Creative Education
2011. Vol. 2, No. 1, 47-55
Copyright © 2011 SciRes. DOI:10.4236/ce.2011.21007
Discussion Article: Disciplinary Boundaries for Creativity
Stuart Rowlands
Centre for Teaching Mathematics, University of Plymouth, Plymouth, UK
Received October 25th, 2010; revised February 9th, 2011; accepted February 23rd, 2011.
Creativity is a very topical issue and indeed a political one. For example, the very notion of ‘little c creativity’
seems to be a reflection of the requirements of what could be described as a ‘Post-Fordist’ economy. However,
the call to develop creativity in education is largely based on the idea of creativity as the production of novel
ideas. The central argument of this article is that creativity cannot be seen purely in terms of novel ideas but that
it is intrinsically bound with the teaching of the academic disciplines. It is within the context of creativity in the
sense of transforming the disciplines that two paradoxes are discussed. The first paradox is that the truly creative
act is not the preserve of the genius but the potential for the whole of humanity. Secondly, creativity involves
both thinking within the constraints of the discipline and challenging those constraints. This implies the need for
students to engage in meta-discourse, involving the nature and history of the subject-m att er taught.
Keywords: Art, M athematic s, Scie nce, Novel Ideas, Transforming the Disciplines
Introduction
Although creativity in education was a concern that began
in the 1950’s (Craft, 2001a), this concern has become a
highly topical issue over the past fifteen or so years. Recently
in the United Kingdom there have been many conferences
and meetings on creativity and education with input from
museums and the arts and examples include the British Edu-
cational Research Association’s Special Interest Group
‘Creativity in Education’, the Economic and Social Research
Council’s Creativity in Education Seminar Series, the Na-
tional Advisory Committee for Creative and Cultural Educa-
tion (NACCCE) and the plethora of books and articles on the
subject. In all this interest, however, there is a tendency to
promote creativity in terms of the creation of novel or inno-
vative ideas that are independent of the teaching and the
content of the formal academic disciplines such as mathe-
matics, science, history and art. This paper argues the claim
that the vast majority of humankind is capable of truly crea-
tive acts, not merely in the sense of novel ideas but also in
the sense of transforming the very disciplines themselves.
This is indeed a very bold and paradoxical claim because it
states that what is normally attributed as acts of genius can
also be attributed to nearly every learner. It is within this
context that the further paradox between having to think
within the constraints of the discipline and thinking crea-
tively by rethinking those constraints is explored. This arti-
cle’s main concern is that if we ignore creativity in the sense
of Boden’s transformation of the subject matter of the disci-
plines then we will fail to unlock the creative potential of the
child.
Section 1 begins with the boom in popularity of creativity
and argues that the current discourse undermines what it is to
be truly creative. Creativity has been reduced to the produc-
tion of novel ideas separate from the learning of academic
subjects. This section argues that no separation should be
made between promoting creativity and the teaching of those
subjects.
Section 2 discusses the link between creativity and intrin-
sic motivation. This section argues that creativity is a prime
need of every human being, is incompatible with extrinsic
rewards and that the whole of humankind is capable of being
creative in the sense of being truly original.
Section 3 discusses how creativity in the transformative
sense of changing the disciplines can be fostered. It shall be
argued that such creativity requires learning and exploring
the domain of knowledge comprehensively and, paradoxi-
cally, by playing with the constraints of the system of
knowledge. This can be encouraged by a meta-discourse that
reflects upon the nature of the theoretical objects the learner
is expected to think in and work with. It will be argued that a
cultural-historical approach can provide the stepping-stone to
the relevant level of abstraction by immersing the learner in
the relevant problem space. This section also discusses why
an objectivist approach to creativity is fundamental compared
with an inter- or intra-subjectivist approach and why, in
terms of creativity, the teaching of science should not be
reduced to making sense of experience (as there has been a
tendency to do).
Section 4 discusses knowledge synonymous with mental
representation and argues that creativity is a construction
process involving rational insight, symbolic thought and
metaphoric perception
What Creativity Isn’t: Creativity as Novel Ideas
The Ideology of Creativity
Given the possible links between creativity and education
and the implications that one has for the other – whether it be in
terms of educational policy, curriculum development or class-
room practice - the interest in creativity is not surprising. What
does seem a little surprising, however, is its boom in popularity.
Perhaps most educators would agree that the fostering of crea-
tivity in the classroom is a priority, but it would appear that this

S. ROWLANDS
48
increase in the promotion of creativity has more to do with
promoting the ability to be novel in an everyday context than
with developing the potential to be creative within the disci-
plines. As Rowland (2009) states in connection with school
mathematics, creativity can be an elusive and over-hyped no-
tion.
Similar to the concepts paradigm and life-form, Creativity
may not admit to being well-defined but having characteristics
that can be discerned. In a negative article ‘What Creativity
Isn’t’ Gibson (2005) attempts to problematise the term by giv-
ing many examples of the various discourses in which the term
is used. With reference to the UK he states:
Certainly it’s a term now widely used, full of promise, a tonic
for some after a decade of national over-governance of the
school curriculum, a glimmer of hope and a word with which
everyone can agree. Another hurrah word. (Gibson, 2005,
p.148, emphasis added).
From the many examples given, Gibson identifies two
emerging themes that have questionable assumptions: a roman-
tic individualism of a bygone age with a Rousseau-like
child-centeredness, and the future needs of the workforce. Gib-
son may well have hit upon the two underlying causes of crea-
tivity’s popularity: creativity as an opportunity to advance
learner-centred pedagogy (and, as will be shown, giving the
opportunity to downplay the disciplines) and creativity as a
necessity if British capital is to survive the global economy.
The two causes of creativity’s popularity may be linked.
