A Cognitive Matrix of Continua

A Cognitive Matrix of Continua
Information we deal with can be placed on a learning continuum extending
from new material for which there are very limited schemas available to
well-learned material with its elements incorporated into an extensive
schematic framework. The first row of Fig. 1 indicates the two extremes of
this learning continuum.
The second row is concerned with schemas. While the characteristics and
functions of schemas were discussed previously, they have one additional
function that is less commonly considered: Schemas held in long-term
memory provide working memory with a central executive. Furthermore,
they may be the only structure available to provide a central executive for
working memory. The second row of Fig. 2 indicates the two extremes of the
schema-based, central executive function continuum.
A schema, acting as a central executive, coordinates information. It
indicates which information can be ignored, which information is
significant, and how the elements of significant information relate to each
other. A well-established, automated schema acts exactly as we would
expect an eVective central executive to act. Both incoming information and
the responses to that information can be governed and coordinated by
schemas. Provided schemas are available, no other central executive
function is required for humans to process information. Of course, schemas
must be learned and activated and so are not always available.
Evidence for the central executive function of schemas comes from one
of the conditions under which problem solving fails. If a problem solver
learns to solve a class of problems using a particular technique, he or she
will continue to attempt to use the technique even when presented a
problem with a similar surface structure for which it is inappropriate. This
mental set, or Einstellung, was demonstrated by Luchins (1942) using his
well-known water jar problems (see also Ben-Zeev & Star, 2001; Fingerman
Evolution of human cognitive architecture 227& Levine, 1974; Levine, 1971; Ross & Kilbane, 1997; Sweller, 1980a,b;
Sweller & Gee, 1978.) The eVect occurs because a schema is acquired
when learning to solve an initial set of structurally similar problems. That
schema then directs the solution of all subsequent similar problems in
exactly the manner to be expected of a central executive. On the one hand, it
permits the solution of problems that would be quite insoluble without an
appropriate schema. On the other hand, it continues to organize the
elements and solution procedures of other, structurally dissimilar problems
that have similar surface features even when the solution procedures are
quite inappropriate. As a consequence, the solution will either be delayed or
fail entirely. In contrast, a person presented such a target problem without
first having acquired the inappropriate schemas will have no diYculty
solving it. The frequently spectacular contrast between the performance of
people with and without inappropriate problem solving schemas demon-
strates Einstellung. In the process, the central executive function of schemas
is revealed graphically.
While schemas held in long-term memory provide a central executive for
working memory at the well-learned end of the learning continuum, it can
be argued that there is no available central executive at the other end of the
continuum when dealing with new, yet-to-be-learned material. Two
arguments can be put forward against the notion of a coordinating central
executive when dealing with new, yet-to-be-learned information for which
no schema is available. The weaker argument simply states that the
characteristics of a central executive have not been suYciently well specified
to be assured that it exists and, in any case, there is no real empirical
evidence for any possible central executive-type construct. This argument is
not pursued further because it is overridden by the stronger argument,
which is that the very concept of a central executive dealing with yet-to-be-
learned material in a nonrandom manner leads to an infinite regress and so
is logically impossible.
Consider a central executive coordinating new information in a
nonrandom manner. The executive must make decisions on how infor-
mation is to be dealt with in that it must decide which elements will be
combined, coordinated, or related in some fashion. In other words, it must
decide on how the information will be processed. That information is both
new and infinite in scope. It is new in the sense that the executive has not
dealt with such information before and it is infinite in that there is no limit
on the types of information or how that information will have to be
combined or processed. Other than randomly, how does the central
executive decide how to deal with this potentially infinite range of new
information? It cannot draw on previous knowledge because the material is
new. It could use biologically programmed or ‘‘hardwired’’ procedures for a
228 John Swellerlimited number of activities but not for the infinite range of information that
humans can potentially deal with. (It will be assumed that we are not
hardwired to deal with each of the procedures of complex mathematics, for
example.) If these assumptions are correct, there is only one other way a
nonrandom central executive can arrive at a decision. If the information is
to be dealt with in an orderly fashion, it must have another executive
function available to direct it. However, the logic of a second executive will,
of course, be identical to the logic of the first, requiring a third executive, etc.
This infinite regress indicates that the entire concept is flawed and requires
replacing. Mechanisms other than a schema-based central executive are
required to coordinate new, unlearned information.
