I ntuition and I ntegration:
Insights from intuitive students
Presented for the MPhil in Educational Research Richard Brock, Homerton College.
Supervised by Dr Keith Taber.
July 2006
20,197 words
(Exclusive of tables of contents, footnotes, tables, appendices, reference list and bibliography.)
" T he roads by which men arrive at their insights into celestial matters seem to me almost as worthy of wonder as those matters
in themselves." (Kepler, quoted in Gentner et al. 1997, p. 403)
L
ist of tables and figuresF
iguresFigure 1: Argument structure...
Figure 2: Cognitive conditions for meaningful learning...
Figure 3: The deliberate and tacit systems...
Figure 4: The blending process...
Figure 5: Partial identity in the higher order (meta) structure of two domains...
Figure 6: Relationships constructed between the meta-structures of two domains...
Figure 7: The properties of intuition...
Figure 8: Ausubel's model of analytical and intuitive thinking...
Figure 9: An illustration of Bastick's model of intuition...
Figure 10: Levels of redundancy...
Figure 11: Conceptual halos and conceptual slippage of the concept of 'a'...
Figure 12: Synthesis of concepts that construct the model of intuition...
Figure 13: Models of methodological structure...
Figure 14: Analogy between philosophical and substantive arguments...
Figure 15: Analogies between the sociological imagination and grounded theory...
Figure 16: The relationship between conceptual structure and methods...
Figure 17: Outline of the stages in the research method and questions asked...
Figure 18: The process of drawing links between two concept maps...
Figure 19 : Outline of concepts emerging from analysis...
Figure 20: Representation of David's model...
Figure 21: Colearner's analogies...
Figure 22: The overlap between meta-structures and possible paths between them...
Figure 23: Fran's Concept Map...
Figure 24: Chris's Concept Map...
Figure 25: David's Concept Map...
Figure 26: Ahmed's Concept Map...
Figure 27: The meta-structure of Ahmed's concept map...
Figure 28: Processes of intuition...
Figure 29: Meta-structures of teaching and concepts...
T
ablesTable 1 : Levels of metacognitive process...
Table 2 : Target and example base concepts for concept mapping exercise...
Table 3 : Code Descriptions...
A
bstractThis investigation is a case study of physical intuition. It aims to investigate how some students appear to find learning complex concepts in physics obvious and easy. Scientists report intuition as central to their work yet it is poorly understood and seldom encouraged in the
classroom. A model of intuition as perception of meta-structures between concepts was constructed through the use of the literature of Gestalt psychology, analogical structure mapping and mental fluidity. Using this model, a new tool to investigate conceptual integration based on the connecting of two concept maps was developed. The empirical part of the investigation consisted of interviews and the use of the concept mapping tool with six sixth form students who had been identified by teachers as intuitive. The research was positioned in a constructivist paradigm and grounded theory influenced the collection and analysis of data. Most of the students reported intuitive experiences and a catalogue of learning skills was begun including the ability to deconstruct ideas and to make links to the real world. The process of analogy construction was examined and the ability to perceive underlying meta-structures noted as important. Finally, ways of encouraging intuitive learning in the classroom are proposed.
A
cknowledgementsI am very grateful to Keith Taber for his advice encouragement and support throughout this project.
The research would not have been possible without the assistance of Steve Martin and Jacqui Smith.
My thanks also to Tom, Ute and Martin Brock, Jenny Symonds and Dunstan Roberts for their advice and assistance.
C
ontentsD
eclarationA
bstractA
cknowledgementsC
ontents1.0 I
ntroduction1.1
A
necdotal observations1.2
T
he foundations of the research program 1.3T
he Argument2.0 L
iterature Review: Building a model of intuition- from 'intuition' to intuition 2.1L
earning and Insight2.2
W
hat is learning?2.3
H
ow do we learn?2.4.0
'E
ureka!' Experiences and Insight Learning 2.4.1O
ther Models of Insight2.4.2
G
estalt Theories2.4.3
C
riticism of gestalt theories2.4.4
I
nsight: 'Nothing Special' or multiple processes?2.4.5
S
ummary of ideas about insight learning 2.5.0T
heories of information processing2.5.1
C
aveats to cognitive models 2.5.2A
ssimilating, Accommodating.2.5.3
C
onceptual Integration, Conceptual Blending 2.5.4W
hat is a concept?2.5.5
C
oncept Change: Revolutionary Intuitions?2.5.6
A
nalogy and the construction of conceptual connections 2.5.7T
he wider conceptual architecture- higher order patterns 2.5.8L
earning from the experts, learning about the experts 2.6A
bove and beyond: thinking about thinking2.7.0
S
uddenly, It all makes sense: Intuition2.7.1
I
ntuition in Science: The metaphysical bathwater 2.7.2E
ducating Intuitions?3.0 T
he nature of research3.1
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ntuition in research: where ideas come from 3.2.0C
onstructing the case3.2.1
S
electing the case: sampling and comparison 3.2.2M
aking the Case: Introducing Rigour 3.2.3E
ntering the Case3.3
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esearch as dialogue: the ethical imperative3.4.0
M
ethods: tools for reflecting cognitive structure- an new approach for capturing intuition 3.4.1D
escribing thoughts3.4.2
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he interview: Access to metacognition?....3.4.3
T
hinking aloud or Listening to thoughts?3.4.4
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ketching the mind- Concept Maps 3.4.5T
ranscribing and Analysis Procedure4.0 F
indings and Discussion4.1
D
escriptions of the Intuitive Process 4.2T
oward a Catalogue of Learning Skills4.3
U
nderstanding the Principles: Laying the Foundations 4.4B
reaking Down into Steps: Building Bridges4.5
M
aking Links to the 'Real World' 4.6M
ental Fluidity?4.7
A
Restructured Problem 4.8B
en's Model and David's Model.4.9
C
olearners' Analogies4.10
F
eedback Between Analogical Domains 4.11W
atching concepts slip.5.0 E
ducational Implications6.0 C
ritique, Evaluation and suggested Improvements and Future Directions.7.0 C
oncluding Remarks8.0 R
eferences9.0 A
ppendices9.1
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etter requesting permission 9.2I
ntuitive definitions9.3
I
nterview Prompts 9.4S
timuli9.5
I
nterview Transcripts and Concept Maps 9.5.1A
hmed9.5.2
B
en9.5.3
C
hris9.5.4
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avid9.5.5
E
dward9.5.6
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ran1.0 I
ntroduction1.1
A
necdotal observationsI have spent many hours in physics classrooms: in schools and universities, as a student and as a teacher. During this time, I have watched many people engaged in the act of learning physics.
One observation in particular struck me: for some students, physics seems effortless, obvious and intuitive, whilst for others it is obscure to the point of meaninglessness. For a small group of people, the abstract laws of physics seem to make automatic sense; they can understand new ideas almost instantly, as if they knew them already. They may find solutions to some problems without apparent conscious effort and solve them rapidly and easily.
I am extremely fortunate to have an older brother who belongs to this first category. As well as being a talented physicist, my brother was also a gifted and patient teacher. He taught me how to be critical of new information, to take concepts apart and to make sense of them in the context of what I already knew. His teaching made me wonder whether it is possible to foster the skill of intuitiveness in physics.