Outside the content knowledge of the disciplines qua discipline
but within the context of subject matter specifically related to
the everyday, creativity becomes how the individual faces life’s
challenges and the requirement to foster a ‘creative’ labour
force specific to the requirements of capital at any time. Having
to survive the everyday also includes having to survive frag-
mented labour markets (which fluctuates. Studying for a degree
in golf course management, surfing studies or perfumery –
courses that presently exists at some UK universities - may
subsequently require the creativity of finding alternative forms
of employment). For example, Craft (2001a) proposes the de-
velopment of ‘little c creativity’ (LCC – the ability to think in
novel ways in an everyday context) that would enable the indi-
vidual to face the reality that we can no longer expect a job for
life. Creativity thus becomes flexibility in response to the frag-
mented labour markets of what some may describe as the
‘Post-Fordist’ economy. The then Secretary of State for Educa-
tion and Skills, Estelle Morris, stated:
The best employer knows that his future depends on the
imagination and the creativity of his current workforce and of
his future workforce. And it is innovation and creativity and
risk taking that actually will give us economic prosperity and
economic survival. The demand it puts on schools is different
than it was before (Morris, 2002, p. 3)
Eight years hence and the employer’s future has more to do
with the current economic crisis than it has the imagination and
creativity of his workforce. This raises the question as to
whether schools should serve the interests of the employer or
the educational interests of the learner. The then prime-minister,
Tony Blair, might have responded that these two interests are
the same in the sense that a healthy economy is necessary for
healthy individuals; but with hindsight we can see that a crea-
tive workforce does not necessarily make a healthy economy. A
Marxist (e.g. Matthews, 1980), however, would argue that this
interest in creativity reflects the need of capital to transform
labour-power (the capacity and ability to do work) relevant to
these fragmented labour markets, in addition to any concern for
the individual to function in a society that is in crisis. For the
Marxist we have a contradiction with respect to the emphasis
on learner-centred pedagogy: The emphasis on LCC and the
reduction of creativity to novel and innovative ideas independ-
ently of the disciplines may serve the interests of capital, yet it
will also undermine the true creative potential of the vast ma-
jority of children with respect to learning the disciplines. It is
this sense of creativity that will be explored throughout this
article.
The educationalist who wants to foster creativity in the
classroom may question the promotion of creativity for the
market. For example:
But, how desirable is the norm of innovation that the global
economy demands? To what extent is it desirable to encourage
and sustain the ‘disposable’ culture, where obsolescence is built
in at the design stage of many consumer goods? To what extent
do we, in the marketplace at any rate, encourage innovation for
innovation’s sake and without reference to genuine need? (Craft,
2003, p.121).
The promotion of creativity in a global economy that cares
more for rates of profit than it does people’s needs might be
unpalatable to the liberal or left-wing educator. Nevertheless,
despite any good intention, to promote LCC might be in the
long run to promote the economy of the nation state and hence
to promote profitability for the employer rather than for peo-
ple’s needs.
Creativity and the Subject Matter
Creativity has become the banner for romantic individualism
and this can be seen in the way many educationalists have
promoted creativity as if it were bipolar to the content knowl-
edge of the disciplines. For example:
Although some teachers or observers want to concentrate on a
particular subject or type of classroom activity, others might
wish to address a more overarching notion like ‘creativity’, or
the extent to which children are able to use their imagination
and ingenuity, irrespective of the subject being taught. (Wragg,
1999, p. 28, emphasis added)
Creative teachers are interested in knowledge, but they are
more interested in skills and even more interested in attitudes
and values (Lucas, 2001, p. 39).
From the creativity SIG at BERA’s 2005 annual conference
we have one speaker stating as a major theme of his presenta-
tion: “ambiguity is important”. This is problematic with regards
to the well-defined concepts of science and mathematics. The
following speaker stated “It is better to be eclectic than to im-
pose the disciplines” uttered within the context of listening to
the child. Ironically, this could serve to undervalue the child:
the promotion of creativity at the expense of knowledge content
has the potential to create a two-tier system whereby the chil-
dren of the rich will learn the disciplines (and, with quality
teaching, learn how to become creative within it) and the chil-
dren of schools in the state sector learn to become eclectic, such
as learning consumer arithmetic as a creative life-skill and fig-
uring out why the school pond has turned green.
The NACCCE (1999) report emphasises the need for a bal-

S. ROWLANDS 49
ance between content knowledge and creativity as if this bal-
ance would satisfy Aristotle’s golden mean. Sometimes, the
importance of the former is referenced but subsequently ig-
nored in the emphasis on the latter (e.g. Haigh, 2003; see sec-
tion 3). An exemplar of this bi-polar view in practice is the UK
National Curriculum for mathematics, whereby creativity is a
‘requirement’ but amounts to little more than the teaching of
the content of the National Curriculum on the one hand and a
few excursions into open-ended investigations on the other.
This bi-polar view is consistent with the constructivist philoso-
phy in mathematics education that emphasises the process of
‘doing’ mathematics (sometimes referred to as ‘mathematiza-
tion’, e.g. Jaworski, 1994) while at the same time downplaying
the learning of the content of mathematics (any emphasis on
content deemed ‘absolutist’).1 This section argues that creativ-
ity and content knowledge ought to be inseparable. If separated,
creativity is reduced to novelty and content knowledge merely
becomes something to be learnt – if at all.
This recent explosion of interest in creativity has tended to
reduced creativity to the production of novel ideas, but this is
not something new. Over the years there have been many dif-
ferent approaches to creativity (for an overview, see Craft,
2001b) and although diverse, many approaches have treated
creativity in terms of ideas that are novel (Boden, 2001), such
as, for example, books on how to develop your creative poten-
tial (e.g. Adams, 1988) or creativity tests that measure diver-
gent thinking. According to Diakidoy and Kanari (1999), crea-
tivity has been traditionally defined in terms of a characteristic
or cognitive process that results in a novel outcome and some-
times without reference to whether those ideas are right, wrong
or relevant. Not surprisingly, the authors have found a tendency
for student teachers to see creativity in terms of novel ideas that
were not necessarily appropriate or correct. This seems to be
quite consistent with the educational philosophy (e.g. radical
constructivism) that emphases how the pupil conceptualises the
problem or task as opposed to whether the pupil has correctly
conceptualised the problem or task (e.g. Jaworski, 1994). Of
course, creativity does involve novelty, but most approaches to
creativity have focussed on novelty to the exclusion of the
background knowledge that gives the novelty its meaning
(Ward et al., 1997). To refer to an idea that is either inappropri-
ate to the problem or resolves the problem incorrectly as
‘novel’, is to lose the meaning of novelty with respect to the
problem. If it is possible to develop creativity amongst pupils
independently of and without reference to the context and con-
tent of ordinary school learning then there is still the problem as
to how they can use their creative abilities in the context of
ordinary school learning (Adey & Shayer, 1994). Conversely,
in the absence of participation in mature cultural conversations,
namely the disciplines, any developing potential for creativity
has little to work with. For Ward et al (1997), creativity is not
the formation of novel ideas as a single process but an outcome
of several processes such as conceptual combination, concep-
tual expansion, metaphor, analogy and mental model construc-
tion. Creativity, they argue, cannot occur without some mean-
ingful link to what has come before, that is, prior knowledge
has an overwhelmingly powerful role in creative endeavours.2
Prior knowledge, that is, in the sense of academic knowledge,
not everyday knowledge.