If there is no central executive available to coordinate new, yet-to-be-
learned elements, how are these elements dealt with? Research into problem
solving provides an answer and also provides the third row, the problem-
solving search continuum of the matrix of continua. Problem solving search
is required precisely when we are faced with new information for which we
have yet to acquire appropriate schemas. Critical research in the early 1980s
on expert–novice distinctions (e.g., Chi et al., 1982; Larkin et al., 1980)
clearly established that when faced with a novel problem for which a learned
solution is not available (i.e., a problem being dealt with by a novice with
respect to that class of problems), people engage in problem-solving search
using a weak strategy such as means-ends analysis (Newell & Simon, 1972).
Using this strategy, problem-solving moves are generated by attempting to
find operators that will reduce diVerences between each problem state
attained and the goal or a subgoal. In other words, faced with a novel
situation, people use general problem-solving search strategies in an attempt
to impose some order and choose between various element combinations.
The purpose of those search strategies is to attempt to coordinate yet-to-be-
learned elements with the external environment. This process of matching is
only required when faced with new material for which adequate schemas
have yet to be acquired. With respect to the cognitive matrix of continua of
Fig. 1, problem-solving search flows directly from the left side of the first
two rows of the matrix of continua. That is, it occurs because a person is
dealing with new, unlearned material for which there is no schema-based
central executive.
At the well-learned end of the continuum, problem-solving search is
unnecessary. On the right side of the matrix, when dealing with well-learned
material for which well-established schemas are available, the schemas
themselves generate problem-solving moves (Larkin et al., 1980). Problem-
solving search to coordinate and establish relations between elements is
unnecessary because schemas provide all of the necessary relations. In
between the two extremes of the third row of the matrix, search becomes less
Evolution of human cognitive architecture 229and less important, moving from the point where moves are generated by
problem-solving search to the point where they are generated by schemas.
Thus, the third continuum, the problem-solving continuum, has been
established and related to the learning and central executive continua.
The first three continua lead to the critical fourth continuum that provides
a direct explanation for working memory characteristics when dealing with
both new and well-learned material. On the left side of the matrix, operators
and problem states must be chosen during problem-solving search in the
absence of schemas and their executive function. A major function of
problem-solving search is to impose a degree of order on otherwise disordered,
more or less random, combinations of elements. This order is imposed by
attempting, as far as possible, to use the environment to provide appropriate
relations between elements. Random combinations of elements are held in
working memory, and attempts are made by problem-solving search to order
them in a manner that reflects the environment. Once an appropriate set of
relations has been established, the goal of the problem has been attained.
It is frequently forgotten that by necessity, problem-solving search
conducted without solution knowledge of moves or element combinations
must include a random component. Consider means-ends analysis as an
example of a strategy that does not rely on a heavy knowledge base. This
strategy requires considerable control and has a relatively small random
component. Nevertheless, a random component cannot be fully eliminated.
The strategy involves first choosing a move and then testing it to see whether
it reduces diVerences between a current problem state and the goal or a
subgoal state. Checking whether a move reduces diVerences between the
current problem state and the goal state cannot occur before the move has
been chosen. It must occur after the move has been chosen. If there is no
prior knowledge concerning the eVect of the move (in the form of schemas
or partial schemas), it must be chosen randomly. Only after it has been
chosen can it be assessed for eVectiveness. There is a high degree of control
in that diVerences between current and goal states are extracted before
moves are chosen and moves that do not reduce diVerences between the
current and goal states are rejected. Nevertheless, in the absence of prior
knowledge, which moves are chosen for testing using the means-ends
heuristic must be random. In the absence of a central executive, there is no
other technique available. Other than a random mechanism, there can be no
knowledge-free procedure for initially choosing moves to test to see if they
reduce diVerences between current and goal states. As a consequence, on the
left extreme of the element combinations continuum, random combinations
of elements are necessarily the norm.
With random choice, the greater the number of alternative subgoals and
operators from which to choose while problem solving, the less likelihood
230 John Swellerthat an appropriate choice will be made. As the number of choices available
increases, the probability of a choice leading to a dead end also increases.
With increased choice, problem-solving search becomes decreasingly
eVective and, indeed, with even a moderately large number of choices,
search becomes pointless. Making an appropriate choice out of two or three
at each choice point is feasible. Choosing out of several dozen or more
alternatives at each choice point would render the process futile. Problem-
solving search is more likely to be eVective if it can be limited, and our
cognitive architecture had to evolve to ensure that it is always limited
because anything beyond a small search space reduces the probability of
arriving at a solution to almost zero.