In contrast to such 'natural' learners, there also exists, certainly at the school level, a group of students who appear to find physics virtually meaningless. These students find what they are asked to learn 'makes no sense' or is 'too difficult.' The symbols and equations they are taught hold little meaning for them; the more motivated survive exams by rote learning and sticking to learned problem solving heuristics. Culturally this attitude to physics seems to have become acceptable and widespread. In my own experience, introducing myself as a physics teacher will often elicit the response: 'Oh, physics it's so hard isn't it? I never got it at school.'
1.2
T
he foundations of the research programThese observations about the nature of physics learning inspired my initial thinking about this research program. Its aim is to try to understand why a distinction should exist between students who find physics natural and those who do not. The investigation will look, in detail, at specific learners who seem to fall into the first category of students and try to understand how they learn. For the moment, the two groups are loosely defined and terms such as 'intuitive' and 'natural' are used in their everyday senses. It is also worth noting that I will use the Gestalt psychology term insight synonymously with intuition. For a justification of this decision section 2.7. In order to build a more rigorous definition of such concepts and tighten the focus of the research it is necessary to examine what research and theories have already been constructed in this area.
1.3
T
he argumentThis paper will attempt to build a model of the 'intuitive' process, however until this is achieved in section 2.7 'intuitive' and 'intuition' will be used in quotation marks indicating their everyday usage given above. The paper will begin by examining theories of learning: first the insight theories of Gestalt psychologists and then theories of conceptual integration of cognitive science (See figure 1). This will lead to a discussion of the role of meta-structures in learning and the development of a model of intuition based on the perception of higher order patterns. The role of intuition in science and science education will be investigated before the presentation of results.
Figure 1: Argument structure (Relevant sections given in brackets)
2.0 L
iterature Review: Building a model of intuition- from 'intuition' to intuition.2.1
W
hat is learning?In order to understand the process of how students learn, one must define what is meant by learning. However, definition of learning is difficult. Any analysis of learning must take into account epistemology, an examination of the nature of knowledge. Indeed the study of learning may be considered as experimental epistemology (Bower and Hilgard 1981, p1). Given the wide range of competing theories of knowledge, the nature of learning is necessarily contested (Mower and Klein 2002, p1). Reviewing definitions of learning Mower and Klein claim:
'All of these definitions seem to share the common theme that learning is a relatively permanent change in the probability of exhibiting a certain behaviour resulting from some prior experience' (Italics added for emphasis) (Ibid, p2).
I have chosen this definition as it makes no assumptions about cognitive processes, it usefully leaves a space between cause and effect for the researcher to fill. Many examples of definitions from the behaviourist or cognitive research programs necessitate the acceptance of various theoretical structures. For example, although I will use his work in other places, Ausubel's definition limits learning to intergratory processes of conceptual learning (Ausubel 2000, p1). The definition above implies a broader focus on behaviours not just events in the hypothetical structures in a learners' mind.
2.2
H
ow do we learn?The definition given above, defines learning as the 'black box' that links changes of
behaviour with prior experience. This 'black box', contains the phenomena, processes and effects that need to be elucidated in order to understand how it is that students come to an understanding of physics. Different research programs have offered different constructs to model what the learning 'black box' may contain. In the constructionist paradigm, meaning generation is a collaborative process (Patton 2002 , p97). It is tempting to see the marked differences of behaviourism, Gestalt and cognitive theories as being incommensurable. However when considering something as complex and subtle as the human mind, it is unlikely that a single model will map its processes perfectly. The anarchist philosopher of science, Paul Feyerabend writes: 'Proliferation of theories is beneficial, while uniformity impairs its critical power' (Feyerabend 1993, p24). I will therefore critically consider a number of different theories of learning and consider how they may inform an understanding of the observed effects discussed in the introduction.
2.3
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hrondike's connectionismOne of the pioneers of learning theory, whose theories dominated the field in America for nearly half a century, is Edward Thorndike (Bower and Hilgard 1981, p21). One of his early experiments involved placing a hungry cat in a puzzle box, from which it could escape only by learning to manipulate some kind of unlatching device that opened the door to the box. He constructed 'learning curves' showing how the time taken for the cats to escape varied with each successive trial. He interpreted the curves as showing no sudden drops in the time taken to escape, rather they showed a gradual decline. This conclusion was taken to support the theory that animals do not have the understandings or insights which he believed to be an important part of human learning, rather they merely use trial and error. He proposed that the best way to promote the occurrence of insights in a learner is to '...teach them many connections relevant to the problem' (Bower and Hilgard 1981, p47). Thorndike developed a very crude theory of the way insights occur by 'habitual associations and analogies' (Ibid p48). These associations between sense impressions and responses came to be known as 'bonds' or 'connections' and hence Thorndike's system has sometimes been referred to as 'connectionism' (Ibid p21).
2.4
'E
ureka!' experiences and insight learningPutting aside ethical considerations, Thorndike's experiments were criticised by his contemporaries for the artificiality of their setting; the results being merely an artefact of the contrived nature of the trials (Chance 1999, p433). Köhler criticised the puzzle boxes arguing that if the animals could not see the entire apparatus, they could not be expected to use their intelligence to solve the problem (Köhler 1927). Köhler carried out his own experiments on problem solving using chimpanzees. He observed that the apes might suddenly 'see' the solution to a problem. He called such sudden moments 'Eureka!' experiences and developed the idea of insight learning (Bower and Hilgard 1981, p301).
In his later work in the Gestalt research program Köhler discusses the nature of being a physicist: 'The exactness of definitions in physics cannot result from the alleged fact that in science definitions are independent of direct experience; for there is no such independence' (Köhler 1947, p27). This is an early acknowledgement of the important role personal experience and construction of meaning play in the sciences. Köhler defines insight as the direct awareness of determination and discusses the absence of the concept from psychological theories (Ibid, p343). He builds a theory of insight that, like Thorndike's, is based on connections but doesn't have the rigidity of the 'bond' concept: 'Interaction in physics, we remember, depends throughout on the “characteristics in relation” of the interesting facts' (Ibid, p121). Whilst Köhler's work is undeniably prescient, its vague nature, and poorly defined terms leave the reader wondering how much substance is actually present in the work. Furthermore much of the theory is based on speculation and anecdote; there is
little rigorous empirical support for his relations model.
2.4.1
O
ther models of insightEven before Köhler's research with chimpanzees, several writers had touched on the idea of discontinuous leaps in learning. In the early decades of the twentieth century, Otto Selz, developed a schematic anticipation model of problem solving (Mayer 1995, p8). Selz argued that insight occurs when a new component is integrated into a larger system or complex; a problem is a gap in a set of coherent information and insight is the process by which it is filled. However as Mayer reports, 'Selz's wrtings are vague and his research methodologies are imprecise by modern standards' (Mayer 1995, p9). Another of these early theories is described by Metcalfe in the foreword to 'The Nature of Insight' (1995). Donald Hebb's belief that insight was central to comprehension and the extraction of meaning and led him to develop a complex model of the process (Metcalfe 1995, pxii). Hebb
described how concepts may have some elements that are central and necessary, whereas others can be described as 'fringe elements' (Ibid, pxii). Two concepts may have fringe elements in common, but this commonality may not be due to past learning or memory, but simply to some inherent
similarity. If these two concepts emerge into consciousness, the overlap of the similar fringe elements may result in a restructuring of the relationship between those elements. Hebb described this process as an insight and believed it to be the most common form of adult learning. Though, as with many cognitive theories, it is impossible to say whether Hebb's ideas are empirically plausible, they certainly provide an exciting base for further research (Metcalfe 1995, pxii).
