Mathematics Education Research Journal, VolA, No.1, 1992.
SOME COGNITIVE FAcTORS RELEVANT TO
MATHEMATICS INSTRUCTION
John Sweller and Renae Low, University of NSW
Our understanding of cognitive processes has progressed
sujficientLy in the last few years to enable us to generate novel
instructional techniques that can enhance substantially learning of
subjects such as mathematics. This paper will review briefly
some research intended to contribute to this process. There are
two relevant aspects. Firstly, recent work has thrown light on
schema acquisition while learning mathematics, and on
techniques for detecting schemas in mathematics learners.
Secondly, other research has assessed the distribution of
cognitive resources while learning mathemarics and other related
subjects leading to the design of instructional techniques to
facilitate schema acquisition.
The Relevance of Schemas to Mathematical Expertise
Cognitive research findings have revealed the importance of schema
acquisition in successful learning and problem solving in mathematics (see
Sweller, 1988). A problem schema can be defined as a cognitive construct
that allows problem solvers to recognise a problem as belonging to a specific
category that requires particular moves for solution. This definition includes
problem states, problem-solving operators, and their relations.
Consider a problem such as, "If a boat travelled downstream in 120
minutes with a current of 5 kilometres per hour and the return trip against the
same current took 3 hours, what was the speed of the boat in still water?"
Irrespective of the person's skill in computation and their basic linguistic and
factual knowledge (e.g., what "still water" means, that rivers have currents
that run only downstream), this problem is beyond the capability of someone
who does not know the specific relations between speed of boat, rate of
current, and time involved in river current contrast problems. In the absence
of such schematic knowledge, there is no context for computation and
calculation. The person needs to be able to identify the problem as belonging
to a given type (e.g., river current contrast) in order to determine what
information from the text should be used, in what sequence, and through
what operations (see Mayer, 1987, pp. 345-373).
84 Sweller and Low
Novices and experts have been shown to differ not only in solution
rates, but in schematic knowledge. When asked to remember problem states,
experts who have acquired appropriate schemas can use them to encode
information with reference to problem structure. A novice, not possessing
such schemas, will find problem configurations more difficult to reproduce
(Chase & Simon, 1973). Similarly, because a schema classifies problems
according to solution steps, it is likely to be used as a mode for classification.
Experts can categorise problems as similar or different on the basis of
underlying principles, but novices are readily deceived by context (Chi,
Glaser & Rees, 1982).
Definitions of a Problem
Readers familiar with the area will recognise that we are using a
particular conceptualisation of a problem. This conceptualisation
distinguishes between experts and novices but not problems and exercises. It
is assumed that we cannot define a particular task as constituting a "real"
problem in isolation, without reference to the person who is attempting to
solve it. A problem for one person may be an exercise for another and only
problems unsolved by anyone (such as unresolved research questions) could
universally be classified as "real" problems for everyone. Once a solution to
such problems is discovered and learned by sections of the population, these
people can be classed as experts with respect to the relevant problems while
the rest of us are novices. From this discussion it can be seen that by
distinguishing between experts and novices we have defined all tasks as
problems. We know how to solve some problems and so are experts; other
problems we do not know how to solve and so are novices. No task can be
classed as either an exercise or problem simply by referring to its structure
and components. Alone, the structUre of a task cannot reveal its "problem"
status. Its status is revealed fully by the novice-expert distinction.
From an educational perspective, obviously we should be concerned
with helping novices become experts and this paper is concerned primarily
with that issue. We know also that some people are much better than others at
solving novel problems and it is equally legitimate to conduct research on
techniques for improving people's ability to solve such problems.
Nevertheless, we do not believe that these techniques have been discovered as
yet. They may not exist. We do believe that techniques for facilitating the
transition from novice to expert in domain-specific areas are available and
some of these procedures are discussed below.
Measures of Schematic Knowledge
Schematic knowledge has been studied mainly through three kinds of
tasks: problem classification, problem recall, and problem construction.
