Problem recognition, definition, and
representation are metalevel executive processes, called metacomponents in
Sternberg’s (1985) triarchic
theory of human intelligence. This theory proposes that metacomponents guide
problem solving by planning, monitoring, and evaluating the problem-solving
process. The metacomponents include such processes as (1)
recognizing the existence of a problem, (2)
defining the nature of the problem, (3)
allocating mental and physical resources to solving the problem, (4)
deciding how to represent information about the problem, (5)
generating the set of steps needed to solve the problem, (6)
combining these steps into a workable strategy for problem solution, (7)
monitoring the problem-solving process while it is ongoing, and (8)
evaluating the solution to the problem after problem solving is completed. In
this theoretical context, the processes of problem recognition, definition, and
representation correspond to the first, second, and fourth metacomponents,
which are used in the planning phase of problem solving.
Problem recognition, also referred to as
problem finding, is one of the earliest stages of problem solving. Getzels (1982)
classified problems based on how they were “found.” According to Getzels, there
are three kinds of problems: those that are presented, those that are
discovered, and those that are created. A presented problem is one that is
given to the solver directly. In this case, there is no need to recognize or
find the problem; it is stated clearly and awaits solution. A discovered
problem, however, is one that must be recognized. Such a problem already
exists, but it has not been clearly stated to the problem solver. In this case,
the problem solver must put together the pieces of the puzzle that currently
exist and seek out a gap in current understanding in order to “discover” what
the problem is. In contrast to presented and discovered problems, the third
class of problems comprises those that are created.
Created problems are those in which the
problem solver invents a problem that does not already exist in the field. For this reason, one
can argue that a created problem will, in some sense, always produce a creative
solution, simply because its problem statement deviated from the usual way of
thinking about the problem. Getzels and Csikszentmihalyi (1976)
found that artists who spent more time in the problem-finding stage while
creating an artwork were judged to have more creative products than did artists
who spent less time in problem finding. In fact, the artists who spent more
time also remained highly creative seven years later. For the
purposes of this chapter, problem recognition refers to both discovered and
created problems.
Problem definition is the aspect of
problem solving in which the scope and goals of the problem are clearly stated.
For example, a presented problem may be easy to define if the problem statement
has been prepared for the solver. However, some presented problems are not clearly
stated, requiring the problem solver to clarify the precise definition of the
problem. Discovered problems usually require definition because the problem
solver has identified the problem in his or her field. Defining a created
problem is likely to be a challenge, given that the problem solver has gone
beyond the current field in inventing the need for a solution in the first
place. Problem representation refers to the manner in which the information
known about a problem is mentally organized. Mental representations are
composed of four parts: a description of the initial state of the problem, a
description of the goal state, a set of allowable operators, and a set of
constraints.
By holding this information in memory in
the form of a mental representation, the problem solver is able to remember
more of the problem by chunking the information, in order to organize the
conditions and rules of a problem to determine which
strategies are useful, and to assess progress toward the goal state (Ellis
& Siegler, 1994; Kotovsky,
Hayes, & Simon, 1985;
Newell&Simon, 1972).Aproblemmaybe
represented in a variety of ways, for example, verbally or visually. Even a
presented problem may require the generation of a new representation in order
to be solved. For example, given the problem of finding your way to a new
location, you may find it much easier to follow a map than to read a set of
directions. If you have trouble following the map, then
it may be worthwhile to write out a description of the route in words, re-representing
the information in a way that makes it easier to get to your destination. It is
important to note that these three aspects of problem solving are not discrete,
sequential stages in the solution process, but rather are interactive and often
difficult to tease apart in a real problem-solving situation. When a problem is
represented in a new way, the problem solver may decide to redefine the goal
accordingly. Similarly, a redefinition may lead to a new representation. It is
useful to consider the roles of problem recognition, definition, and
representation in the solution of well-defined versus ill-defined problems.