2.4.2
G
estalt theoriesKöhler, with Wertheimer, is seen as one of the pioneers of Gestalt psychology (Bower and Hilgard 1981, p301). A common theme in the Gestalt theories is the wholist nature of perception, that is the form emerges as result of the relationship between the parts. The 'law' of prägnanz (meaning 'compact and significant') states that perceptual organisations tend to form 'good Gestalts' or 'good figures', for example the famous reversible figure that can be made to switch between the Gestalts of two faces or a goblet. (Ibid, p302). The theorists make the assumption that the laws of perception are applicable to learning and memory, and hence coming to understand something is the sudden process when the relationships between ideas form a good Gestalt. Wertheimer proposed the twin concepts of reproductive thinking, based only on existing associations and productive thinking, which involves insights beyond what is already known (Sternberg 2006, p407). The Gestaltists hypothesised that insights might be some form of unconscious leaps in thinking or short-circuitings of the normal reasoning process but provide few details and less evidence to back up such ideas (Ibid p408).
Wertheimer also applied Gestalt principles to memorisation. He claimed that people learned
most effectively by seeing or understanding the principles or patterns that structured the information (Bower and Hilgard 1981, p317). That is they grasped the fundamental Gestalt of the information.
Rote memorisation is therefore only to be adopted as a strategy of last resort when natural organising factors are absent. This important role of meaning and structure in learning was demonstrated by Katona in 1940 (Svensson 1997, p60). He distinguished between learning by memorising and learning by organising. Volunteers were asked to learn strings of digits, if the volunteer grasped the underlying structure of the string, then the string was easier to memorise and less prone to decay than a string which was perceived to be random. For example the string:
581215192226 becomes much easier to reproduce if one notices the difference between the figures oscillates from 3 to 4 (Ibid, p61). This may be the kind of process occurring during intuitions, where the perception of a connection can suddenly make previously meaningless information, almost trivially obvious. However, in the domain of physics, patterns, and the way we come to understand them, are not as straight forward as in this arithmetical example. Even so, it may be that 'intuitive' students are able to 'see' certain higher level patterns, that make the concepts they are learning seem as obvious as the number sequence above. This ability to switch to a different level of thinking will be discussed further in the section on metacognition.
2.4.3
C
riticism of Gestalt theoriesA brief review of Gestalt and pre-Gestalt models of insight learning has brought up some provocative ideas, but the entire gestaltist research program has frequently been criticised for its lack of methodological rigour (Mayer 1995, p26). It is claimed that Gestalt theorists did not provide convincing evidence for any of their models nor did they produce a coherent account of cognition (Sternberg 2006, p408). However Mayer argues that such criticism is mistaken - the real challenge lies in reformulating and investigating in more rigorous ways the questions and ideas that have been inherited from the Gestalt school (Mayer 1995, p27). Indeed cognitive psychologists (for example Davidson and Sternberg) and even neuroscientists (Dehaene et al 1999 and Vogel 1997) are now studying such 'soft' phenomena as insight and intuition.
2.4.4
I
nsight: 'Nothing special' or multiple processes?Some researchers claim that the Gestalt school's failure to convincingly theorise insight is because no separate, independent process of insight exists (Weisberg 1995, p157; Davidson 1995;
p127 and Sternberg 2006, p408). Such writers claim that the failure to understand insight occurs because insight is 'merely an extension of ordinary processes of perceiving, recognising, learning and conceiving' (Davidson 1995, p127). However, it is dangerous to assume that because we cannot understand a process it does not exist: some research suggests that psychologists' failure to isolate insight is becuase it involves more than one single, discrete process.
The three process view of insight (Davidson 1995, p127; Sternberg 2000, p409) proposes that there are three kinds of insight, correlating to three different processes: Firstly, selective encoding insights occur when a person suddenly sees a pattern or patterns in information that previously had not been obvious. This seems to be similar to the work of Katona discussed above. The second type of insights are selective comparison insights. These involve the novel integration of new elements of information with knowledge acquired in the past. The final category of insight is selective
combination, whereby a relationship between units of information that have already been integrated by selective comparison, is suddenly discovered. Davidson and Sternberg (Davidson 1995, p128;
Sternberg 2000, p411) mention the importance of analogy in these processes and this will be discussed in greater detail below. The division of insight into three processes may make it easier to observe the separate processes in action (Davidson 1995, p132) but it does not of itself advance our understanding of how these processes occur.
2.4.5
S
ummary of ideas about insight learningCommon sense experience and the Gestalt literature suggest that in certain environments learning can be experienced as a sudden phenomena: an insight occurs. Many of the models of this process have in common that the integration of a new concept causes a pattern to become suddenly clear or a new meaning obvious. The perception of patterns and the creation of links between concepts is central. These ideas will play a role in the construction of a model of intuition later in this paper. None of the theories however give any details of the mechanisms that drive such cognitive revolutions. It will therefore be instructive to examine in detail theories concerned with the way new information is processed.
2.5
T
heories of information processingIn the 1950's researchers including Abby, Walter and Hoffman began to construct robots that were designed to mimic certain principles of animal behaviour (Bower and Hilgard 1981, p353). This field became known as 'robotology'- the designing of artificial machines to further our understanding of cognitive processes (Ibid, p353). These constructions influenced cognitive psychology to the extent that models of cognition began to be expressed in the form and language of computer programs.
Stimuli could be considered as inputs, behaviour as an output with a logical set of processes acting on the data as it is converted from input to output. Two examples of such 'programs' are shown below:
2.5.1
C
aveats to cognitive modelsBefore considering such cognitive models it is worth sounding a note of caution. Cognitive psychologists and educators are naturally keen to understand how and why thinking and learning processes function. Sternberg claims this occurs in a Popperian manner: models of cognitive
processes are proposed, subjected to a test and then either accepted, rejected or modified (Sternberg 2006, p3). However the new research program of post-cognitive psychology is encouraging
researchers to examine to what extent such practices are valid (Potter 2000). This program has arisen out of Wittgenstein's attack on the notion that private mental objects can correlate with public
Figure 2: Cognitive conditions for meaningful learning (Adapted from Figure 2: in Meyer 1992 p248)
Figure 3: The deliberate and tacit systems.(Adapted from Exhibit 15 The dotted lines indicate functions of the tacit system. Hogarth 2001 p196)
cognitive theories (Ibid, p32). These ideas have been reinterpreted by John Rogers Searle in the Chinese room thought experiment (Searle 1980). Searle pictured a person in an isolated room, who sorts the Chinese characters he receives into a meaningful order by following a set of formal procedures. The person has no 'understanding' of Chinese yet the characters emerge in meaningful order. This analogy forms the basis of Searle's argument that an information processing model is insufficient to fully understand the processes of consciousness.