With increasing knowledge, the random choice of elements decreases. At
the right extreme of the element combinations continuum, well-learned
material has schemas to coordinate elements, and problem-solving search is
unnecessary with all element combinations ordered by previously acquired
schemas. It is only after learning has occurred that problem-solving search is
not needed to order elements because they are ordered by schemas.
We are now in a position to consider the last continuum, the working
memory limitations continuum (the fifth row of the cognitive matrix of
continua), and to indicate why working memory must be limited when
dealing with new, yet-to-be-learned material. The need for a random
component when combining elements through problem-solving search
leads directly to a requirement for working memory to have a severely
limited capacity. Consider someone dealing with two new elements. While
the manner in which elements should be combined will vary depending on
the material being dealt with, assume that they must be combined using
the logic of permutations. There are two (2!) unique ordered permutations
possible for two elements (ab or ba). As the number of elements increases,
the number of permutations rapidly becomes very large (5! ¼ 120). The
way in which these elements should be combined can be handled easily by
a system with a schema-based central executive determining the
appropriate combination, as occurs on the right side of the matrix of
continua, dealing with well-learned material. Without a central executive,
on the left side of the matrix dealing with new material requiring problem-
solving search and its attendant need to combine elements randomly, no
more than two or three elements can be handled because any more
elements would result in more potential combinations than could be tested
realistically.
It may be for this reason that we have evolved with a limited working
memory. When dealing with new, interacting elements that have not been
learned (i.e., have not been formed into schemas), there is no structure that
can indicate the manner in which the elements should be combined and so
Evolution of human cognitive architecture 231the need to combine any more than two or three elements can result in a
huge number of possible combinations that could not be tested properly
against reality. A limited working memory ensures that combining a large
number of elements in the absence of a controlling schema cannot occur.
Such combinations of many elements would rarely reflect reality. The
proposal that working memory is limited in order to limit the number of
element combinations that require testing constitutes a central core of this
chapter.
The suggestion that a limited working memory may have advantages
when processing information under some conditions has been made
previously. Both Dirlam (1972) and MacGregor (1987) provided a formal
analysis indicating that search is most eYcient when the number of items
being dealt with closely approximates the number of items that can be held
in working memory. Elman (1993) and Newport (1990) suggested that by
constraining the search space for grammatical forms, a limited working
memory is an advantage when learning a language. Kareev (1995) indicated
that when dealing with correlations, a smaller sampling size increases the
probability of the sample having a correlation stronger than the population.
Thus, if a relation exists, a limited capacity working memory could have the
eVect of increasing the probability of its being detected. Kareev, Lieberman,
and Lev (1997) provided data indicating that people with smaller working
memories were more likely to perceive a correlation than people with larger
working memories. Taken together, these suggestions all indicate that there
may be advantages to a limited working memory when dealing with new
material, and the commonsense view that a larger working memory should
be advantageous may be erroneous.
In summary, the manner in which our cognitive architecture interacts
with information can be represented by a matrix that incorporates five
parallel continua: (1) a yet-to-be-learned to well-learned continuum in which
the extent that individuals have learned the material (i.e. acquired schemas)
that they are faced with increases; (2) an uncontrolled to schematically
controlled central executive function continuum in which the degree to
which schemas control working memory processing increases; (3) a problem-
solving search continuum in which the need to solve problems by
problem-solving search varies from essential to unnecessary; (4) a random
to ordered combination of elements continuum in which the manner in
which elements combine varies from random to ordered; and (5) a working
memory limitations continuum with working memory limitations critical at
one end and irrelevant at the other.
These five continua are linked causally providing a matrix. On the left side
of the matrix, new material that is still to be learned has no central executive
coordinating high interactivity elements. Some degree of coordination only
232 John Swellercan be provided by problem-solving search that incorporates testing the
eVectiveness of random combinations of elements. When dealing with these
element combinations, a limited capacity working memory is essential to
prevent a combinatorial explosion. In contrast, on the right side of the
matrix, well-learned material has schemas providing a central executive
function. Problem-solving search is not required because schemas provide
ordered combinations of elements. Interacting elements are incorporated
within schemas, resulting in no eVective working memory limits when
dealing with such well-learned material. Examples demonstrating the
relations incorporated in the matrix are discussed in detail in the next two
sections