Although this article argues that creativity should not be re-
duced to the formation of novel ideas, the individual who can
formulate novel ideas that are appropriate and correct with
respect to the subject domain can be said to manifest creativity.
The ideas formulated may only be novel with respect to his or
her own understanding of the subject; in other words, the ideas
may have been thought of before by someone else, so the ideas
are not truly novel. Nevertheless, the construction of a concep-
tual understanding of a subject domain, as opposed to the
memorisation of rule-of-thumb procedures, manifests the con-
ditions for development towards creativity. For example, the
pupil in mechanics who changes her conception of force and
motion from what may be described as an Aristotelian notion to
a Newtonian one manifests creativity (see Carson & Rowlands,
2005). To think of problems concerning force and motion
within possible world contexts of frictionless surfaces or where
gravity can be an option, without resorting to casting the prob-
lem algebraically, is to think in a creative context. The teacher
who invites the class to think in terms of possible worlds: ‘How
would you set in uniform motion a puck resting on a fric-
tionless horizontal surface?’ or ‘Gravity is switched off and I
throw a ball in the air. Describe its motion’ (see Carson &
Rowlands, 2005), has created the conditions for creative de-
velopment whereby the class uses its imagination but is con-
strained to think within the domain of the subject. This is
elaborated further in the section after next.
A critique of constructivism’s notion of ‘absolutism’ can be found in
Rowlands et al. (2010).
2According to Kneller (1965), creativity can be introduced into education
in one of two ways, either by teaching creativity on its own and in its
own right, or we draw on the creative potential in all the subject matter
we teach. Dunbar (1997) goes so far as to ask how is creative though
t
p
ossible if the notion of creativity is reduced to novel ideas, especially
since people make so many reasoning errors in experimental tasks in-
volving arbitrary concepts with no background knowledge. Cropley
(1971) doubts whether creativity can be taught as a subject in the school
curriculum at all, but that the teacher can develop ‘divergent’ heuristics
by encouraging the finding of solutions through inquiry, curiosity, inde-
p
endence and the drawing
t
ogether of domains of relevant experience:
“Creative thinking occurs when the boundaries of the known are first
mastered, through convergent processes, and then extended, by the ap-
p
lication of divergent processes” (Cropley, 1971, p.29). According to
Parnes (1970), there exists much research that shows that mastery o
f
subject matter increases, along with creative ability scores, as a result o
f
weaving creative problem solving into existing courses. Whether creativ-
ity in education is seen in terms of either a two-stage approach or an
interweaving approach, Kneller (1965), Dunbar (1997), Cropley (1971)
or Parnes (1970) made no separation between creativity on the one han
d
and subject matter on the other.
Creativity and Intrinsic Motivation
Creativity requires work, either in learning a skill or an aca-
demic subject, which is suggested by the old say ing that creativ-
ity is 1% inspiration and 99% perspiration. The 99% perspiration
also suggests learning through intrinsic motivation. Despite the
tendency in much of the literature to reduce creativity to novel
ideas, there is also the support for the ‘intrinsic motivation prin-
ciple of creativity’ (see Hennessey & Amabile, 1988). For exam-
ple, Cropley (1971) states that “material which is learned because
it is in itself satisfying to the student’s curiosity and ingenuity,
and which is seen as a challenge to the learner’s mental agility,
needs no external rewards and punishments to keep the learning
process in motion.” (p. 82). Amabile (1983) argues that intrinsic

S. ROWLANDS
50
motivation is conducive to creativity whereas the extrinsically
motivated state can be detrimental. This is supported by Bohm
and Peat (2000), who argue that creativity is actually incompati-
ble with external and internal rewards and punishments which are
arbitrary requirements extraneous to the creative activity itself.
They make the point that if creativity is made subservient to ex-
ternal rewards then the whole activity involved degenerates into
something mechanical and repetitious. Of course, the pupil may
want to learn for external reasons such as to pass an examinatio n
or please the teacher, or whatever, and the teacher may teach with
respect to the class passing an exam to the best of their ability.
However, to teach the subject in a way that develops the awe and
intrinsic motivation of the pupil for the subject is the real chal-
lenge facing the teacher (which, ironically , may well develop the
ability to do even better in examinations).
According to Bohm (1998), creativity involves the act of
learning for its own sake, which is not to be confused with rote
learning. “Creativity is a prime need of a human being and its
denial brings about a pervasive state of dissatisfaction and bore-
dom” (Bohm and Peat, 2000, p.232) and the biggest struggle in
overcoming this pervasive state would be to convince most peo-
ple that they are capable of being creative:
Most people, however, tacitly suppose that they do not have
the necessary passion and courage to act in a truly creative way
and are doomed to forever ‘play false’ with the more subtle fea-
tures of their knowledge. They believe that, not being geniuses,
they are restricted to the tacit infrastructures of subliminally held
ideas. But suppose that this assumption is false, and that everyone
is potentially capable of truly creative acts in various fields that
accord with his or her particular abilities, skills, and knowledge.
Clearly a prerequisite for this creativity is that we must cease to
take for granted that we are incapable of creativity. (Bohm &
Peat, 2000, p. 51, emphasis added)
The reduction of creativity to novel ideas indepen dently of the
subject matter seems to rest on the tacit assumption that the vast
majority of us are incapable of truly creative acts in the sense
outlined. As stated above, this paper takes a more optimistic (and
paradoxical) view that nearly all of us are capable of truly crea-
tive acts that are normally attributed to the genius.
For the teacher to draw on the creative potential of the learner
is not to imply that creativity is innate in the sense that some
people are and some are not creative. What it does imply is that
anyone can become creative in the sense that we all have that
potential. Creativity may well be a possibility for all, just as all
chess players, according to Hestenes (1992), could become mas-
ters if only they would reflect on the reasons why they lose. En-
couraging pupils to reflect so as to engender creativity will be
explored in the next section. The point here is that creative qual-
ity is not the preserve of the geni us but can reside in any kind of
human activity and can be present in many different levels of
ability or intelligence (Lytton, 1971). In art, for example, differ-
ent but conventional forms of representation operate across a
wide spectrum from children’s’ art to famous artists (Gombrich,
1996) and originality is possi ble for anyone at any level of repre-
sentation. One prerequisite for originality is the inclination not to
impose preconceptions on the facts as they are seen - a principle
that is common to the whole of humanity (Boh m, 1998) to which
Bohm includes children learning to walk and talk just by trying
something out an d seeing what ha ppen s. As eloquently expressed
by Boden: “The creative thinker (potentially, every one of us) has
the ability not to be rigorously limited by the pre-existing rules.