Hinsley, Hayes and Simon (1977) assessed schematic knowledge by
requiring relatively experienced students to group algebraic story problems
Cognitive Factors 85
into clusters, categorise a given problem after hearing only part of the text,
solve problems in which content words were replaced by nonsense words,
and solve problems when irrelevant information in the text had produced
ambiguity. They found extensive agreement between subjects in the
categorisation of problems. Further, after hearing the words "A river
steamer. .. ", many subjects judged that the problem was going to be one
involving the river current type. From a number of converging
demonstrations, Hinsley et a1. (1977) concluded that the encoding and
retrieval of information in the process of solving algebraic word problems is
governed by a person's schematic knowledge.
In further research emphasising the role of problem schemas, Mayer
(1982) required adults to recall problems to which they had been exposed
earlier, and to construct problems within a theme (for example, "trains leaving
stations"). Recall was best for the types of problems that occur frequently in
textbooks, and schema-relevant material was more likely to be recalled
accurately than schema-irrelevant information. In addition, the problems
constructed by students tended to approximate standard rather than nonstandard formats. These results, Mayer argued, suggest that while students
possess schematiC knowledge for simple versions of a problem category, their
schematic knowledge for the more complex versions are less accessible.
Text Editing as a Measure of Schematic Knowledge
Schematic knowledge also can be assessed by requiring students to edit
problem texts to specify what information is necessary and sufficient for
solution. It has been claimed that schemas provide the basis for filJing gaps
and for selecting relevant information (Chi & Glaser, 1985). To be able to
supply information that is missing from a problem text but is necessary for the
problem to be soluble, or to tease out irrelevant information contained in the
problem text, one has to know the minimal elements that have to be present in
the text in order for the problem to be open to solution. Since the content of a
problem schema includes principles, formulae, concepts, and characteristics
of the problem situation (De long & Ferguson-Hessler, 1986), it can be
assumed that successful text editing of algebraic word problems requires
schematic knowledge. To illustrate, when presented with a problem that is
incomplete in the sense that further information is needed before a solution is
possible (such as "A rectangular lawn is 72 square metres. What is the width
of the lawn?"), the student needs to know that the area of a rectangle is the
length of one side multiplied by the length of the adjacent side in order to
identify the essential component missing from the problem. Similarly, when
presented with a problem such as "A truck leaves Melbourne for Sydney at
1.00 a.m. The distance between Melbourne and Sydney is 1200 kIn. If the
truck travels at 80 km/h, how long will it take to reach Sydney?" a student is
able to specify which information in the text is irrelevant for solution only
through knowing about the structure of this class of problem. If solution
requires schematic knowledge, a student who can solve a given problem
should be able to identify the minimal text that will allow the problem to be
86 Sweller and Low
solved.
In a series of three studies in which tenth grade students were required
to classify algebraic word problems in terms of whether a problem contained
sufficient, insufficient, or irrelevant information for solution, relationships
were demonstrated between text editing and solution, as well as between text
editing and several indices of schematic knowledge (Low & Over, 1990).
Text editing performance correlated with judgments of similarities and
differences between problems and with memory for problems. Further, text
editing as a skill was predictive of solution rate, and of mathematical
achievement in general. Students with higher scores on tests of general
mathematical ability were more accurate in cClassifying algebraic word
problems in terms of whether they contained sufficient, insufficient, or
irrelevant information. This relationship applied even after allowance was
made for contributions from verbal skill.
As noted earlier, measures that have been used in past research to index
schematic knowledge include identification of similarities and differences
between problems and recall for problems that were previously presented.
Low and Over (1990) have shown that scores on the text editing task
correlated not only with the probability that problems were solved, but with
recall memory for problems and discrimination of whether problems are
similar or different. These relationships are to be expected if text editing, in
common with problem solution, problem classification, and recall memory,
assesses the extent to which a student has processed information based on
understanding of the structure of the problem.
Hierarchical Ordering of Schematic Knowledge
Low and Over (1990) noted that the nature of errors that students make
in text editing suggest that individuals who lack knowledge of the higherorder template appropriate for the problem in hand will conceptualise the
problem as one entailing a lower-order template. Consider, for example, the
problem, "The length of a rectangular window pane is twice its width. If the
area of the pane is 98 cm2 , what are the dimensions of the pane?" In order to
classify this problem as one containing information that is necessary and
sufficient for solution, a student needs knowledge of what Mayer (1981)
termed the "area relative" template. Most students who classified this
problem as providing insufficient information indicated that either the width or
the length of the pane must be provided in the text in order for solution to be
possible. Since these students specified the length of one side as the
information that was missing, they were clearly employing the "simple area"
template.
