the crucial
process is insight instead of search. This view also has a rich
tradition, rooted in Gestalt psychology. According to the
insight theory, the interesting moment in problem solving is
when the subject suddenly “sees” the solution, in a moment
when an “unconscious leap in thinking” takes place. Instead
of gradually approaching the goal, the solution is reached in
a single step, and reasoning efforts before this step have no
clear relation to it. In this sense problem solving is often
divided into four stages: exploration, impasse, insight and
execution.
Insight can be viewed in two ways: as a special process, or
as a result of ordinary perception, recognition and learning
processes (Davidson, 1995). Despite the intuitive appeal of a
special process, the latter view is more consistent with the
modern information-processing paradigm of cognitive psy-
chology, and much more open to both empirical study and
computational modeling.
Both the search and the insight theory select the problems
to be studied in accordance with their own view. Typical
“search”-problems involve finding long strings of clearly
defined operators, as in the eight puzzle, the towers-of-hanoi
task and other puzzles, often adapted from artificial intelli-
gence toy domains. “Insight”-problems on the other hand,
can be solved in only a few steps, often only one. Possible
operations are often defined unclearly, or misleadingly, or
are not defined at all. A typical insight problem is the nine-
dots problem, in which nine dots in a 3x3 array must all be
connected using four connected lines. The crucial insight is
the fact that the lines may be extended outside the 3x3
boundary. Other insight problems are the box-candle prob-
lem and several types of matchstick problems (see for exam-
ple Mayer, 1995). Due to this choice of problems, both
evidence from insight and search experiments tend to sup-
port their respective theories. Both theories ignore aspects of
problem solving. The search theory seems to assume that
subjects create clear-cut operators based on instructions
alone, and that subjects do not reflect on their own problem-
solving behavior, while the insight theory assumes all pro-
cessing that happens before the “insight” occurs has hardly
any relevance at all. So probably search and insight are both
aspects of problem solving, and the real task is to find a the-
ory of problem solving that combines the two (Ohlsson,
1984). One such view sees insight as a representational
change. Search is needed to explore the current representa-
tion of the problem, and insight is needed if the current rep-
resentation appears not to be sufficient to solve the problem.
In this view, search and insight correspond to what Norman
(1993) calls experiential and reflective cognition. If someone
is in experiential mode, behavior is largely determined by the
task at hand and the task-specific knowledge the person
already has. In reflective mode on the other hand, compari-
sons between problems are made, possibly relevant knowl-
edge is retrieved from memory, and new hypotheses are
created. If reflection is successful, new task-specific knowl-
edge is gained, which may be more general and on a higher
level than the existing knowledge.
The scheduling task
An example of a task in which both search and insight are
necessary is scheduling. Figure 1 shows an example of a
scheduling task used in our experiments. The goal is to
assign a number of tasks (6 in the example) to a number of
workers (2 in the example), satisfying a number of order
constraints. A solution to the example in figure 1 is to assign
Problem
There are 2 workers with
each 6 hours
Task A takes 1 hour
Task B takes 1 hours
Task C takes 2 hours
Task D takes 2 hours
Task E takes 3 hours
Task F takes 3 hours
The schedule has to satisfy
the following constraints:
Task C must be before A
Task E must be before B
Task F must be before B
Task D must be before C
Readq
Clear
Correct!
Figure 1: Example of a scheduling experiment