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Educational Review
ISSN: 0013-1911 (Print) 1465-3397 (Online) Journal homepage: http://www.tandfonline.com/loi/cedr20
The Role of Representation in Teaching and
Learning Critical Thinking
Jean McKendree , Carol Small , Keith Stenning & Tom Conlon
To cite this article: Jean McKendree , Carol Small , Keith Stenning & Tom Conlon (2002) The Role
of Representation in Teaching and Learning Critical Thinking, Educational Review, 54:1, 57-67,
DOI: 10.1080/00131910120110884
To link to this article: https://doi.org/10.1080/00131910120110884
Published online: 01 Jul 2010.
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Article views: 483
Citing articles: 17 View citing articles
Educational Review, Vol. 54, No. 1, 2002
The Role of Representation in Teaching
and Learning Critical Thinking
JEAN McKENDREE, CAROL SMALL & KEITH STENNING, Human
Communication Research Centre, University of Edinburgh, Edinburgh, UK
TOM CONLON, Faculty of Education , University of Edinburgh, Edinburgh, UK
ABSTRACT Curriculum design in recent years re ects the growing belief in the
importance of developing thinking skills. In this paper, we focus on a particular
theoretical approach to the study and teaching of thinking: cognitive science. We
rst give a very brief review of the recent research on critical thinking. We then
concentrate on what cognitive science can add by looking at models of how people
learn and how they transfer what they learn from one context to another. The main
concept we focus on is representatio n and the crucial aspects of systems of
representation and the meaningful transformation of information. We present exam-
ples of how representations can support, or hinder , problem solving and
communication. We also discuss the social aspects of representation, the challenges
of language use, and the paradoxes thrown up by attempting to guide students to be
more critical thinkers.
Introduction
In recent years thinking skills have been placed rmly on the map by edu-
cational policy makers. Curriculum design throughout school programmes re ects
the growing belief in the importance of learners’ developing thinking skills, not
only as a tool with which to maximise potential in individual subjects but also as a
generic skill to be learned in classes and transferred from one to the other in all
directions.
One example of the growing recognition of the importance of thinking skills has
been the inclusion of problem solving in the Higher Still curriculum in Scotland,
covering the 16–18 stages. Problem solving has been divided into three elements, the
rst of which is critical thinking (though it could be argued that it should be the other
way around). Problem solving can be seen as a skill which binds together the other
core skills identi ed in the Higher Still curriculum: communication, maths and
information technology.
While this paper concentrates on critical thinking, because of the breadth of this
topic, we have chosen to concentrate on a particular approach to thinking about
critical thinking that emerges from cognitive science research: an approach based on
the crucial concept of representation. We outline a theoretical background to promote
a deeper understanding of the issues and to tie this concept into teaching practice.
ISSN 0013–1911 print; 1465–3397 online/02/ 010057-1 1 Ó2002 Educational Review
DOI:10.1080/00131910120110884
58 J. McKendree et al.
The paper includes references to further readings about critical thinking for those
who would like to delve deeper into other aspects of the area.
A Brief Discussion about Critical Thinking
Background
The notion that critical thinking is a valuable teaching and learning tool has been in
circulation at least since the time of Socrates. However, we will concentrate here on
the recent resurgence of interest since the beginning of the twentieth century. A
detailed review of recent research ndings can be found in Kennedy et al. (1991).
For a brief review of the more ancient history of the roots of critical thinking, see
the online paper by the Centre for Critical Thinking, http://www.criticalthinking.org/
university/cthistory.html.
Over the decades, educationalists have recognised both the academic and non-aca-
demic bene ts of critical thinking. Although research has demonstrated the value of
teaching thinking throughout this protracted period, it is only since the early 1980s
that there has been a renewed interest in critical thinking as an educational compo-
nent whose relevance continues to increase. Turn-of-the-millennium life-styles,
including the information overload that accompanies the growth of electronic
networks, mean that being able to think critically is essential to be able to respond
appropriately to rapid and complex changes in modern society. This is perhaps
particularly true of the employment ‘ exibility’ dictated by recent economic trends.
Equally important, critical thinking is also viewed as an essential tool with which to
effect a working and meaningful democracy (Dewey, 1909, 1997).
