A Rational Analysis of Alternating Search and Reflection Strategies in Problem Solving



DEA to the first worker, and FCB to the second. An experi-
ential strategy for this problem is to take one of the con-
straints, for example C must be before A, and assign them to
a worker directly following each other, so for example assign
CA to the first worker. A reflective strategy involves some
inference. For example, from the facts that both E and F have
to be done before B, that E and F each take 3 hours, and that
each worker has only 6 hours, it can be deduced that E and F
cannot be assigned to the same worker. Furthermore, since
each worker has 6 hours, either each worker gets a task of 3,
2 and 1 hours, or one worker gets both 3 hour tasks and the
other the rest. Since the latter option has already been ruled
out by the fact that E and F can’t be assigned to the same
worker, the former has to be correct. When the subject has
made these inferences a few times for different problems,
they can become part of the experiential strategy.

Protocol analysis of subjects solving these problems show
that all subjects start with an experiential strategy, and only
later on switch to a reflective strategy. So in a sense this
reflects the explore-impasse-insight-execute pattern
described in the literature about insight. Some, but not all of
the subjects show some sort of impasse, during which they
stop searching, just stare at the screen for a minute, and then
try a new approach. Furthermore, there is no difference
between the explore and the execute stage: the subject just
searches on, using the knowledge gained by reflection.
Sometimes further reflection is needed to reach a solution.

In this paper two models are presented that explore the dis-
tinction between search and reflection. Both models are
based on Anderson’s theory of rational analysis (Anderson,
1990). According to rational analysis, subjects choose strate-
gies based on a cost-benefit analysis: the strategy that has the
lowest expected cost and the highest chance of success is
selected in favor of others. The first model is a dynamic
growth model, in which the trade-off between search and
reflection is modeled in a course-grained way. Dynamic
models are used in developmental psychology to describe
developmental paths, for instance a model that describes
stage-wise increases in knowledge (Van Geert, 1994). The
second model is an ACT-R model, in which the competition
between individual strategies is modeled on a number of
concrete tasks.

A dynamic growth model

Why would subjects initially prefer the experiential strategy
in the scheduling problem? The reflective strategy seems to
be much more powerful. There are several reasons for this. A
first reason is that reflective reasoning takes more effort. To
be successful, several aspects of the task must be combined
and kept in memory. Additional knowledge must be retrieved
from memory and it may be necessary to seek analogies with
other problems. A second reason is that it is not immediately
evident to the subject that the experiential strategy will be
unsuccessful. The problems in the experiment were chosen
so that the experiential strategy alone wouldn’t work, but
subjects didn’t know this. So why not try the strategy which
takes the least effort first? A third reason is, that as a subject
starts with a new type of problem, he has only read instruc-
tions and has maybe seen an example problem. So he first
has to learn the basic rules and operators by experience,
before he can attempt any higher level strategies.

The model

According to rational analysis strategies are chosen with
respect to their expected outcome, according to the following
equation:

Expected outcome of strategys = PsG - Cs

In this equation, P is the estimated chance of reaching the
goal using this strategy, G is the expected value of the goal,
and C is the estimated cost of reaching the goal using this
strategy.

The model will attempt to predict how search and reflec-
tion will alternate while solving a problem. This model is
quite course-grained in the sense that the knowledge of the
system with respect to a certain task is summarized in two
variables
L1 and L2 . L1 is a measure for the amount of
basic task-knowledge in the model, for example in the case
of the scheduling task an operator to add a task to an existing
plan and knowledge to judge whether a solution is correct.
L2 corresponds to the amount of higher-level knowledge in
the system, for example the fact that it is a good idea to look
how the tasks add up to the amount of time the workers have
available. If a subjects starts with a new problem, we assume
that both variables have a small value. They can however
increase, because the subject builds up knowledge while
problem solving. The assumption of the model will be, that
search will increase the amount of basic knowledge, repre-
sented by
L1 , and reflection will increase the amount of
higher-level knowledge, represented by
L2 . The following
equations show how
L1 and L2 grow in time, and are based
on the equation used by Van Geert (1994):

If the strategy in step i-1 is search:

L1(i-1)

L ι( i) = L ι( i-1) + R ι 1--L-------I

L1 max

else L1 keeps its value, so L1(i) = L1(i- 1). R1 is a con-
stant that controls the rate of growth, and
L1 max is the maxi-
mum possible value for
L1 . The fraction at the end of the
equation ensures that
L1 doesn’t exceed its maximum value.

The equation for L2 is slightly more complicated, because
the increase in value depends on the current value of
L1 : we
can only gain higher-level knowledge if we have enough
basic knowledge.

If the strategy at step i-1 is reflection:

L2(i-1)

L2(i) = L2(i-1) + S12 L 1(i-1 )l 1--L-------I

L2max

else L2(i) = L2(i - 1). Again, L2max is the maximum possi-
ble value for
L2 . The constant S12 (support) controls the
influence of basic knowledge on the increase of higher level
knowledge.

Whether the strategy at step i will be search or reflection is
determined by their respective expected outcomes:

Expected outcome of search = Psearch(i) ■ G - Csearch



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