It is important to remember that processing models are exactly that - highly simplified, partial models of complex processes. Whilst they may never be able to clarify what consciousness is, formulating some processing hypothesises may produce some insight into the nature of learning. If these produce new and effective teaching approaches then, it can be argued, their philosophical limitations can to some extent be ignored.
2.5.2
A
ssimilating, accommodatingOne of the earliest, but still one of the most influential investigators of knowledge acquisition was Jean Piaget. The methodological weaknesses of his work, at a time when psychology was still an evolving science are frequently noted: little information on the children interviewed is reported, he relied heavily on cross-sectional rather than longitudinal data and he placed a strong emphasis on the child's failures rather than successes (Smith et al 2003, p412). Despite these shortcomings Piaget provided a powerful model of cognitive growth that has important implications for education.
Piaget proposed that ideas, beliefs, thoughts and similar mental operations were organised in sets, he called schema. Throughout life, the invariant quality of intelligent thought, is the organisation of schema and their adaptation through the processes of assimilation and
accommodation (Ibid, p391). Organisation is the inborn capacity to manipulate and combine existing schema; more complex operations are acquired by the combination of simpler, existing schema. As well as manipulating the schema themselves the organism has the ability to adapt schema through the complimentary processes of assimilation and accommodation. Such adaptation is seen as mechanism for creating equilibrium between the objects and concepts in the mental and physical environments.
The first process, assimilation, allows an organism to 'take in' a new experience and fit it into an existing schema. Accommodation, the balancing process, modifies an existing schema to fit with the nature of some new stimuli. Learning therefore could be seen to advance through the twin channels of consolidation and modification; through schema equilibrium and disequilibrium. One may speculate whether these two processes relate to analytical and 'intuitive' thinking. The former being a gradual accumulation of knowledge, the latter a sudden and discontinuous 'jump' which results in the creation of new meaning. Piaget's ideas have been developed by Ausubel into a theory of meaningful learning (c.f. Figure 1 above):
'Meaningful reception learning is inherently an active process because it requires,
at the very least, (1) the kind of cognitive analysis necessary for ascertaining which aspects of existing cognitive structure are most relevant to the new potentially meaningful material; (2) Some degree of reconciliation with existing ideas in cognitive structure-that is, apprehending similarities and difference, and resolving real or apparent contradictions, between new and already established concepts and propositions; and (3) reformulation of the learning material in terms of the idiosyncratic intellectual background and vocabulary of the particular learner.' (Ausubel 2000, p5)
If these three processes are indeed those that underlie meaningful learning (surely an oversimplification) then it may be that 'intuitive' learners are those that are particularly gifted in controlling such mechanisms. This idea will be discussed further below in an examination of learning skills, metacognition and conscious versus tacit thinking. Of the three mechanisms described above perhaps the most powerful is the second, the reconciliation of new and existing ideas; it lies at the heart of what it means to understand an idea.
2.5.3
C
onceptual integration, conceptual blendingThe nature of scientific knowledge is a 'highly linked and largely coherent body of knowledge' and therefore education should encourage learners to develop 'integrated conceptual structures' (Taber 2005, p1-2). Integration of new concepts to form a coherent conceptual organisation is vital to scientific progression and also to science education.
The cognitive operations of conceptual integration have received much attention in the domain of linguistics. In particular the 1998 'blending' theory of Fauconnier and Turner is an attempt to understand the way in which the human mind can integrate new concepts (Fauconnier and Turner 1998). They propose that conceptual blending is composed of three processes, composition, completion and elaboration (see figure 4).
Composition involves importing elements from the two input concepts into a new integrated concept. During completion additional structure is 'recruited' into the blend; Grady discusses that this pattern completion may arise out of 'spreading activation' (Grady 2000, p339).
Spreading activation is the process by which 'the activation of one neural ensemble leads to the activation of another' (Ibid, p338). For example, in trying to recognise an object, below a certain threshold we might just perceive something silver, or something round, but once a threshold of low- level cues is reached the higher-level of object recognition is triggered. Grady writes that 'activation propagates from structures underlying the conceptualisation' and may 'subsequently, in a way which may feel instantaneous and effortless, a more complex pattern is evoked and superimposed' (Ibid, p338). This appears to be very much like the sensation experienced during insight. Indeed pattern completion seems to be similar to the Gestalt concept of noticing 'good Gestalts.' The final of the three blending processes is elaboration, where new connections and linkages are formed to the newly produced blended concept.
2.5.4
W
hat is a concept?Before considering how conceptual learning has been theorised in the science education literature, it will be useful to consider what a concept is. Concepts have been defined as: 'cognitive entities:... the furniture of the conscious mind' (Pines 1985, p108), 'packets of meaning' (Ibid, p108)
Concept 1 Concept 2
Concept 1 Concept 2
Blended Concept
d1 d2
d1 d2
d
Figure 4: The blending process. The riddle of the Buddhist monk: A monk begins walking up a mountain at dawn, meditates at the top and begins to walk back down at dawn of the following day. Prove that there is a place on the path he occupies at the same time on both days. The solution arises from imaging a situation where two monks walk up and down the
mountain on the same day. This imagined situation is formed by a blend of elements of the two different days. (Fauconnier and Turner, 1998)
a) Composition
b) Completion
and 'internal representations or notions about things people etc.' (Miles and McLetchie, 2004). In fact, some commentators would argue that the concept in itself has no real substance rather it is 'the theoretical point where meaningful relationships converge' (Pines 1985, p109) or 'all concepts begin with relationships' (Miles and McLetchie, 2004). Concepts can be mapped onto the real world processes as follows: Gaining a concept is contiguous with learning, possessing a concept with knowing and linking concepts with understanding (Adamczyk et al 1994, p117). Such ideas are similar to the constructivist ideas of Ausubel discussed above and will inform the discussion of intuition later in this paper.
The wide range of definitions of 'concept' has caused some commentators to consider whether 'concept' is 'a useful scientific concept' (Barsalou et al 2003, p84). It may be that nothing as discrete as a concept exists, however its continued widespread usage demonstrates its problematic usefulness. Over the last twenty-five years there have been three major paradigms in the
representation of concepts (Gelman and Diesendruck, G. 1999, p81). The 'traditional' view as assumed by Inhelder and Piaget holds that concepts can be represented by a list of properties that are 'singly necessary and jointly sufficient' (Ibid, p81) for identifying all the instances of a particular concept. There followed a shift to a more probabilistic paradigm which claimed that 'category boundaries are not clear cut; rather, probabilistic models (such as prototypes) determine category membership' (Ibid p81). In the first model, necessary features for belonging to the class 'birds' would be having wings, a beak, feathers etc. The second model of a concept would suggest the likelihood of such features being present and could mark some examples (e.g. a robin) as being better prototypes, more typical of the class, than others (e.g. a penguin). The final paradigm discussed by for example Murphy and Medin, Keil and Heit and Rubinstein (Ibid p82) claims that probability information alone is not enough to delineated a concept; theoretical belief systems are highly influential during the categorisation process. Within a constructivist model of knowledge (see section 3.1), this is a natural assumption to make. These three models underline what will be the central thesis of this research: it is not concepts themselves that define understanding, rather the way connections are constructed between them. Moreover as the third conceptual paradigm suggests, theories held will influence the way in which concepts are organised and linked. Such ideas will be discussed further in the section on metacognition.
This model that 'the links [between concepts] may actually be seen as representing the way we make sense of our worlds' (Taber 2005, p4) is not to be accepted too readily. The philosopher Jerry Fodor's well known puzzle of concept acquisition asks how it is ever possible to acquire a concept given that acquisition must involve some manipulation of a concept that is yet to be acquired (Fodor 1983). A similar argument is made by Plato in the Meno: if one knows what one is trying to learn, then no learning is necessary or one does not know, in which case it is impossible to tell if learning has been successful or not (Vosniadou and Brewer, 1987, p51). In response to this argument Fodor developed an atomist theory of concepts, that is, 'what makes a concept the very concept that it is, is not how it is related to certain other concepts, but how it is related to the world' (Margolis, 1998, p348). Therefore it is not the conceptual structure or 'inferential disposition' that is necessary to a
concept; it is only important that 'the concept stands in the right mind-world relation' (Ibid, p351).
Fodor's puzzle is highly problematic for constructivist theories of learning, however atomist theories also raise deeply troubling issues. It is difficult to believe that all concepts are either innate or constructed from innate components; that there is no ability to acquire concepts and we are born with all the necessary mental vocabulary to understand everything in our worlds. However, some theorists have suggested that this is indeed the case that our cognition in guided by 'in built' constraints most famously Chomsky theory of universal grammar (Preece 1984, Sebastia 1989). The idea of whether there exists a native, 'intuitive' understanding of physics will be discussed in a later section.
2.5.5
C
oncept change: revolutionary intuitions?Since the growth of the 1970's the 'children's science' research program has been attempting to construct an understanding of the ways in which children make sense of the world (Duit and Treagust, 2003, p671). The subsequent research into how students' concepts change has taken the form of several overlapping research programs as reported by Duit and Treagust in their 2003 review of the subject. I will use concept change in the same sense they do, meaning not an exchange of pre- instructional concepts for scientific ones, but rather a restructuring of those initial conceptions to allow the understanding of the intended knowledge. They cite the Posner, Strike, Hewson and Gertzog model of conceptual change as being the best known and describe it as initiating 'dramatic or revolutionary conceptual change' (Ibid, p673). In contrast authors such as Vosniadou and Ioannides and Limon view conceptual change as a more gradual, continuous process. In an earlier review of the topic, Vosniadou and Brewer present the dichotomies of global versus domain specific and weak versus radical restructuring (Vosniadou and Brewer, 1987). It seems unlikely that there are in reality two such distinct cognitive processes for learning rather there may be a continuum of effects occurring at different times in different learners. Yet the radical, discontinuous alterations may share some similarities with the process of intuition.
2.5.6
A
nalogy and the construction of conceptual connections'A native talent for perceiving analogies is...the leading fact in genius of every order.' (William James, The Principles of Psychology quoted in Mitchell 1993, p1)
Dedre Gentner, in her discussion of the discovery process of Kepler, suggests that 'cognitive processes inherent in analogy can promote conceptual change' (Gentner 2002, p22). It seems likely that the process of concept alteration must at some stage involve the comparison of different ideas and hence some form of analogous reasoning must take place. The mechanism of producing analogies can be compared to the conceptual blending theory discussed above. This link between
two distinct concepts is itself a kind of weak analogous thinking and highlights the importance of connection formation in thinking processes, or as Gentner and Markman write: '..an understanding of similarity processing may provide general insight into human thinking' (Gentner and Markman 1997, p45). Most importantly for this research, as I have argued above, it may be that 'Comparison processes foster insight' (Ibid, p54) and even that the ability to form analogies is 'a sign of
intelligence' (Ibid, p47)
Analogy can be defined as a device for highlighting the structural similarities between ideas despite arbitrary differences between the concepts (Gentner and Markman 1997, p46). Analogies convey the 'overlap in relations among objects' (Gentner and Gentner 1983, p101) and allow diverse knowledge sources to be integrated to model a new situation (Holland et al, 1986, p287). Analogies are ubiquitous in physics (Podolefsky, p2) and are often linked with scientific discovery (Gentner 2002, p21) . Arthur Koestler coined the term 'bisociation' (Koestler 1964. p35) to refer to the creative act of linking two different associative contexts. He cites examples from the history of science:
Gutenberg's invention of moveable type (the connection of the wine press with punches for making coins) and Kepler's explanation of planetary motion (Conceptualising gravity with light or the holy spirit to theories action at a distance).
Dedre Gentner conceptualises the process of analogy as a product of structure mapping (Gentner 1983, Gentner and Markman 1997). An axiom of structure mapping is that conceptual organisation has some defined structure (Morrison and Dietrich, 1995, p1). Analogy is then the mapping of one structure onto another based on the similarity of the relations between the concepts.
James Clerk Maxwell held a similar idea: 'The similarity is a similarity between relations, not a similarity between the things related' (James Clerk Maxwell quoted in Podolefsky, p2) or as Gentner and Gentner put it: 'The relational structure is preserved, but not the objects' (Gentner and Genter, 1983, p102). This is illustrated in figure 5, demonstrating the similarities in what I will call meta- structure of concepts between the atomic and solar domains.
Figure 5: Partial identity in the higher order (meta) structure of two domains. Adapted from figure 6.2 (Gentner and Gentner 1983, p106-107)
F=GMm
r2 F=
Qq kr2
Conceptual organisation plays a guiding role in creating new analogies; 'predicates are more likely to be imported into the target if they belong to a system of coherent, mutually constraining relationships' (Gentner and Gentner, 1983, p104). This extends Ausubel's notion that conceptual absorption will occur most effectively when there is similarity between concepts; the detection of such similarity may be more likely if conceptual structure is more coherently organised (see section 2.5.8 for a discussion of conceptual organisation). The importance of metacognition is also
highlighted in this model. Figure 6 illustrates how connections may be created between two meta- structures. Such a process is possible if a learner is able to operate on a 'meta' level that allows the manipulation of meta-structures. The ability to recognise higher order structures is closely related to Gestalt ideas of recognising patterns and the work of Katona on learning by organising (see above).
The structure mapping model is also reflected in two other models which are also worth consideration. Holland and colleagues discuss the role homomorphism, that is the mapping of elements in the world to elements in a mental model, plays in problem solving (Holland et al. 1986, p31). Their model agrees with Gentner's in the importance of drawing connections between similarities in structures. However, it is reminiscent of Fodor's argument discussed earlier claiming that connections are drawn not between concepts themselves as Gentner does, but between concepts and the world. Such an argument raises similar problems to Fodor's atomism: how is it possible to build a mental model of an atomic system if we have no direct experience of it.
A third position on the construction of meaning between concepts is presented by Douglas Hofstadter's theory of analogy as high-level perception (Chalmers et al 1995, Morrison and Dietrich 1995). This theory conceptualises analogy making as the interplay between the low level process of situation perception and the high level process of mapping the similarities between such situation perceptions. It is claimed that the two processes interact in a deep and subtle way: perception feeds mapping but mapping requires perception to check its links (Chalmers et al 1995, p181). Morrison Dietrich claims these two models, structure mapping and high level perception, are not mutually exclusive but represent horizontal and vertical perspectives respectively of the same effect (Morrison
Figure 6: Relationships constructed between the meta-structures of two domains
and Dietrich 1995, p4). Similarly, the phenomenological primitive research program (identifying 'subconceptual' 'feelings of naturalness' (diSessa 1996, 715)) mentioned below and the program interested in higher order patterns can be seen to be working on different ends of the same problem.
DiSessa's program focuses on the atomic, phenomenological experiences of the world, whereas Gentner and colleagues are interested in how higher order patterns are constructed. This research alignment with a model based on perceiving meta-structures is seen as complimentary to the phenomenological program. If we are to accept a constructivist view of learning, the process of constructing links and hence drawing analogies between concepts will play an important part in gaining new concepts. It would be particularly interesting to understand better how connections are constructed between concepts as in figure 6. This is the motivation for the concept mapping tool discussed in more detail below.
2.5.7
S
pontaneous analogiesProfessor John Clement has published extensively on the use of analogy (Clement 1994, 1998, 2004). His work on spontaneous analogy is of particular interest due to the rapid nature of the process. He highlights the importance of evaluating the validity of a new analogic relationship after its generation (Clement 1998, p1274). One method for doing this is matching 'key relationships' whereas it is claimed that 'novices are more likely to attend to surface features' (Ibid, p1271). The differences between expert and novices will be considered in the next section, the ability to attend to higher level relationships in section 2.6. Duit and colleagues also investigated students' spontaneous analogies but rather than the interview situations of Clement they recorded those produced in the classroom (Duit et al 2001). They highlight that analogies can be a 'two-edged sword': they can support learning but also lead to misconceptions. The studies discussed are fascinating as they record the sudden way in which learning can proceed and give some indication of the way in which links between concepts are constructed. However, the reliance in both cases on verbal reports and students' diagrams give limited evidence for drawing conclusions. The concept mapping tool described below gives a richer, more visual representation of the creation of conceptual links.
2.5.8
T
he wider conceptual architecture- higher order patternsSo far in discussing the integration of new concepts, consideration has only been given to connections between two concepts. However, several authors suggest that for some high level learners a highly interconnected cognitive structure is to be expected (Taber 2005, p3; Tsai and Huang 2001). The integration of new concepts will be affected by the connections of current concepts with their neighbours thus it is impossible to ignore the influence of the wider conceptual structure.
Indeed many authors report that the ability to reason at a high level in the natural sciences is due to the creation of 'elaborate, strongly hierarchical, well-differentiated and highly integrated frameworks
of related concepts' (Pearsall et al 1997, p194). As has already been discussed in the section
concerning concepts, it is likely that concepts acquire their meaning through their relationship with other concepts (Novak and Musonda 1991). Therefore an understanding of the entirety of a learners' conceptual relationship would be necessary to understand the way even a single concept is
integrated. This is clearly not a practical aim, especially in a small scale investigation such as this one.
Therefore the integration of two small topics will be investigated but it must be remembered that many important conceptual linkages will be missing. However, as the aim of this part of the study is to investigate how conceptual links are formed, not to map them exhaustively, this may not be a major issue. Never-the-less, an examination of what is known about the knowledge structure and processes of integration of experts in general will be enlightening.
2.5.9
L
earning from the experts, learning about the experts'Observe carefully what guides the actions of the wise, and what they shun or seek' (Marcus Aurelius in Meditations, 1964 Penguin Books book iv:38 p48)
The goal of this research, as indicated by the title, is to use the seemingly special class of highly 'intuitive' students as a resource for uncovering techniques and processes that may be taught to the less 'intuitive'. This process can be compared to the technique of knowledge engineering developed in the 1980's to construct 'expert' computer systems through the study of the behaviour of human experts (Smith 1996). Ericsson and Smith describe the original expertise approach in which they attempt to capture 'superior performance' in the laboratory (Ericsson and Smith 1991, p12). They discuss the use of think-aloud-protocols and ways of modelling expert behaviour which inform the methods used in this investigation. Some of the earliest work on expertise by de Groot and Chase and Simon (Ibid, p8) studied Chess Grandmasters. They demonstrated that chess masters did not have superior memory capacities or faster inherent processing systems. Rather, their higher ability derived from the way they perceived groupings of pieces on the board. The usual reported limit for holding 'chunks' of information in short term memory is 7±2 (Ibid, p11). The grandmasters did not have a larger capacity but, due to the way their knowledge structures were organised, they could perceive more complex chunks, that is chunks of many more pieces. These superior perceptual skills provide a larger set of patterns that 'serve as an index or access route' to the knowledge in long term memory (Larkin et al, 1980, p1336). The recognition of Gestalts that drive insight discussed above, may have the added benefit of organising concepts in memory into easily accessible chunks.
Perception of information and its organisation and retrieval seem to be closely linked. Chalmers, French and Hofstadter quote Kant: 'Concepts without precepts are empty; precepts without concepts are blind' (Chalmers et al 1995 p192-193). There may be a positive feedback situation in learning; the better formed and indexed a learner's knowledge is the more easily it will acquire new concepts.
Piaget notes: ‘any scheme of assimilation tends to feed itself’ (Piaget 1978 quoted in Meadows 1993,
p138). It is possible that the existence of a positive feedback loop may account for the apparent wide difference in ability between 'intuitive' and other students. The structure and indexing in cognitive structure determine the way concepts are linked and may play an important role in Intuition (see figure 12)
The previous few sections have involved discussions of the way concepts are compared, integrated and indexed. The processes that act on concepts seem to be of more interest than the concepts themselves. This ascent to a 'higher' level, from concepts to meta-concepts motivates the following investigation of metacognition.
2.6
A
bove and beyond: thinking about thinking.It is more than thirty years since John Flavell introduced the term 'metacognition' and since then many writers have reported its importance in many fields of education (Georghiades 2004).
Necessarily, different authors have given different definitions of metacognition: 'knowledge and cognition about cognitive phenomena' (Falvell 1979, p906) 'cognition about cognition' or 'second- order cognitions' (Papaleontio-Louca 2003, p9-10). There also appears to be consensus that good metacognitive skills play a significant role in good learning (Ibid 2003, p10; Martin et al 2000 p304).
In the educational context, where the key cognitive process can be thought of as learning, a frequently used taxonomy classifies metacognitive skills on a continuum of the particular to the general:
Level Description
Learning Skill Single actions, almost like 'tricks' designed to do one job and can be taught.
Learning Strategy A number of skills used together for a particular purpose.
Learning Style A deep-rooted preference for a particular type of learning.
Operates across activities and subject areas.
As an educator, I have chosen to focus mainly on the specific teachable learning skills. Whilst there is a large literature on students learning styles and strategies in science (For example Conner and Gunstone 2004) little has been published about specific skills (Baird and White 1982 and Baird 1986 are notable exceptions). At the more fundamental level of styles, Carl Jung labelled one of the axes of his personality type space as ranging from sensing to intuition (Goldbeg 1985, p103). It may be that a propensity for 'intuitive' or analytic thought is deeply ingrained in our personalities and not available for modification. Never-the-less, it is possible that some of the specific skills used by 'intuitive' learners may be taught to others to move them towards the 'intuitive' pole (and perhaps
Table 1 : Levels of metacognitive process. Adapted from Adey et al 1999, p2
also analytical skills may be taught to the 'intuitive'). Metacognitive skills may also be conscious or subconscious and the literature is divided between those who claim experts will be especially aware of their cognitive processes (Anzai 1991, p72) and those who think experts' thinking will be
automatic and subconscious (Olson and Biolsi 1991, p243).
There has been considerable research into the role metacognition plays in problems solving (see for example Gearce and Beatty 2005; Taconis et al 2001) and in the UK, the CASE project has made the teaching of thinking skills in science common place (Adey and Shayer 1994). However, the metacognitive literature is largely focused on the domain of problem solving: there is little work on the role metacognition may play in regulating learning and concept acquisition (the work of Baird noted above is again an exception). The work on problem solving is illuminating in that it postulates the idea that more able students are able to perceive the 'deep structure' of a problem and rather than just using surface features of a problem are able to combine these into 'second order' cues that facilitate the selection of a solution strategy (Gerace and Beatty 2005; Chi et al, 1991).
Metacognition can be seen as rising up a conceptual level, this may be a key process in intuition.
To use a metaphor, it is difficult to understand the layout of a town when you are actually wandering the streets but if you climb up a tower the relations between streets becomes clear.
Metacognition enables the manipulation of not only concepts but relations between concepts and relations between those relations:
"Good mathematicians see analogies between theorems or theories. The very best ones see analogies between analogies." (Banach 1990, pix)
Ohlsson and Lehtinen claim that abstraction is necessary for the acquisition of complex ideas (1997, p37). They quote Bertrand Russell: 'Curiosity about general propositions shows a higher level of intelligence than curiosity about particular facts; broadly speaking, the higher order of generality the greater is the intelligence involved' (Ibid, p37). In another paper, Ohlsson and Noakes argue that though 'surface features of the problems change' there is a 'deep', 'abstract and generative'
knowledge that remains unchanged' (2001, p1). These ideas are related to Katona's theory of learning by organising and Gentner and Gentner's discussion of the transfer of structural similarities between analogies, both discussed above. This idea is mentioned by other authors. Gick and Holyoak propose that relationships are mapped based on 'an abstract “core idea”' (Gick and Holyoak 1983 , p7). This is conceptualised in the schema, '..an abstract category that the individual analogs
instantiate in different ways' (Ibid, p8). Clement models physical intuition with 'schemas of modest generality' rather than 'specific episodic memory' (Clement 1994, p221).
The concept of a 'core idea' or 'schema' is poorly defined and minimally theorised in the papers discussed above. This work aims to build an understanding of how students' are able to make use of such knowledge which I call meta-structure. The first stage of this research, therefore aims to examine students thoughts about their thoughts about their learning, to try and understand how they construct their cognition and their metacognition. The second part, the concept mapping
exercise, attempts to 'catch them in the act' of switching to the meta level as they integrate two structurally related concepts.
2.7
S
uddenly, it all makes sense: IntuitionThe one point of consensus between writers is the difficulty of defining 'intuition': 'Intuition has been a baffling and elusive subject for generations. The lack of clear cut definition and the loose usage of the word has only added confusion to this nebulous matter' (Guiora in Bastick 1982, p9).The word derives from the Latin intueri which means 'to look upon' or 'to see within' but this brings us no nearer to an understanding (Goldberg 1985, p31). Despite emphasising its importance in
education Bruner declares 'intuition' 'elusive' (Westcott 1968, pv) and Bastick himself calls 'intuition' a semantic riddle (Bastick 1982, p1). The term is used in psychology, literature, theology and in an every day sense, indeed even in 1938 Wild reported thirty-one different conceptualisations of the term (Bastick 1982, p32).
This confusion is not necessarily a problem: Pring argues that some words are 'essentially contestable' and offers a number of solutions to this dilemma (Pring 2000, p9). In the fourth of these he urges the researcher to 'think of the different ways of understanding which are brought together under this one label' and use these disparate conceptions to construct a richer interpretation of a concept (Ibid, p11). This is reminiscent to the discussion of concepts not being defined by a set of sufficient and necessary properties but rather by probabilistic relations. Similarly Wittgenstein argues that it is not possible to provide an absolute list of similarities between members of a class only to indicate their 'family resemblances' (Wittgenstein 1952, p32). One way to define 'intuition' is therefore:
Intuition
Preconscious/Subconscious ‘Intuition is...a perception therefore which comes by way of the unconscious’ Messer (p26) ‘Those intuitive judgements...depend on less fully instantiated cues. i.e. Unconscious’ Vernon (p28) Intuition is ‘perception by way of the unconscious irrational aimed at pure perception’ Jung (p26)Contrast with analytic thought Intuition is opposed by ‘abstractive cognition’ William of Ockham (p25) ‘Intuition is....an irrational function’ Messer (p26) ‘The brute power of logic is useless somehow...The moment it comes there is no question about the right solution’ Green (p27) Intuition: .’reaching a conclusion on the basis of little information which is normally reached on the basis of significantly more information’ Westcott (p34) Emotional ‘It [Insight] carries with it a strong feeling of certainty’ Garard (p27) ‘A feeling of evidence such that the need for a mathematical proof is not felt’ Fischbein (p29) ‘A sense of feeling of pattern’ Bigge and Hunt (p31)
Recentering ‘Returning information in a radically different form to the input’ Guilford (p26) ‘The occurrence of complete solutions with reference to the total organisation of the field’ Hartmann (p30) ‘The perceptual structure “frees“ the key elements to be shifted into new patterns’ Watson (p30)
Influenced by experience ‘Most skills depend largely on “intuitive experience”’ DeGroot (p27) ‘He must have a rich background of knowledge and experience in it’ Sinnott (p27) Quick, Immediate, Sudden ‘The breakthrough type of insight comes by sudden arrival’ Haefele (p31) ‘Immediate perception’ Drever (p24) Sense of relations Insights are a ‘feeling for, basic sense of relationships’ Bigge and Hunt (p31) ‘Insight is not only the perception of relations, but also awareness of relations’ Resse (p32) Figure 7: The properties of intuition (Adapted from table 1.3/1 Bastick 1982 p25-34)
It is necessary to caution that some literature uses the term intuition whilst other authors prefer insight. Bastick carried out a meta-analysis of the uses of these terms and concludes there is no conceptual difference between them and proposes to use them synonymously (Bastick 1982, p47). I will do the same.
Whilst Pring asserts it is important to remember the influence all the terms in the conceptual halo exert on meaning, it is necessary to have an operational definition: the ability to learn or solve a problem without explicitly going through all the steps that appear to be logically necessary. This is not to claim that those steps have not been carried, merely they occurred at some level of
subconsciousness, or over a previous incubation period.
2.7.1
I
ntuition in science: The metaphysical bathwater'Somewhere in the process of formulating the positivist project, the intuition baby was thrown out with the metaphysical bath water' (Laughlin 1997, p22)
The sciences have at times tried to distance themselves from intuition seeing it as 'unbridled subjectivity....even mysticism' (Claxton 1998, p217). Many authors have seen a sharp distinction between intuition and science, intuition being 'unscientific' (Welling 2005, p3). Yet it has been claimed that 83% of scientists surveyed admitted to 'frequent or occasional assistance' from intuition (Koestler quoted in Goldberg 1985, p28). In a fascinating study Marton, Fensham and Chaiklin analysed broadcasts made by Nobel Laureates in the sciences between 1970 and 1986 (Marton et al 1994). Seventy-seven percent of the laureates either 'declare explicitly their belief in scientific intuition or they make comments about it, thereby taking its existence for granted' (Ibid p460). Max Planck claimed that scientists 'must have a vivid intuitive imagination' (Wolpert 1992, p56) even Einstein is quoted as 'The really valuable thing is intuition' (Goldberg 1985, p15).
Some authors go so far as to suggest there is a special form of intuition that applies to our understanding of the real world: physical intuition. Singh argues that 'physical intuition is elusive-it is difficult to define, cherished by those who posses it, and difficult to convey to others' (Singh, 2002, p1103). Larkin and colleagues mention the role physical intuition plays in problem solving ( Larkin et al 1980 p1335) as do Simon and Simon (1978 p326). John Clement describes experts' reports of a 'seat-of-the-pants feeling...a "sense" of what will happen to a physical system' (Clement 1994, p204).
He discusses the importance of 'imagistic simulation' or thought experiments in problem solving.
Some authors even go so far as to suggest that we may be 'endowed with some innate
physics'(Wisniewski 1998, p123) and even that babies may have an 'intuitive grasp of simple laws of physics' (Myers 2002, p18).
2.7.2
E
ducating intuitions?Lev Vygotsky's proposition of 'spontaneous concepts' (sometimes also called intuitive concepts) (Vygotsky 1934 p360) prefigured the birth of the 'children's science' program (Driver et al
1994). The field of constructivist science education can be seen to consist of competing research programmes (Taber 2006, p135). Some seek to catalogue the intuitive ideas children form about the world, others to build models of intuitive understanding. Recent emphasis has been on the valuable work of understanding individuals' alternative conceptions and it is hoped that in the future emphasis will shift to producing 'models of wide-spread applicability' (Ibid, p135).
Andrea diSessa moved the argument to a more basic level by introducing the concept of phenomenological primitives or p-prims (diSessa 1996, p715). These are claimed to be
'subconceptual' and 'account for our feelings of naturalness' for example 'heavy things need harder pushes to move' (Ibid, p715). These primitives have much explanatory potential but again the research program is still in its infancy: concentrating on cataloguing p-prims. The really interesting work will begin when we understand how p-prims interact and are triggered by choice and context.
In their paper 'What has happened to intuition in Science Education?', Fensham and Marton (1992) report the Woods Hole Conference in 1959 that brought together 35 eminent scientists to discuss the perceived crisis in American science education. Bruner's 'The Process of Education' (1960) reports the four main themes discussed. Among the themes introduced were the now widely used ideas of 'key concepts' and the 'spiral curriculum' (Fensham and Marton1992, p114). The last theme, which 'seems to have been largely ignored' was the Nature of Intuition. Bruner indicated the importance of intuition in science and encouraged teachers to model intuitive thinking to their students. Wolaver claims there is 'a clear consensus that teaching...intuition is of the highest
importance' (Wolaver 2005, p1). Unfortunately as Bruner reports a sentiment that is perhaps still true today: 'the formalism of school learning has somehow devalued intuition' (Bruner 1960, p59). Thirty years on Fensham and Marton claim there is little evidence that the situation has improved (Fensham and Marton1992, p120.). Similarly, Goldberg asks that we 'give students a more inspiring and
realistic picture of how discoveries are born' (1985, p220). Understanding intuition is a challenge that science education cannot avoid any more: this challenge is what drives this research. Having
established its importance, I will now present an outline of the few models of intuition that do exist.
2.7.3
M
odels of Intuition: Fluidity and Redundancy.In his book Educational Psychology (Ausubel et al 1978) David Ausubel proposed a model of the differences between analytical and intuitive thinking: analytic individuals' thinking moves primarily between subordinate concepts whereas more intuitive thinkers' cognitive drift occurs at the level of superordinate concepts.
This model, despite it's obvious simplicity, is fascinating in that it connects with the issues raised in the metacognition and analogy sections above: intuitive thinking is the ability to think at a different level. For Ausubel, intuitive thinking implies not having to think through the safe, multiple steps of the 'lower' analytical path but rather be able to 'step up' and think at the level of
superordinate concepts. As with the process of analogy formation it involves an ability to think with larger units of thought.
Tony Bastick appears to have used Ausubel's model as a base for perhaps the most extensive theory of intuition in the literature (Bastick, 1982). He criticises Ausubel for ignoring the important role of emotions in intuition and cites the work of Köhler who reported how an observer's emotional state affects the Gestalt of an object, for example patterns of dots will be grouped in different ways depending on the mood of the observer (Bastick, 1982, p88). He claims that perceptions become connected to a particular emotional state, called an emotional set (Bastick, 1982, p107). These sets are labelled A,B,C in figure 9. Over time there is a drift between different emotional sets governed by the transition probability between the sets. This process of gradual drifting between different sets allows new linkages to be created between previously unrelated concepts: the process of intuition.
Figure 8: Ausubel's model of analytical and intuitive thinking (Adapted from figures 2.1/4, 2.1/5 Bastick 1982 p58)
Bastick also introduces the concept of redundancy which he describes as 'duplication of information' and claims is central to the process of intuition (Bastick 1982, p355).
Figure 9: An illustration of Bastick's model of intuition (Adapted from figures 5.5/7, 5.5/8 Bastick 1982 p242)
Figure 10 illustrates how duplication in connections facilitates the ability to move between concepts. This echoes the previous ideas outlining the highly connected nature of expert conceptual structures and again points to a cumulative effect in intuition: the more redundant the structure, the easier it is to form new intuitions and so increase its redundancy. The idea of how the mind moves between concepts is a significant focus of this work, and is reflected in the choice of methods.
Douglas Hofstadter is also interested in this concept. He coins the term 'conceptual halo' for the 'fuzzy' way that concepts are defined (Hofstadter 1995 p198; Hofstadter 1985 p246). This has been discussed previously and a conceptual halo was used to define intuition above. Due to the overlap of conceptual halos the mind can 'slip' between concepts that are only to some extent similar.
Hofstadter boldly declares that 'making variations on a theme is really the crux of creativity' ( Hofstadter 1985 p233). That is, gradual conceptual slippage is the path to insights. In his later collaboration 'Fluid Concepts and Creative Analogies' (Hofstadter 1995), the contributors discuss the role of 'mental fluidity' and how it is constrained by the higher order process of analogy. In figure 11, the letter form slips through various different forms but they are all analogous: the meta 'a' is
unchanged. Gabora suggest that creativity is linked with 'high conceptual fluidity' as well as the
a a a a
Figure 11: Conceptual halos and conceptual slippage of the concept of 'a'.
(Adapted from Hofstadter 1985, Figure 12-3 p243 and Figure 12-5 p246) Figure 10: Levels of redundancy (Adapted from figures 9.3/1, 9.3/2
Bastick 1982 p364)