Using four templates for area-of-rectangle (adjacent sides, ratio of sides
and one side, perimeter and side, and diagonal and side) in three different task
formats (text editing, knowledge of formula, problem solution) Low and Over
(l992b) demonstrated hierarchical organization of the schematic knowledge
underlying the solution of the four problems. Solution, text editing, and
. knowledge of formula task were most accurate for the adjacent sides
Cognitive Factors 87
templates, followed by the ratio and side template, the perimeter and side
template, and then the diagonal and side template. For all three tasks, students
rarely were successful on a problem requiring a template above one on which
they had failed. Performance on problems involving a lower-order template
could thus be predicted by performance involving a higher-order template.
Although the basis for the hierarchical ordering of templates found in
the Low and Over (l992b) study is unclear, this investigation, together with
Mayer's (1981) taxonomy offer a starting point for exploring the manner in
which schematic knowledge is organised and integrated. The methodology
used by Low and Over (1992b) can be used to establish relations between
families, categories, and templates, particularly whether there is hierarchical
ordering of templates within families or categories. Such work may have
important implications for classroom instruction. For example, if the
acquisition of competence in the use of one template proves to be dependent
on the mastery of another, instruction should be programmed to ensure that
students are not exposed to superordinate templates until they have
demonstrated understanding of subordinate templates.
Gender Differences in Schematic Knowledge
Following Marshall and Smith (1987) who characterised gender
differences in mathematical performance in terms of differences in information
processing, Low and Over (1992a) compared boys and girls in terms of
schematic knowledge. In one experiment, boys and girls in tenth grade
classified algebraic word problems in terms of whether the problems
contained sufficient, irrelevant, or insufficient infonnation for solution.
Among students with similar levels of general mathematical ability, girls were
less likely than boys to identify missing information or irrelevant information
within problems. More girls than boys perceived irrelevant information within
the text of a problem as being necessary for solution. It was found in a
subsequent experiment that girls who were as able as boys to solve algebraic
word problems containing sufficient information had lower solution rates than
boys on problems containing irrelevant infonnation. On these latter problems
the girls more often than the boys incorporated the irrelevant information into
their attempted solution. Low and Over (1992a) suggested that the results
point to differences between boys and girls in their schematic knowledge.
Marshall and Smith (1987) suggested that traditional instructional
procedures initially produce, and later maintain gender differences in the
cognitive processes that are involved in learning mathematics. There may be
scope for modification of educational practices so that neither boys nor girls
are disadvantaged in acquiring skills that are basic to competent performance
in mathematics. The results obtained by Low and Over (1992a) point to the
need for instruction designed to ensure that all students (giris as well as boys)
acquire schemas pertinent to the solution of mathematical problems, and that
students adopt a schema-oriented approach to problem solution.
88 Sweller and Low
Cognitive Load Theory and Techniques to Design Instruction to
Facilitate Schema Acquisition
If schema acquisition is a major component of mathematical problem
solving expertise, clearly it is important to determine ways of facilitating the
acquisition of schemas. Cognitive load theory (Sweller, 1988; 1989; see also
Ayres & Sweller, 1990) has been developed for this purpose. The theory
assumes that instructional designs must be structured in a manner that focuses
attentional resources on problem states and their associated moves because it
is familiarity with problem states and their associated moves that provides the
basis of schemas. If instructional designs require students to devote cognitive
resources to other aspects of a problem or other features of instruction, a
heavy, extraneous cognitive load may be imposed that interferes with learning
and problem solving. As it happens, many conventional instructional
procedures are inappropriate from the perspective of schema theory and
cognitive load theory. The remainder of this paper will be concerned with
some ofthese difficulties and techniques for overcoming them.