The research in critical thinking has gained momentum from studies which
demonstrated that a signi cant number of students, and adults, experience dif culties
when faced with complex reasoning tasks (Kuhn, 1991). Lack of critical thinking
skills meant that few students were able to apply appropriate reasoning beyond the
super cial in order to reach solutions when presented with even moderately novel
problems.
Teaching Critical Thinking: Subject Speci c or Independent?
The question then emerges of how to teach such skills. There are generally three
models for teaching thinking skills: embedded within a subject, subject-independent,
and a mixed model where a set of generic attitudes and skills are applied to speci c
knowledge and experience across the curriculum. It is arguable that, theoretically at
least, the mixed model approach provides opportunitie s for practice in a variety of
contexts and at the same time reinforces those skills taught in subjects. However, all
three approaches would appear to be effective in so far as their participants obtain
better results than learners who have received no explicit instruction in thinking
(Fisher & Scriven, 1997; Nisbett, 1993; van Gelder, 2000). It is more a matter of
nding a programme that best matches the needs of the students, teachers and school
than identifying ‘the one best’ programme for everyone. A summary of many
different programmes is available in Hamers and Overtoom (1997). The present
paper instead of outlining another programme of instruction, is an attempt to focus
on a particular area that can enhance thinking and problem solving skills and can
apply to teaching and learning of any subject: the concept of representation.
59Teaching and Learning Critical Thinking
A Cognitive Science View of Critical Thinking and Communication and its
Relation to Constructivism
The last section was a whirlwind tour of some of the general issues about critical
thinking that have been discussed for the last several decades. In this section, we
focus on a particular theoretical approach to the study of thinking. In the last half
century, cognitive science has begun to give us a more precise scienti c understand-
ing of thinking processes. The cognitive revolution made strange bedfellows.
Linguistics, computer science, arti cial intelligence, electronics, architecture, anthro-
pology, psychology, logic and philosophy turned out to have important contribution s
to make toward understanding aspects of human communication and reasoning. Yet
the theoretical insights developed by cognitive science have only recently begun to
be applied to educational practice in a substantive way (McGilly, 1994). While there
are many paths into cognitive science, the main concept we will focus on here is
representation, the meaning of which will be expanded in the next section.
First, however, we will brie y outline the connections that we believe exist
between representations in cognitive science and the principles of constructivis t
learning. These two approaches have often been characterised (or, we believe,
caricatured) as being in opposition in various ways (Lave & Wenger, 1991;
Anderson et al., 1996, 1997). However, more recently there has been a more
reasoned attempt to draw out the more subtle relationships between the two
approaches (Anderson et al., 2000; Stenning et al., in press). We will not go into the
details of this debate, but rather will brie y outline the ways in which we see
representational abilities as seen by cognitive science to be compatible with construc-
tivist principles.
The primary commonality between these two approaches is that of agency. As we
hope to show in this paper, representational ability is about gaining the skills of
uency in building and transforming information in a personally meaningful way.
Without these skills, knowledge remains inert and in exible. The best representation
almost always lies beneath the surface of the given information and requires learners
to engage in a deep way, often in collaboration with others, to impose their own
framework on the problem. Further, representational systems are often very local to
a particular problem or problem type and must be reinterpreted each time in the
current context. We would suggest, as we hope to show in the remainder of the
paper, that this viewpoint has much to offer constructivist perspectives on learning
(Stenning & Sommer eld, 2000).
These insights are of particular relevance to the core skills of communication,
reasoning and information understanding—the very skills that are the focus of
cognitive science research as well as goals of constructivist learning. So what can
cognitive science tell us about these skills? Cognitive science concentrates on
building models of how people learn and, perhaps more importantly, how they
transfer what they learn from one context to another. We shall look more closely now
at one of the key concepts of cognitive science that may allow us to teach thinking
skills more effectively, representation.