But that is not to say that the rules are irrelevant (Boden, 2001,
p.97, emphasis added).
If the majority of people have the potential to become truly
creative in the sense of the scientist, then with respect to devel-
oping creativity in the classroom it may be insightful to ponder
the awe and motivation of the creative scientist. For Bohm (1998):
the scientist searches for somethi ng ne w that ha d previously been
unknown, but this search has a certain fundamental kind of sig-
nificance, a hitherto unknown lawfulness in the order of nature,
which exhibits unity in a broad range of phenomena. Thus, he
wishes to find in the reality in which he lives a certain oneness
and totality, or wholeness, constit uting a kind of harmony that is
felt to be beautiful. In this respect, the scientist is perhaps not
basically different from the artist, the architect, the musical com-
poser, etc., who all want to create this sort of thing in their work.
To be sure, the scientist emphasizes the aspect of discovering
oneness and totality in nature. For this reason, the fact that his
work can also be creative is often overlooked. But in order to
discover oneness and totality, the scientist has to create the new
overall structures of ideas which are needed to express the har-
mony and beauty that can be found in nature. (Boh m, 1998, p. 2,
emphasis given)
Perhaps we can ‘open the eyes’ of the child to th e oneness and
totality of nature before we teach the structures of ideas that
would enable the child to understand nature and to eventually
express that oneness and totality. ‘Opening the ey es’ could begin
in the first couple of years of schooling by providing an envi-
ronment similar to the ‘creative school’ in Jeffrey and Woods
(2003). Subsequently, immersing pupils in the relevant concep-
tual space and introducing the m t o the history of ideas that gener-
ated the conceptual space may help to develop that intrinsic awe
and motivation. This is discussed in the next section.
Creativity, Instruction and the Generative
Structure
The Cultural-Historical Approach
Mental development is dependent ‘from the outside in’
(Bruner, 1974) which is to say that, despite innate abilities,
mental development is dependent on learning processes which
are in turn related to instruction. According to Vygotsky (1978),
learning processes are not synonymous with mental develop-
ment but consist of the former leading the latter. Creativity in
art, for example, is not a natural process that requires little or no
teaching and practice - people have to learn to draw. Gombrich
(1960) demonstrates how culture enables the construction of art
by providing the necessary symbolic codes and schemata. The
implication for the art teacher is that her class has to be encul-
turated into these symbolic codes and schemata prior to any
possibility of the class becoming creative. How can the teacher
do this? One possibility is an historical-cultural approach
whereby the class examines the work of predecessors. For any
discipline, the pupil reliving those cultural conversations that
gave rise to the subject matter can become part of the creative
process.
A truly creative idea, not one that is merely novel but inno-
vative with respect to a background of ideas, does not come out
of the void; it must rely on the insights of predecessors.
Gombrich (1960) gives many examples, one of which is Con-

S. ROWLANDS 51
stable whose paintings appear to ‘mirror’ reality as if he copied
faithfully one-to-one what he actually saw - yet his style was
developed from Cozens’ study of clouds and Dutch seventeenth
century landscape paintings. This suggests a point by Bohm
and Peat (2000) that the ‘inward perception’ of the artist or
scientist is affected by everything that the artist or scientist
holds important about the history of art or science. An example
by Bohm and Peat is Manet’s ‘Olympia’ which owed much to
Goya’s ‘The Naked Maja’ and, in turn, inspired Cézanne to
paint ‘A Modern Olympia’.
To be aware of the history of something is to be aware that
nothing is ‘given’ as such – that is, to be aware of development
as human ingenuity to which one can belong. An awareness of
the history of ideas as well as the ideas themselves can en-
courage a meta-discourse of those ideas, such that the pupil can
become part of the creativity that was originally involved. A
meta-discourse may encourage the pupil to think about the
theoretical objects of the subject matter as well as thinking
within those theoretical objects. This elaborates the point made
earlier that creativity can be engendered by encouraging pupils
to reflect upon their ideas in relation to the subject matter. For
example, when asked ‘what forces act on a thrown ball?’ many
pupils will respond that in addition to gravity and air resistance
there has to be a force pushing the ball in order to overcome
gravity. The teacher can respond by saying ‘that is not the right
answer but it is a good answer because that was the answer
given by Aristotle and accepted for two thousand years until
challenged relatively recently by Galileo’ etc. The teacher can
then invite the class to consider thought experiments similar to
Galileo’s so as to reflect upon and reconsider their initial re-
sponse. This has nothing to do with any recapitulation theory,
but learners at nearly all levels of development can relive those
cultural conversations and if immersed in the relevant problem
space can become part of the creative process that transformed
the discipline in the first place.
An historical-cultural approach is a possibility in developing
creativity because it provides the opportunity for reflection on
the concepts of a formalised system. For example, as well as
applying the ‘rules’ of mechanics, pupils can understand where
those rules came from and develop a qualitative as well as a
quantitative understanding of those rules. The people that were
involved in the creation of these formalised cultural systems
were creative people and if we, as educators, are to work to-
wards developing this kind of creativity then it may be instruc-
tive to examine the creative person, not in terms of personality
traits, idiosyncrasies, dispositions and the like, but in terms of
what it is about them that is creative in terms of the subject
matter.
The Creative Person
According to Gardner (1994), the creative person has two
counteracting tendencies: one of scepticism that challenges
assumptions and rejects conventional wisdom and one that
explores a domain of knowledge comprehensively and deeply
to the point of exhaustion. This suggests that if all or nearly all
people have the potential of becoming creative then perhaps
these two counteracting tendencies should be encouraged in the
classroom. The first tendency implies giving the pupil the right
to demand a reason for something, for example, why is Py-
thagoras’ theorem how it is, and this can encourage critical
thinking. The final authority residing in the discipline itself:
that something is the case is not because the teacher, or indeed
the academic community, says so (the fallacy of the argument
by authority). The second tendency is encouraged by teaching
the domain of knowledge in a way that the class ‘makes it their
own’ (internalisation). Of course, an over packed curriculum
may make this very difficult, but developing a conceptual un-
derstanding is shorter in the long run than memorising ‘rule of
thumb’ procedures (Skemp, 1976).3 Creativity requires the
development of critical thinking, but the conceptual space has
to be thoroughly explored prior to any creative transformation
of it (Boden, 1994b).