Using Problem Solving as a Learning Technique
The solution of many problems or exercises as a means of gaining
experience in a particular area probably is a universal procedure. Most
teachers have their students solve problems, most texts have a series of
problems to solve at the end of each section. The strategy most of us use to
solve novel problems for which we do not have a schema is called meansends analysis. This strategy requires us to attempt to reduce differences
between each problem state we encounter and the goal state. If we are
presented with a geometry problem for which we must find a value for Angle
X (as in Figure 1), the diagram provides the initial problem state, Angle X is
the goal, and the theorems of geometry are the problem operators that we
must use to transform the initial state into the goal state. Thus, using meansends analysis, we may work backward from the goal angle attempting to find
theorems that will allow us to connect the goal angle to the givens of the
diagram. Using means-ends search we can establish that since we have a
value for Angle CDE, if we had a value for Angle DEC we could find X using
the theorem, external angles of a triangle equal the sum of the opposite
internal angles. Angle DEC becomes a subgoal. We can find a value for
Angle DEC from Angle FEG using the theorem, vertically opposite angles are
equal. Once we have succeeded, in constructing this chain we can then
reverse direction and substitute values until we have a value for the goal, thus
solving the problem. Searching for relevant angles and theorems is the major
intellectual process of.the task.
This means-ends strategy is an excellent technique for attaining a
problem goal but except in tests, we do not present students with problems to
solve in order that they will attain the goal. We give students problems so that
they will learn and it is learning that is the real goal of problem solving in
most educational ·contexts. From the perspective provided above, we know
Cognitive Factors 89
that schema acquisition is a major aspect of learning. Means-ends analysis
does not place an emphasis on learning to recognise and categorise problem
states and their associated moves. Instead, it requires a complex search
process involving relations between problem states and the operators that can
be found to reduce differences between states. We can predict that solving
problems by means-ends analysis imposes a heavy extraneous cognitive load
that interferes with learning. In other words, under many commonly found
conditions, learning and problem solving may be incompatible. (See Owen &
Sweller, 1989, for a more detailed exposition of this position, Lawson, 1990
for a response, and Sweller, 1990, for a reply to Lawson).
o
_ _ _ _ _r#-
~r_._,:_-F
A
G
II
Solution: J. Angle DEC = Angle FEG
(verlically opposile angles are equal)
= 50'
2 Goal Angle X = Angle CDE 'Angle DEC
(ExternaJ angles of a triangle equaJ the sum
of the opposite interoal angles)
= 60', 50·
= 110·
Figure 1. A conventionally structured worked example
demonstrating a procedure for finding a value for Angle X.
Two techniques have been used to simultaneously provide evidence for
this suggestion and to test alternatives to conventional problem solving that
are more in accord with the requirements of schema acquisition. The
90 Sweller and Low
techniques are a heavy use of (a) goal-free problems and (b) worked
examples. These will be discussed next.
Goal-Free Problems
If means-ends analysis imposes a heavy cognitive load that interferes
with learning, then preventing students from using a means-ends strategy and
having them attend to problem states and their associated moves should
facilitate schema acquisition. Goal-free problems should have this effect.
Consider the previous geometry problem again in which the goal is to find a
value for Angle X. Assume this goal is replaced by the statement,"Find the
value of as many angles as you can." With this statement, it is no longer
possible to reduce differences between current problem states and the goal
state because the goal state is insufficiently specified to be used in this
manner. All students can do is attend to each problem state and attempt to
find moves that can be made from that state; precisely what we might assume
they need to do to acquire a schema. In many experiments comparing goalfree with conventional problems, this effect was obtained with learning
substantially enhanced by goal-free problems (See Owen & Sweller, 1985;
Sweller, Mawer & Ward, 1983; Tarmizi & Sweller, 1988. In addition,
Silver, 1990, gives useful examples of goal-free problems.)
Worked Examples
'While worked examples are vastly different in structure from goal-free
problems, from a cognitive load perspective they should have similar effects.
Consider a worked example in elementary algebra:
Solve for a
(a + b)c = d
a + b = dlc
a = (dlc) - b
To process this worked example, students must do no more than
concentrate on each problem state (line) and the operator (rule of algebra)
needed to transform the problem state into the next state. In contrast, if the
problem is presented without the solution (first line only), students must
engage in the usual means-ends search to fmd a solution. They must consider
the difference between (a + b)c and a and try to find operators to reduce the
differences. Cooper and Sweller (1987) demonstrated that this can be a very
difficult task for novice algebra students. The cognitive load associated with a
means-ends strategy interfered with learning. Both in that paper and in
Sweller and Cooper (1985), results of many experiments demonstrated that
giving students many algebra worked examples to study was substantially
superior to having them solve the equivalent problems.