Representation: the Tool of Thought
A representation is a structure that stands for something else: a word for an object,
a sentence for a state-of-affairs, a diagram for an arrangement of things, a picture
for a scene. Although there are cases in which such pairings of representation and
60 J. McKendree et al.
represented are just isolated idiosyncratic pairings—such as when a beer mat
stands in for a car in a pub discussion about a recent car crash—they usually operate
in systems of representation, and these systematic cases are the ones that concern
us. Even the beer mats standing for a car and a bus represent systematic spatial
relations between the two, though the choice of a beer mat or a pencil is arbitrary.
Such a set of representations makes an ephemeral, local system. The English
language is rather more systematic and less local, though even here interpreting
English requires sensitivity to local context. Diagrammatic systems such as maps are
somewhere in between—more universal than the beer-mat-as-car, but perhaps less so
than English.
The second critical concept is transformation of representations. In a system of
representations, particular operations can be used to manipulate the structures into
new ones which are systematically related to the one they were derived from. If our
pub chum moves the beer mat relative to the ash tray in certain ways, then he will
be ‘saying something different’ about the relationship of the cars in the crash. ‘Max
is the dog’ can be rearranged into ‘The dog is Max’ according to the rules of English
and can be understood without requiring that you actually know anything about dogs
or Max. The rules of algebra allow you to turn x13x512 into x53. Some
operations are not legitimate in the local system. Thus, turning ‘Max is the dog’ into
‘Is Max dog the’ yields word-salad. Turning x13x512 into x3x12 51is not a
recognised transformation in the system of algebra. In other words, both of these
transformations fail to preserve meaning in the representational system, be it English
or algebra.
The meaningful transformations of representations are at the core of understandin g
human information processing. Reasoning with an abstract representation of a
situation can be much more effective than reasoning with a concrete situation alone.
Thus, a good representation system captures exactly the features of a problem
that are important rather than representing everything. The operations that create
meaningful transformations can manipulate those critical features in useful ways,
and the pay-off is more effective and ef cient reasoning about a particular
problem or situation. The next section presents some examples of how getting the
representation right can make a seemingly dif cult or impossible problem much
easier.
Representations as Friend or Foe
In this section, we present examples to illustrate some properties of representations.
The rst important realisation is that there are many different forms of representation
and for any particular problem, some forms of representation will be good and some
will be bad. A common example of a bad representation system for a particular
problem is Roman numerals for long division. The modern Arabic system enables far
more rapid calculations. The abacus is another representational system that is even
more rapid than the Arabic number system for arithmetic for those who have learned
and practised it. Any representation must be learned and practised for pro cient use,
but some representations are better than others for particular problems.
Another example of representations that are good or bad for different purposes
relates to the speci cation of prescriptions. (This example is from Day, 1988 as
presented in Norman, 1993.) The pharmacist likes to have all the different medicines
listed as the primary category of organisation, with doses, frequencies and times
61Teaching and Learning Critical Thinking
TABLE I. Tabular representatio n of medication s for a stroke patient
Breakfast Lunch D inner Bedtime
Lanoxin Ï
Inderal Ï Ï Ï
Quinaglute Ï Ï Ï Ï
Carafate Ï Ï Ï Ï
Zantac Ï Ï
Coumadin Ï
sub-listed with each item. This representation facilitates their work in measuring out
quantities of medications needed. One actual list of medications for a stroke patient
looked like this:
Inderal: 1 tablet three times day
Lanoxin: 1 tablet every a.m.
Carafate: 1 tablet before meals and at bedtime
Zantac: 1 tablet every 12 hours (twice a day)
Quinaglute: 1 tablet four times a day
Coumadin: 1 tablet a day
You can probably see that this representation is rather poor for communicating to
patients about what to do. For the patient, it is much better to have a tabular
representation with, for instance, times of day as columns, drugs as rows, and table
entries as doses, such as shown in Table I.
Mathematics is replete with examples of problems which are easy to solve with
one representation, but very hard with another. A famous example is the bird-on-the -
tracks problem.
Two engines, one travelling at 60 mph and the other at 40 mph face
each other on a track. They start at 12 noon from 100 miles apart. A bird
also starts ying at 12 noon, starting at the front of the faster engine
at 100 mph, ying along the track, until it meets the other engine. At
this point it reverses (instantaneously) and ies, always at 100 mph in the
other direction until it meets the rst engine and reverses again. Unfortu-
nately the inevitable occurs—the engines crash into each other squashing
the bird (instantaneously). How far did the bird y before its untimely
demise?