Recall that a well-defined problem is one
whose path to solution is straightforward, whereas an ill-defined problem is
one that does not lend itself to a readily apparent solution strategy. Consider
the following well-defined problem, referred to as the Tower of Hanoi problem:
There are three discs of unequal sizes,
positioned on the leftmost of three pegs, such that the largest disc is at the
bottom, the middle-sized disc is in the middle, and the smallest disc is on the
top. Your task is to transfer all three discs to the
rightmost peg, using the middle peg as a stationing area, as needed. You may
move only one disc at a time, and you may never move a larger disc on top of a
smaller disc. (Sternberg,
1999)
The problem here is easy to recognize: One
needs to move the discs onto the rightmost peg. The problem is also defined
clearly; the relative sizes of the discs as well as their locations are easy to
distinguish. Also, the solution path is straightforward based on this
representation. Working backward, one realizes that the largest disc must be
placed onto the rightmost peg, and in order to do so, the other two discs must
be removed. So that the mediumsized disc does not end up on the rightmost peg,
the smallest disc must first be moved to the far right. Then the medium disc is
placed on the middle peg; the small disc is placed on top of the medium disc.
The large disc is then free to be placed on the rightmost peg. Finally, the
small disc is moved to the left so that the medium disc is free to move to the
rightmost peg. The last step is then to move the small disc atop the other two
and the problem is solved. Note that this well-defined problem can be expanded
to include many pegs and many discs of varying sizes, but its solution will
always proceed according to the algorithm described in this, the simplest case.
For the most part, well-defined problems
are relatively easy to recognize, define, and represent. However, a
well-defined problem may entail some degree of “problem finding,” in the sense
that a problem exists but must first be discovered. For example, a scientist
may struggle to identify a gap in the existing literature on a problem, but the
actual process of filling that gap may come easily once the problem itself has
been identified.
The solution to the discovered problem may
follow a path similar to that of other problems in the field (e.g.,
experimental methods). For example, much early psychological research was
conducted using male participants. When a researcher questioned the validity of
the results for females, a new problem had been discovered. Given this new
problem, the path to solution was well defined: Simply use the same
experimental method but include female participants in the study. In this
sense, this well-defined problem was somewhat difficult to recognize, yet once
identified, it was easily defined and represented in familiar terms. The
representation of well-defined problems is not necessarily easy, however.
Consider another problem: Three five-handed extraterrestrial monsters were
holding three crystal globes. Because of the quantum-mechanical peculiarities
of their neighborhood, both monsters and globes come in exactly three sizes,
with no others permitted: small, medium, and large. The small monster was
holding the large globe; the medium-sized monster was holding the small globe;
and the large monster was holding the medium-sized globe. Since this situation
offended their keenly developed sense of symmetry, they proceeded to transfer
globes from one monster to another so that each monster would have a globe
proportionate to its own size. Monster etiquette complicated the solution of
the problem since it requires that: 1.
only one globe may be transferred at a time; 2.
if a monster is holding two globes, only the larger of the two may be
transferred; and, 3. a globe may
not be transferred to a monster who is holding a larger globe. By what
sequence of transfers could the monsters
have solved this problem? (See Kotovsky et al., 1985)
Hanoi problem (Newell & Simon, 1972).
However, it is actually directly isomorphic to (i.e., its structure is exactly
the same as that of) the Tower of Hanoi problem. In this case, it is the
difficulty of representing the problem correctly that increases the level of
difficulty of the problem as a whole. After you are told of the isomorphism
between the two problems, the solution is simply a matter of mapping
relationships from one problem to the other. In summary, problem definition is
usually easy for the class of well-defined problems; however, accurate problem
recognition and representation are not necessarily straightforward, even when
the scope and goals of the problem are clear. In the case of ill-defined
problems, however, it is often the case that all aspects of problem formulation
are relatively challenging. Perhaps the easiest stage in attempting to solve an
ill-defined problem is that of problem recognition. It is often relatively
simple to identify a fuzzy problem. For example, it is easy to identify the
problem of developing a test of creativity. It is hard, however, to define the
exact contents of such a measure. The real difficulty in solving an ill-defined
problem is in clarifying the nature of the problem:howbroad it is, what the
goal is, and so on. Although well-defined problems have a clear path to
solution, the solution strategy for an ill-defined problem must be determined
by the problem solver. To develop a problem-solving strategy, it is first
necessary to specify the goals of the task. For example, if we take on the task
of designing a creativity test, we must decide whether the goal is (a) to
estimate the creativity of undergraduate psychology majors or (b) to measure
creative potential among people of all ages and educational and cultural
backgrounds. Before the path to solution can be constructed, the goal must be
clear.