The Generative Structure
Understanding creativity requires understanding the genera-
tive structure – the conceptual space – that makes creativity
possible. For Boden (1994b), the study of creativity requires
asking the sort question that involves the structure of a genera-
tive system, such as ‘Could that be proven using this method?’
‘Is that a sonnet?’ Generative systems include the rules for
chess, the axioms for a vector space, the axioms of the Newto-
nian system, the schemata of the impressionists, grammar, a
rhyming schema for sonnets etc. Questions about the generative
processes involved (e.g. what inspired X to come up with Y)
has to make reference to the generative system:
A merely novel idea is one that can be described and/or pro-
duced by the same set of generative rules as the other, familiar,
ideas. A genuinely original or radically creative idea is one that
cannot. It follows that the ascription of creativity always in-
volves tacit or explicit references to some specific generative
system. (Boden, 1994b, p.78. emphasis added).
The exploration of a conceptual space can lead to novel ideas,
but a novel idea can be said to be truly creative if it transforms
the conceptual space. According to Boden (1994b), constraints
are necessary for creativity and dropping a constraint can be a
general heuristic for transforming a conceptual space (her ex-
amples include Schoenberg dropping the home-key constraint
to create the space of atonal music and the dropping of Euclid’s
fifth axiom resulting in the development of non-Euclidean ge-
ometry). This dropping of a constraint is compatible with
Johnson-Laird’s (1988), ‘non-determinism’ – creativity as de-
pending on arbitrary choices but also on the criteria or con-
straints of the framework. This tension between freedom and
constraint in the development of creativity is a paradox facing
the educator: “On the one hand, they must train logical, rational
thought; on the other, they must foster fluency and freedom of
mind and the ability to tap the springs of the creative subcon-
scious. Mental discipline vs. mental freedom: in every class-
room a balance must somehow be struck. The more we know of
creativity, both its scope and its limits, the more enlightened
this balance will be.” (Kneller, 1965, p.79/80).4
In many respects the problem of creativity and the problem
of free-will are one and the same in that they can both be solved
together (Johnson-Laird, 1988). What gives us freedom of will
but which occurs in acts of creation, according to John-
son-Laird, is the ability to reflect about how we shall make a
decision and thus to choose at a meta-level of choice. For
Skemp’s, 1976, distinction between relational understanding and in-
strumental understanding in the learning of mathematics expresses this
eloquently.

S. ROWLANDS
52
Johnson-Laird, creativity on the one hand depends on arbitrary
choices as opposed to deterministic procedures yet, on the other
hand, creativit y in art is carried out within the conventions of an
existing genre and creativity within science normally occurs
within the constraints of an existing paradigm.
The reference to the generative system involved is crucial if
we are not to overstate any claims regarding generative proc-
esses. This is certainly true in the history of ideas. For example,
scientific theories often have consequences that were unfore-
seen and unintended by the original proponents of the theory
(Chalmers, 1982). Chalmers cites the example of Maxwell’s
introduction of the concept of a displacement current to Fara-
day’s concept of an electric field – the consequences of which,
that is, the prediction of radio waves, was not realised by Max-
well and not realised by the community until two years after
Maxwell’s death. This is not to undermine the creativity of such
a move, or indeed to undermine the genius of Maxwell, but just
how fruitful a creative move is can only be ascertained with
reference to the generative system itself. Another example is
Sacchieri, who, without realising the significance of what he
had done, had contributed to the proof of a number of theorems
to an entirely new type of geometry (Barker, 1964). In other
words, the study of creativity requires an analysis of the devel-
opment of the discipline, as a generative system, prior to any
analysis of the psychology of the creativity of the individual
concerned and prior to any analysis of the intersubjectivity of
the relevant community (Chalmers, 1982). Subjective or in-
ter-subjective considerations ought to be in the light of objec-
tive considerations – that is, the body of knowledge, the claims
that it makes, its logical structure, its theoretical objects etc.5
The Need for Educators to Understand the Genera-
tive Structure
Without understanding the structure of a conceptual space,
any notion of creativity becomes redundant or reduced to the
creation of novel ideas. Put another way, if educators do not
understand the nature of the discipline to be taught, then they
cannot advise on developing creativity within the context of the
discipline. For example, in the book ‘Creativity across the Cur-
riculum’, Hodgson (1980) describes science as beginning with
observation using enhanced senses from which information is
gathered and generalisations made. This empiricist description
of science has become quite prevalent amongst constructivist
educators (for example, the New Zealand science curriculum
discounts subject-matter competence for ‘making sense of the
world’. See Matthews, 1995) yet it offers no insight into sci-
ence and creativity. If science is based on observation then the
scientific revolution would not have began in the 1600’s but
within the great civilisations of thousands of years ago. The
hallmark of science is abstraction, involving the construction of
theoretical objects relevant to their respective domains, it is not
the observation of physical objects: “The semantic reference of
physical theories is not constituted by the objects perceived by
direct observation, but by ideal objects which, in their recipro-
cal links, form the so called ‘physical models’.” (Lombardi,
1999, p. 222). Indeed, observations are theory laden (Chalmers,
1982) and no scientist performs an experiment without some
theoretical point in mind (Toulmin, 1967). The implication is
that to understand how creativity is possible within science is to
understand the abstract nature of science and how this abstrac-
tion can model the real world (which is suggested by Hestenes’,
1992, argument that experimental ‘games’ are model deploy-
ment ‘games’). Indeed, Galileo was heavily criticised by his
Aristotelian colleagues because his theoretical ideal pendulum,
through which he elucidated the laws of nature, implied per-
petual motion and real pendulums do not conform to this (see
Matthews, 1994, chap. 6). The teacher therefore has to thor-
oughly understand the abstract nature of science if she is to
encourage creativity in science lessons. Consider the following:
The purpose of this chapter [‘Getting Meaning from Experi-
ences: the Child and Science’] is to persuade such teachers
[with little formal science education themselves] that (a) they
have already a considerable knowledge of the science required,
and (b) many of them have probably been teaching science,
without necessarily being aware of the fact, by identifying some
appropriate [sic] science objectives and illustrating ways in
which they may be achieved. (Hodgson, 1980, p. 133)
Hodgson is creating a fallacy and an excuse for science teach-
ers to be ignorant of science. A teacher who is not an expert in
the subject matter cannot begin to develop creativity amongst
the pupils with respect to the subject matter. Indeed, the teacher
should not only be an expert with respect to what is taught, but
should also be a philosopher and an historian of the subject and
be able to engage the pupils in a meta-discourse concerning the
concepts involved. That is a daunting challenge for the science
teacher, but there really is no other option. To regard ‘Big C
Creativity’ (creativity in the sense of Boden’s transformative
creativity) as ‘elitist’ and to push for Craft’s ‘Little C Creativ-
ity’ (‘democratic creativity’) in the teaching of science (e.g.
Haigh, 2003) may result in not actually teaching science.
Within the context of New Zealand’s science curriculum, Haigh
(2003) has downplayed the learning of content knowledge for
the ‘three Ps of science’ (problem posing, problem solving and
peer persuasion). The ‘three Ps of science’ is not unique to
science and fails to capture the nature of science. Haigh’s in-
vestigative approach in the learning of science becomes vacu-
4Interestingly, Kneller’s book was written during the post-reform period
in mathematics education when ‘modern mathematics’ and a progressive
education that encouraged a more child-centred pedagogy replaced the
rote learning of Euclid’s Elements. Kneller argued that the educational
system has neglected or suppressed the natural creativity of the young
but prophetically stated: Although in the past education has neglecte
d
creativity, it would be folly to go to the opposite extreme and extol it to
the detriment of mental discipline and mastery of subject matter. Advo-
cates of creativity who call, for example, for ‘‘permissive’’ and ‘‘crea-
tive’’ classrooms are really recommending that we subordinate formal
education to the development of creative thinking. Yet such a subordina-
tion is inimical not only to education but also to creativity itself, fo
r
successful creation demands both material for the imagination to wor
k
on and techniques for transforming that material into realized form.
Sound creativity, in short, presupposes mental discipline through mastery
of subject matter. As Whitehead has said, ‘‘Education must essentially
be a setting in order of a ferment already stirring in the mind’’, a setting
in order that proceeds from the nature of the subject matter itself.
(Kneller, 1965, p. 88)
5A sociological theory of knowledge that does not take the content o
f
knowledge into consideration would be unable to ascertain just how
creative an idea in the body of knowledge is or can be (a critique of the
sociology of knowledge of the Edinburgh Strong Programme can be
found in Phillips, 1998).

S. ROWLANDS 53
ous if investigation is independent of the theoretical constructs
of the relevant domain (such as gene, force, electron, latent
heat). Such an investigative approach is a form of what Mat-
thews (1995) describes as sensism (or what Carson and Row-
lands, 2005, describe as naïve empiricism). Another example is
given by Leach (2001), whereby pupils pose and discuss their
own questions entered into a database concerning the cockroach
observed in a lab. Questions such as ‘can roaches learn’ and
statements such as ‘I think that roaches can learn, since our
Madagascan giant roaches have learned that if they fall on their
backs they can wave their legs and we will help them” (p.187),
has more to do with making sense of appearances than it has
with scientific reasoning. There is no evidence of the teacher or
indeed of any pupil asking the question as to how we can dis-
prove such a statement. In this example, scientific understand-
ing appears to be a reduction to consensus established by the
pupils – without reference to any scaffolding by the teacher
with respect to the theoretical concepts of biology. This is not
science in the sense of a body of knowledge that has developed
since the scientific revolution. The ‘science’ of the teacher who
has little or no formal science, the very notion of a ‘children’s
science’, pupils posing their own questions about the world and
arguing for a consensus, data collection and trying to make
sense of the data, is not science. If there is no science, then
there can be no scientific creativity.
Creativity, Symbolic Thought and the
Educational Impli ca ti on
Creativity that involves original insight, argues Bohm (1998),
has to do with both rational insight as well as the formation of
new kinds of mental images. Rational insight involves the per-
ception of key questions that help to indicate some of the con-
tradictory or confused features of previous accepted general
ways of thinking (Bohm, 1998). Rational insight usually pre-
sents a challenge to what is considered ‘viable’ and if the po-
tential for original insight is to be promoted then knowledge
must not be seen as making sense of experience. This is simply
because the structure of a discipline may be totally different to
what the pupil sees as ‘viable’. The hundreds and if not thou-
sands of academic papers on pupil ‘misconceptions’, ‘precon-
ceptions’, ‘alternative conceptions’, or ‘intuitive ideas’ of sci-
entific concepts such as force and motion are testament to the
difference between ‘viability’ (e.g. a held and cherished ‘mis-
conception’ of force and motion) and knowledge (e.g. force as
defined and understood within the Newtonian system). If sym-
bol and metaphor are involved in the process of rational insight,
but knowledge becomes synonymous with ‘mental representa-
tion’ (e.g. von Glasersfeld, 1995), then that process will not be
understood (a critique of making synonymous knowledge with
mental representation can be found in Rowlands and Carson,
2001).
Metaphoric perception, according to Bohm and Peat (2000),
is fundamental to all science and in perceiving a new idea in
science the mind is involved in a similar form of creative per-
ception as when it engages a poetic metaphor. Whereas the
latter may remain relatively implicit, in science the meaning of
the metaphor has to be made explicit with a more ‘literal’ detail
(Bohm and Peat, 2000). Innovative ideas are the terms in which
theories are conceived and give rise to specific questions which
are articulated only in the form of these questions – a construc-
tive process in which symbolisation, the essential act of thought,
is the key to that process (Langer, 1957). For example, any
question concerning force and motion in Newtonian mechanics
has an implicit structure which determines the answer accord-
ing to the laws of motion:
Mechanics determines one form of description of the world by
saying that all propositions used in the description of the world
must be obtained in a given way from a given set of proposi-
tions – the axioms of mechanics. It thus supplies the bricks for
building the edifice of science, and it says. ‘Any building that
you want to erect, whatever it may be, must somehow be con-
structed with these bricks, and with these alone’. (Wittgenstein,
1974. proposition 6.341, p. 68).
To ask any question in mechanics is to invite an answer struc-
tured by the axioms, but it usually invites an intuitive response
(‘misconception’). In mechanics, force is the fundamental unit
of analysis and is symbolic of the interaction between objects.
The symbol is invariant to the many different forms of motion,
all of which can be explained according to the axioms. For
many students, however, force is contextualised according to
how the motion is perceived (e.g. for a ball going up, there
must be a force pushing it to overcome gravity). The shift from
having an intuitive response to force to understanding its sym-
bolic function (invariant to the different forms of motion) is a
constructive process aided by how the axioms apply. To think
according to those axioms requires a creative leap from intui-
tive ‘misconceptions’ to understanding impossible idealised
worlds structured by the axioms as a way of explaining the real
one. That shift can be said to be transformative, if only in rela-
tion to the lear ner. To teach in a way that creat es that shift will
have created the possibility for creative development in the
transformative sense.