If you try to solve the problem in terms of distance travelled (the way the question
is posed), it quickly becomes a calculus problem involvin g integration. However, if
you represent the problem in terms of time, it is trivial. The trains travel for 1 hour.
The bird travels for 1 hour. It covers 100 miles.
In the examples, picking the right representation aids understanding and compu-
tation of a solution. It transforms the task from a memory intensive, complex
problem into one in which the answer can almost be ‘read off’ with little effort. Of
course, many problems require a great deal of work even with a very good
representation, but still much less than with a bad one. Imagine, for instance, trying
to do long division with Roman numerals.
62 J. McKendree et al.
Representations and the Relation to Critical Thinking
Most areas of knowledge have ranges of representations tailored to the particular
problems encountered. Quite a lot of the school curriculum can be conceived of as
teaching these representational skills. More or less the whole of mathematics is little
else than exploring representational systems and their inter-relations. In each subject,
problems arise and children are taught how to solve them using representations:
language, graphics, whatever.
What is much less often taught is how to search for a representation for a given
problem or to understand the abstract properties of the representation that makes it
a useful one in a particular instance. It is this which often holds the key to
transferring learning from one problem to another (Singley & Anderson, 1989).
Recognising what is structurally similar between two problems, one about prescrip-
tions and one about properties of mammals lets you transfer a good deal of your
learning from one problem to the other. Similarity is often to be found exactly at the
level of good representations for the problems.
Being able to think about why a representation may or may not be good in a
particular context is a big part of being a critical thinker. If a student can realise that
the problem they are working on is best represented in a particular way, it can help
them identify the most important aspects of a situation and analyse what should be
done next. If they know what the ‘accepted’ transformations of a problem type are,
then they can begin to think at a higher level about why those are the ones that are
used and why, perhaps, it might be interesting to try a different representation. They
can begin to understand why problems are the way they are.
Lacking an understanding of representations can present problems at many levels.
In pre -school children, a lack of representational uency can be identi ed in
individuals by their inability to select which of several representations—e.g. dice,
clock dial, and number line—indicate the same number. This is not due to a lack of
exposure to the representations, but rather an inability to link the different forms of
representation to the underlying common concept of number (Grif n & Case, 1997).
A failure to develop this foundationa l understanding can lead to serious lags in
learning mathematics throughout formal school years (Grif n et al., 1994). Problems
of representational uency are apparent in older student s as well. University students
who can readily translate between and manipulate representations score higher on
measures of reasoning ability than those students whose skilled use of representa-
tions is more limited (Stenning et al., 1995). Monaghan et al. (1999) found also that
higher scores on problem solving tasks were related to ability to use representations
more exibly.
A representation is never the whole solution to any problem, but it usefully factors
problems into two parts: rst, achieving an agreed set of assumptions or premises
captured as the critical features, and second, consequences that follow from them by
applying appropriate operations. These parts or phases may, of course, be inter-
twined. If we nd that we cannot reach a conclusion, we may realise that we have
missed out an assumption. If we do not like the consequences, we may perfectly
reasonably decide that we have not outlined the assumptions correctly.
Try this problem shown in gure 1. (The answer can be found in the Appendix.)
In the join-the-dots problem, most people implicitly have an assumption that you
are not allowed to extend the line beyond the dots. It is not in the problem as stated,
and it happens to make the problem impossible to solve. Sometimes, students may
63Teaching and Learning Critical Thinking
FIG. 1. Draw four straight lines tha t pass through all nine dots without raising the pencil from the paper
be too quick to assume that because a problem looks similar to others they have seen,
it must be the same kind of problem.
For instance, consider this word problem:
On a boat there are six sheep, 24 ducks and 10 pigs. How old is the
captain?
Students often assume this is a maths problem because of the similarity to many they
have seen before and immediately respond by saying ‘40’. When you then get them
to really read the problem—try to represent the actual information asked for—they
quickly realise they have been tricked. Sometimes we read in things that we should
not, just as sometimes we leave out crucial items that are necessary. Making an
explicit representation to aid our reasoning can often help to uncover such problems.