The Split-attention Effect
It might be considered that the appropriate conclusion from the above
studies on worked-examples is that they should universally replace
Cognitive Factors 91
conventional problems. In fact, the only legitimate conclusion is that when
teaching mathematics, extraneous cognitive load should be reduced.
Conventionally structured worked examples in algebra focus attention almost
solely on problem states and their associated moves, and this is important if
schema acquisition is the goal of the exercise. Conventionally structured
worked examples in other areas have quite different properties. Consider a
geometry worked example as in Figure 1. It consists of a normal diagram and
associated statements. Neither the diagram nor the statements are intelligible
until they have been mentally integrated. Once they have been integrated, the
worked example can be processed. The act of mental integration requires
cognitive effort. The sole reason for the cognitive effort is to mentally
restructure the example so that it can be understood. With a slightly different
structure as in Figure 2, the cognitive resources required to mentally integrate
the diagram and statements might not be required for this purpose and so
could be used entirely to better understand the mathematics associated with the
example.
D
G) Go3J Angle X
CDE • Angle DEC
(External angles of a
triangle equal the sum of the
= Angle
opposite internal angles)
60'· 50'
110'
____
=
=
A
CD Angle DEC
Angle FEG
(verticaJJyopposite
angles are equ3J)
= 50'
=
~<#_------------------
i"_~-F
B
G
Figure 2. An integrated worked example demonstrating a
procedure for finding a value for Angle X.
Several studies have demonstrated that if worked examples are
presented in a form that includes multiple sources of information that are
unintelligible until they have been mentally integrated, then such worked
examples are no more effective as a student learning device than solving the
equivalent problems. Mentally integrating disparate sources of information
imposes an approximately equal extraneous cognitive load as solving the
92 Sweller and Low
equivalent problems by means-ends analysis. In contrast, such examples can
almost always be restructured to elimin.ate split-attention between disparate
sources of information. For example, in geometry, the geometric statements
are placed at appropriate locations on the diagram as in Figure 2 rather than
physically separated from it as .in Figure 1. When physically restructured to
reduce the need for mental restructuring and thus reduce extraneous cognitive
load, worked examples are massively more effective than either conventional
examples or conventional problems. It should be noted at this point that the
split-attention effect applies to all instructional material, not just worked
examples. Any instructional material that unnecessarily requires students to
split their attention between disparate sources of information should be
physically integrated if possible. Results suggest that the seemingly minor
alterations involved in physical integration can have enormous benefits unless
we are dealing with redundant rather than essential sources of information.
(Experiments on the split-attention effect may be found in Sweller, Chandler,
Tierney & Cooper, 1990; Tarmizi & Sweller, 1988: Ward & Sweller, 1990.)
Conclusions
Work on cognition and instruction in mathematics has provided us with
a new conception of what is involved in mathematical skill. A skilful
mathematician has a huge knowledge base consisting of tens of thousands of
schemas acquired over many years. For any individual, mathematical
problem solving skill is restricted to those areas of mathematics for which he
or she has access to a large, readily accessed knowledge base. The major task
of someone wishing to acquire mathematical problem solving expertise is not
to learn the rules of mathematics (although the rules of mathematics obviously
are essential) but rather, to acquire the huge range of schemas that the subject
demands.
Frequently, there is a discrepancy between what is taught in
mathematics classes and what must be learned. While mathematics instructors
may place a heavy emphasis on the rules of mathematics, students, to be
competent, must both learn the rules and acquire schemas. For many
students, learning the rules is a relatively simple task. Unfortunately, simply
knowing the rules needed to solve a particular problem frequently is
insufficient to allow us to find a solution. We have difficulty applying
previously learned rules in unfamiliar contexts and our students are no
different. We require the relevant schemas and acquiring a large body of
schemas may be overwhelming for many students. Successful schema
acquisition may require teaching techniques that differ from those traditionally
used. Our knowledge of the processes of cognition may have generated new
procedures to facilitate schema acquisition. Hopefully, these procedures may
make mathematics classes more productive and enjoyable for both students
and teachers.
Cognitive Factors 93
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