Symbolic thought makes provision for the interpretation of
the facts, even those we don’t anticipate from unexpected
problems posed by nature (Cassirer, 1962). This is not to say
that every kind of thought can cover the whole of reality (as
Bohm and Peat, 2000, states: “if reality were ever to cease to
show new aspects that are not in our thought, then we could
hardly say that it had an objective existence independent of us”,
p.8). Nature has a habit of revealing the limitations of our
thought (exemplified in Newtonian mechanics with non-macr-
oscopic objects and objects at high velocities). Those limita-
tions, however, have to be reached, requiring the necessary
complex range of skills embedded in symbolic thought. The
creative act in the transformative sense may require disgorging
ensembles of symbolic thought from their original contexts to
other novel contexts within the parameters of their constraints
(which may also include the dropping of a constraint in the
transformation of a conceptual space).
There are many historical examples in science that can serve
to contextualize, for pupils, their intuitive notions and to under-
stand fundamentally what science is – that is, to understand the
intricate relationship between what is given empirically by
Nature, what is derived logically from reason and what has
been contributed by the artistic genius of some scientist in the
form of a strategic ‘convention’ of thought (a formula, a model,
an idealisation, an analogy, a metaphor or some other concep-
tual heuristics Carson and Rowlands (2005)). Immersing learn-
ers in the relevant problem space structured by the symbolic

S. ROWLANDS
54
thought that transformed the very discipline not only creates the
possibility for understanding the nature of science but also the
possibility to become creative within science.
Conclusion
Creativity is not the same thing as knowledge, but is firmly
grounded in it. What educators must try to do is to nurture the
knowledge without killing the creativity (Boden, 2001, p. 102,
emphasis added).
This article has taken a rather dim view of much of the cur-
rent literature on creativity, but there is much scope for opti-
mism, especially since the emergence of a number of articles
that view creativity in the context of the disciplines. For exam-
ple, according to Brinkmann and Sriraman (2009) some parts of
the creativity literature have identified five principles con-
cerned with creativity: The Gestalt principle (the need for
gifted learners to consider problems over a protracted time
period in the classroom); the free market principle (the taking
of risks and the need to defend solutions to problems); the
scholarly principle (the debating of certain historical ap-
proaches that helped transform the subject); the uncertainty
principle (how the history of the subject can reveal the uncer-
tainty of finding solutions) and the aesthetic principle (the aes-
thetic appeal of a ‘beautiful’ idea that unifies disparate ideas).
Although a minority, such research shows what creativity
should mean and therefore ought to be encouraged, as opposed
to the plethora of articles that appears to support the creativity
of the child but will in fact fail to unlock any truly creative
potential that the child – indeed any child – might have.
Section 1 attempted to show that the divorce made between
creativity and subject matter, that this bipolar view of creativity
and content will render creativity as a political plaything rather
than what the child, any child, may be capable of. Section 2
attempted to highlight the importance of intrinsic motivation in
relation to creativity and section 3 argued that educators and
teachers must understand the subject as a generative structure
and find ways to engage the class with that structure. This en-
gagement requires a meta-discourse between teacher and class
and section 4 elaborates on how creativity in the classroom
relies on making conscious symbolic thought as a cultural con-
vention. All this is quite consistent with Vygotsky’s point that
instruction has to proceed ahead of development if it is to lead
it, which means that the teacher who has embodied the subject
matter can ‘arouse the mind to life’ by creating a cognitive
response to the subject matter. This way, the learner can inter-
nalise the subject matter and become creative within it.
How can the teacher create a cognitive response? The answer
has to be ‘by engaging the pupil with the subject matter’ and
examples might include asking questions that demand a quali-
tative (conceptual) response (rather than simple recall or ‘guess
what’s in my head’ games), the giving of cues, prompts and
hints for consideration and the raising of ‘epistemological ob-
stacles’ that gave rise to the subject matter in the first place. In
short, engaging the pupil with a meta-discourse (for a full dis-
cussion of metacognition in this context see Rowlands, 2009).
Of course, certain concepts have to be defined, explained,
elaborated etc. but the learner has to become engaged with the
way these concepts are related. The subject matter cannot be
simply ‘transmitted’ as such, but constructed by the learner
under the guidance and direction of the teacher. Independently
of specific training programmes, the facilitation of creativity in
the classroom will be dependent on how the teacher structures
the educational environment that makes it conducive to creativ-
ity (Diakidoy & Kanari, 1999, emphasis added) and the struc-
ture should include the opportunity for meta-discourse – in-
volving the nature of the subject and its history.
References
Amabile, T. M. (1983). The social psychology of creativity. New York,
NY: Springer-Verlag.
Adams, L. J. ( 1988). The care and feeding of ideas. L ondon: Penguin.
Adey, P., & Shayer, P. (1994). Really raising standards: Cognitive
intervention and academic achievement. London: Routledge.
Barker, S. F. (1964). The philosophy of mathematics. New Jersey:
Prentice-Hall.
Bohm, D. (1998). On creativity. London: Routledge.
Bohm, D. and Peat, F. D. (2000). Science, order, and creati v i t y (Second
ed.) (first published in 1 98 7). London: Routledge.
Bruner, J. (1974). Beyond the information given. London: Allen and
Unwin.
Boden, M. A. (1994a). Introduction to dimensions of creativity (M. A.
Boden, Ed.). London: M IT.
Boden, M. A. (1994b).What is creativity. In M. A. Boden (Ed.), Di-
mensions of creativity. London: MIT.
Boden, M. A. (2001). Creativity and knowledge. In A. Craft, B Jeffrey
and M. Leibling (Eds.), Creativity in education. London: Continuum.
Brinkmann, A., & Sriraman, B. (2009). Aesthetics and creativity: an
exploration of the relationships. In B. Sriraman and S. Goodchild
(Eds.), Relatively and philosophically ernest: festschrift in honor of
paul ernest’s 65th birthday. Charlotte, NC: Information Age Publish-
ing.
Carson, R., & Rowlands, S. (2005). Mechanics as the logical point of
entry for the enculturation into scientific thinking. Science & Educa-
tion, 14, 473-492. doi:10.1007/s11191-004-1791-9
Cassirer, E. (1962). An essay on man: An introduction to a philosophy
of human culture (first published in 1944). New Haven: Yale Uni-
versity Press.
Chalmers, A. (1982). What is this thing called science? （2rd ed.）Milto
n. Keynes:The Open University Press.
Craft, A. (2001a). Little c creativity. In A. Craft, B Jeffrey and M.
Leibling (Eds.), Creativity in education. London: Continuum.
Craft, A. (2001b). An analysis of research and literature on creativity
in education. Report Prepared for the Qualifications and Curriculum
Authority.
http://www.ncaction.org.uk/creativity/creativity_report.pdf
Craft, A. (2003). The limits to creativity in education: Dilemmas for the
educator. British Journal of Educat ional Studies, 51, 113-127.
doi:10.1111/1467-8527.t01-1-00229
Cropley, A. J. (1971). Creativity. London: Longman Education Today
Series.
Diakidoy, I. N., & Kanari, E. (1999). Student teachers beliefs about
creativity. British Educational Research Journal, 25, 225-243.
doi:10.1080/0141192990250206
Dunbar, K. (1997). How scientists think: on-line creativity and concep-
tual change in science. In T. B. Ward, S. M. Smith and J. Vaid (Eds.),
Creative thought: An investigation of conceptual structures and
processes. Washington DC: American Psychological Association .
doi:10.1037/10227-017
Gardner, H. (1994). The creators patterns. In M. A. Boden (Eds.), Di-
mensions of creativity. London: MIT.
Gibson, H. (2005). What creativity isn’t: the presumptions of instru-
mental and individual justifications for creativity in education. Brit-
ish Journal of Education a l S t udies, 53, 148-167.
doi:10.1111/j.1467-8527.2005.00288.x
Gombrich, E. H. (1960). Art and illusion. London: Ph aidon.

S. ROWLANDS 55
Haigh, M. (2003). Fostering creativity through science education: A
case for investigative practical work. Paper presented at the British
Educational Research Association 2003 conference. Edinburgh:
Heriot-Watt University.
Hennessey, B. A., & Amabile, T. M. (1988). The conditions of creativ-
ity, In R. J. Sternbe rg (Eds.), The nature of creativity: contemporary
psychological perspectives. Cambridge: Cambridge University Press.
Hestenes, D. (1992). Modeling games in the newtonian world. Ameri-
can Journal of Physics , 60, 732-748.doi:10.1119/1.17080
Hodgson, P. (1980). Getting meaning from experience: The child and
science. In M. Poo le (Ed.), Creativity across the curriculum. London:
George Allen and Unwin.
Jaworski, B. (1994). Investigating mathematics teaching: A construc-
tivist enquiry. London: Falmer.
Jeffrey, B., & Woods, P. (2003). The creative school: A framework for
success, quality and effectiveness. London: RoutledgeFalmer.
doi:10.4324/9780203437223
Kneller, G. F. (1965). The art and science of creativity. London: Holt,
Rinehart and Winston, Inc.
Johnson-Laird, P. N. (1988). Freedom and constraint in creativity. In R.
J. Sternberg (Eds.), The nature of creativity: contemporary psycho-
logical perspectives. Cambridge: Cambridge University Press.
Langer, S. K. (1957). Philosophy in a new key: a study in the symbol-
ism of reason, rite, and art (first published in 1942). Mass: Harvard
University Press.
Leach, J. (2001). A hundred possibilities: creativity, community and
ICT. In A. Craft, B. Jeffrey and M. Leibling (Eds.), Creativity in
education. London: Continuum.
Lombardi, O. (1999). Aristotelian physics in the context of teaching
science: A historical-philosophical approach. Science & Education, 8,
217-239. doi:10.1023/A:1008641526822
Lucas, B. (2001). Creative teaching, teaching creativity and creative
learning. In A. Craft, B Jeffrey and M. Leibling (Eds.), Creativity in
education. London: Continuum.
Lytton, H. (1971). Creativity and education. London: Routledge and
Kegan Paul.
Matthews, M. (1980). The marxist theory of schooling: a study of epis-
temology and education. Sussex: Harvester Press.
Matthews, M. (1994). Science teaching: the role of history and phi-
losophy of science. New York: Routledge.
Matthews, M. (1995). Challenging nz science education. New Zealand:
The Dunmore Press.
Morris, E. (2002). Creativity in education. Speech presented at the
national campaign for the arts/ national union of teachers creativity
in education confere nce.
http://www.artscampaign.org.uk/campaigns/education/EMorris.html
NACCCE (1999). All our futures: creativity, culture and education,
national advisory committee on creative and cultural education.
London: DFEE.
Parnes, S. J. (1970). Education and creativity. In P. E. Vernon (Ed.),
Creativity. Harmondsworth: Penguin .
Phillips, D. C. (1998). Coming to terms with radical so cial constructiv-
isms. In M. R. Matthews (Ed .), Constructivism in science education:
a philosophical examination. Dordrecht: Kluwer Academic Publish-
ers.
Rowland, T. (2009). Geometry: Tales of elegance and love. In B.
Sriraman and S. Goodchild (Eds.), Relatively and philosophically
ernest: festschrift in honor of paul ernests 65th birthday. Charlotte,
NC: Information Age Publ ish in g.
Rowlands, S. (2009). The importance of cultivating a meta-discourse in
deliberate support of metacognition. In C. B. Larson (Ed.), Metacog-
nition: new research developments. New York: Nova.
Rowlands, S., & Carson, R. (2001). The contradictions in the construc-
tivist discourse. Philosophy of Mathematics Education Newsletter, 14.
http://www.ex.ac.uk/~PErnest/pome14/rowlands.htm
Rowlands, S. Graham, E., & Berry, J. (2010). Problems with fallibilism
as a philosophy of mathematics education. Science & Education,
online (and in print).
Skemp, R. R. (1976). Relational understanding and instrumental under-
standing. Mathemat i cs Teaching, 77, 20-26.
Toulmin, S. (1967), The Philosophy of Science. First published in 1953.
London: Hutchinson University Library.
Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing
and understanding. London: Falmer.
doi:10.4324/9780203454220
Vygotsky, L. (1978). Mind in society. Mass: Harvard University Press.
Ward, T. B., Smith, S. M. & Vaid, J. (1997). Conceptual structures and
processes in creative thought. In T. B. Ward, S. M. Smith and J. Vaid
(Eds.), Creative thought: an investigation of conceptual structures
and processes. Washington DC: American Psychological Associa-
tion.
Wittgenstein, L. (1974). Tractatus logico-philosophicus, (first pub-
lished, 1922). Londo n : R o ut l e dge and Kegan. Paul.
Wragg, E. C. (1999). An introduction to classroom observation, Lon-
don: Routledge and Falmer.