The examples given (dots and farmyard arithmetic) are suggestive of maths
problem solving. However, we want to emphasise that social sciences, arts, lan-
guages, natural sciences, and humanities all use various forms of representation,
some quite general and some very specialised. Some of these are evidence diagrams,
concept maps, v-diagrams, timelines, grammar notations , and dance notations (No-
vak, 1998; Peterson, 1996). The same costs and bene ts for communication and
problem-solving apply to all of these as well.
This re ning of our assumptions and inferences supported by building and using
a representation is what makes our inchoate understandings into ones that are explicit
and public. In the course of this process, we often change ourselves, our audience,
or both. The process of representation makes explicit and external our reasoning,
which is the rst important step in being able to be critical about our own and others’
thinking.
The Social Aspects of Critical Thinking
Analysing the ‘Critical’ in Critical Thinking
So what does this very particular label critical thinking mean? Here, the topic has
been introduced through the idea of representational systems—both transformations
within systems and the idea of taking the meta-stance and choosing a suitable
representation from a range of systems. This fundamental theoretical distinction is
helpful in systematising many of the things that people say that critical thinking is.
We have seen that choosing a good representation can often help us consider the
important aspects of a problem and can help us be more effective in searching for
a solution. However, reasoning at the meta-level about what systems of representa-
64 J. McKendree et al.
tions there are points us to another important aspect of critical thinking that raises
some tricky issues for teaching. This is the question of whether we have the ‘right’
problem in the rst place.
‘Problem nding’ as this process is known when it is an explicit part of problem
solving, is not merely an issue in science and mathematics. In politics, con icts may
be unresolvable when focused on one question or issue, but progress may be made
if refocused on another. If we accept the question ‘What should the government do
to stem the tide of immigration?’ we are into one kind of problem solving; if we
reject this question and ask ‘Is there a tide of immigration?’ or ‘What problems does
a tide of immigration pose, and what could be done to ameliorate them?’, then we
are into quite different problems and solutions. Students need to understand that such
‘questioning the question’ is sometimes a very good thing to do.
This brings to the social setting of critical thought. If this is school, students may
feel that they cannot question the teacher’s question. They may believe that if teacher
set the question then it must be a good one. They may be right in these beliefs.
Similarly, if it is the geography teacher who set the question, then they may believe
it has to be solved by something learnt in geography class. It may not be allowed to
‘import’ our outside-school knowledge or even our inside-school knowledge from
other classes.
Students have to learn what the conventions are for school-set problems. It is
extremely important for students , as well as teachers, to be clear that there are
conventions of reasoning in all subjects and that they are different in different subjec t
areas. Bringing the literalism of maths problems to the poetry class may not go down
well; likewise, relying solely on personal anecdotes is not suf cient in science. In
both, it is still important to support your conclusions with explicit assumptions and
reasoning, and externalising these by drawing or writing a representation can be a
very good way to do so. A good English teacher can draw out relationships in a
wealth of examples and experiences to help students probe the richness of metaphor
in poetic language. A good science teacher can outline the models or data underlying
a particular claim. What changes between subjects is not the demand to be clear,
concise and critical; what changes is what is acceptable as evidence, what is
considered a proper set of assumptions and representations, and what are the criteria
for a ‘good’ solution.
Authoritative Democracy: the Paradoxes of Teaching Critical Thinking
There is something very paradoxical in the very notion of teaching critical thinking.
We have touched on the rst one above: the paradox of authority. If critical thinking
is thinking for oneself, then how can a teacher justify using authority to direct a
student to do it? Suppose the students just want to follow on in the same old tracks
thinking what they have been told to think and do not want to stand back and look
for new perspectives. Who is the teacher to say that this is not a perfectly rational
response to the students’ situation? Especially if the school operates an assessment
regime that in fact rewards students for doing exactly that, despite the rhetoric about
core-skills?
These are hard question s that have to be taken seriously. Our approach is to argue
that teaching core skills may actually be labour saving, both for student and teacher.
If the easiest way of learning to solve problems is by transferring knowledge from
one context to another, then perhaps there is immediate self-interest to be applied?
65Teaching and Learning Critical Thinking
It may take slightly longer initially to get the process started, but once we have
helped students to become thoughtful and critical learners, the result ultimately
should be faster learning and better transfer.
Perhaps one of the best ways one can resolve the paradox of an authority gure
questioning authorit y is by teaching-by-personal-example. Instead of providing only
an argument from authority that one should think critically, perhaps the most
powerful way to get students to adopt the model is to show the process in action. If
students like what they see, perhaps they will adopt the model themselves.
A second paradox is the fact that students must implicitly know core skills before
they get into the classroom. Perhaps this is most obvious with the grammar of a
student’s native language. If a student did not ‘know’ this grammar, in the sense of
being able to produce and understand speech, the teacher would lack any medium in
which to teach. Yet students palpably do not have an explicit grasp of grammar
simply by virtue of speaking their native tongue. They do not know how they talk.
We can change language abilities by teaching explicitly about language. Perhaps the
most obvious example is how literacy changes peoples’ habits of thought, and with
it their whole culture (Rogoff, 1981; Vygotsky, 1996).
Students know implicitly the principles of problem solving and representation or
again they would hardly be able to function independently at all. Yet they de nitely
do not have an explicit grasp, nor quite often even a well developed skill. How can
we resolve this apparent paradox? One resolution is to deny that explicit instruction
has any impact on peoples’ behaviour. This does not stand up to even the most basic
evidence at hand. An alternative is to realise that students’ ‘native abilities’ in
reasoning, while powerful, and much to be marvelled at from a scienti c point of
view, are also highly context dependent. Learning some of the more abstract
principles, and explicitly discussing how the principles apply to many contexts, can
have a profound effect on students’ ability to generalise reasoning from one situation
to another.
Both the paradox of authority and the paradox of students already ‘knowing’ core
skills at some level can be resolved to some degree by thinking about discussion as
a teaching tool and representation as an artefact to prompt deeper discussion. The
key for the students is to discuss these issues explicitly with them. Besides
representations aiding in individual problem solving, it has been shown that getting
students, and teachers, to externalise their ideas by drawing a representation can
substantially boost expressiveness. Students talk more and talk differently about their
knowledge and thinking processes, resulting often in improved learning and perform-
ance as well as revealing progress and problems (Chi et al., 1994; Novak, 1998;
Umata et al., 2000).
Good dialogue about the process of learning and the importance of representation
will let the students begin to monitor and assess their own learning. Modelling
problem solving by talking aloud and drawing intermediate representations while
working through a problem, and then re ecting on the process afterwards is an
excellent way to show students how you would like them to think, too. They can
see that being knowledgeabl e about something is not magic and that the seeds of
the skills already lie within them. It is also extremely important for students to
realise that language can be very ambiguous. A teacher and a student may think that
they are talking about the same thing when they use the same word, but they may
actually have very different meaning attached. For instance, Pimm (1987) tells the
story of an 8 year old at the end of a maths lesson who when asked if she understood
66 J. McKendree et al.
what ‘common’ meant replied that it is when someone wears too much lipstick.
The point is that we need to not only use a consistent vocabulary, but we need to
discuss explicitly the meaning of words that might seem obvious at rst blush (Gee,
1998; McKendree et al., 2000).
Some current teaching approaches that are based upon constructivist learning
principles, such as cognitive apprenticeship (Brown et al., 1989; Collins et al., 1990),
do embody many of the ideas presented in this paper. What we feel is often lacking
is the emphasis on the skills of selection of, construction with, and reasoning about
commonly occurring forms of representation. A repertoire of such representational
skills can give learners new ways to manipulate and transform information for
themselves. Also, it becomes possible to infer properties of the information from the
structures of the form of representation in which the information is being expressed,
making transfer of learning more likely. Representational skill thus provides an
essential dimension to critical thinking, a cognitive exibility that can become a
powerful tool for transforming raw information into fruitful and personally meaning-
ful knowledge.
Correspondence: J. McKendree, Human Communication Research Centre, Univer-
sity of Edinburgh, 2 Buccleuch Place, Edinburgh EH8 9LW, UK. Email: jean@
LTSN-01.ac.uk
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Appendix
Solution to the nine-dots problem:
