When ‘light’ dawns upon them: Mapping the essence of conceptual understanding … · 2014-12-16 · the subjective understanding of its scholars, this study sets out to investigate

When ‘light’ dawns upon them:

Mapping the essence of conceptual understanding of physics learners

Stefan Yoshi Buhmann

December 16, 2014

Stefan Yoshi Buhmann MEd dissertation

Abstract

Motivated by an interest in the inherent friction between objective knowledge in the natural sciences andthe subjective understanding of its scholars, this study sets out to investigate the essence and developmentof conceptual understanding in physics from a constructivist perspective. In view of current pressuresto reduce the Electromagnetism syllabus, the focus has been laid on ‘light’ as a core topic. Students atthe beginning and end of their undergraduate physics education have been asked to create a conceptmap of ‘light’ and the understanding of some of them has been further probed in phenomenologicalinterviews. Their views have been complemented with a concept map produced by the course instructors.The submitted maps have been analysed in terms of structural parameters, morphologies and content.

The study has created answers to the methodological (How . . . ?), essential (What . . . ?) and de-velopmental (Whither . . . ?) aspects of the question: can concept maps be used to reveal conceptualunderstanding of ‘light’ of physics students at different stages of their education? Methodologically, ithas shown that conceptual understanding can indeed be analysed on the basis of concept maps. How-ever, such an analysis requires a consistent combination of structural, morphological and content aspectsinto a meaningful whole. The essence of conceptual understanding emerging from such an analysis is awell-structured core of interconnected physics concepts, theories and formulas shared by physics learners.This understanding can be interpreted as a result of Ausubel’s key processes of meaningful learning.The development of this understanding is suggested by the study to proceed in three main stages: Aninitial stage of broad interest and receptiveness is consolidated by a focussing and deepening. Expertunderstanding seems to require a final stage of adding hierarchy and cross-connections.

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Contents

Acknowledgements 5

1 Introduction 7

2 Literature review 9

2.1 Visualising understanding: concept maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Theoretical frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Ausubel’s theory of meaningful learning . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Nespor’s chain of mobilisations in physics . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Discipline-specific studies: conceptual understanding in physics . . . . . . . . . . . . . . . 13

3 Methods 17

3.1 Research approach: phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Research site: Imperial College London . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Participants: BSc and MSc students and experts . . . . . . . . . . . . . . . . . . . . . . . 183.4 Researcher context and perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Data generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5.1 Student concept maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5.2 Student interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5.3 Expert interview and concept map . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.6 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6.1 Concept maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6.2 Interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6.3 Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.7 Ethical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Findings 27

4.1 Student concept maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.1 Quantitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.2 Morphological analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.3 Content analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Student interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.1 External influences on the concept-map creation . . . . . . . . . . . . . . . . . . . 354.2.2 Polarisation as a common misconception . . . . . . . . . . . . . . . . . . . . . . . . 364.2.3 Variations of conceptual understanding . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.4 Key elements of the concept ‘light’ . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.5 Learning history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Expert contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.1 Concept map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.2 Interview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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5 Discussion 43

5.1 Concept maps as witnesses and facilitators of meaningful learning . . . . . . . . . . . . . . 435.1.1 Subsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1.2 Obliterative subsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1.3 Superordinate learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.1.4 Progressive differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.1.5 Integrative reconciliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Concept maps as evidence for a chain of mobilisations? . . . . . . . . . . . . . . . . . . . . 46

6 Conclusion 49

A Teaching concept mapping 59

B Interview protocols 61

B.1 Student Interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61B.2 Expert interview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

C Topological normalisation 67

D Transcription 69

E Graph theory 71

F Ethical conduct of research 73

G Structural parameters of participants’ concept maps 75

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Acknowledgements

Many people have contributed to this work in various ways. Firstly, I would like to thank my supervisor,Martyn Kingsbury, for helping this project form and grow, for the exact right degree of guidance andsupervision and for his stimulating input. I thank him and Sarah Worton for encouraging me to embarkon this last step of the Master’s in Education programme; Peter Wren for frequent tea-time discussionsaround and beyond education; Dave Riley for his insights and inspiration; Jo Horsburgh for being veryhelpful in providing materials; Kate Ippolito for her critical perspective; and all of the members of theEducational Development Unit at Imperial College for providing excellent sessions on teaching, learningand educational research.

I would like to thank Ian Kinchin for sharing his insights on concept mapping. In addition, I amindebted to Routledge and in particular Freya Davidson-Smith for providing educational literature.

I am grateful to Simon Bland, Julia Sedgbeer and Andrew Williamson for their enthusiasm for es-tablishing concept maps as a learning tool and for their help in organising my concept-map sessions atImperial College. I thank Rob Nyman for allowing me to conduct most of the interviews in his office andYoshi Buhmann for providing professional audio recording equipment.

I thank all those first-year BSc and MSc students who were willing to contribute their concept mapsto my research and in particular those students who were willing to devote additional time by sharingtheir understanding in interviews. I am grateful to the two leaders of the Electromagnetism course atImperial College for providing their expert insights into teaching this subject.

I am grateful to Carole Evans and Alison Kingsley from Pro Rata Secretarial Solutions for theirtranscription services.

Finally, I would like to thank my father for his interest in this project and the useful advice that hegave; my mother for sharing her expertise in interviewing; my wife for her continuous support, criticalfeedback and help in keeping the project on track; and my children for sharing their everyday joy inlearning.

London, December 16, 2014 Stefan Yoshi Buhmann

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Chapter 1

Introduction

‘I believe the intellectual life of the whole of western society is increasingly being split intotwo polar groups. ... Literary intellectuals at one pole—at the other scientists, and as the mostrepresentative, the physical scientists. Between the two a gulf of mutual incomprehension.’(Snow 1998, pp. 3-4)

Every scientist who is also a teacher bears his own version of that deep conflict between Snow’s famoustwo cultures. To appreciate this, let us start from his image of two opposing camps of knowledge-seekingindividuals, each group defined by ‘common attitudes, common standards and patterns of behaviour,common approaches and assumptions’ (Snow 1998, pp. 9). Later, Kuhn (2012) coined the term paradigmto refer to such an ‘entire constellation of beliefs, values, techniques, and so on shared by the members ofa given community’ (p. 174). Rather tellingly, being a physicist by training, he seemed to see the naturalsciences in a privileged position as each possessing a paradigm, linking ‘the contemporary social sciences ...[to the] pre-paradigm periods of fields that are today unhesitatingly labeled science’ (Kuhn 2012, p. 159).The different paradigms dividing natural and social scientists have their root in almost diametricallyopposing philosophical assumptions about the nature of truth and knowledge, owing to their very differentobjects of study. Natural scientists who investigate phenomena involving unconscious entities typicallyadhere to positivist (or post-positivist) views of the world as an objective reality, absolute truths aboutwhich can be (approximately) found (Psillos 2012). Social scientists study humans and their interactionsand hence often adopt constructivist, social constructivist or post-modern perspectives where knowledgeis constructed by individuals or communities and it is possible for a variety of contradicting truths tocoexist (Savin-Baden & Major 2013).

So far, the described conflict has involved two neatly separated camps whose opposing views couldbe understood from the fact that knowledge about the natural world is very different from knowledgeabout people. Yet there is a point where these two realms intersect and collide, namely: knowledge aboutpeople’s knowledge about the natural world. In other words, how do we study the learning of naturalsciences, and which of the two opposing paradigms might best inform the practice of science teaching?This dilemma lies at the heart of heated inter-departmental arguments such as that sparked by physicistMagueijo’s (2009) recent critique of training courses on natural-science teaching as run by social scientists(Sotto 2010).

As noted above, every teaching scientist or teacher of sciences has to negotiate the opposing paradigmsas part of their professional practice. Being both practising physicist and university teacher, I have madefirst attempts towards a reconciliation by making myself comfortable with both world views and reducingthe gap between them. I have moved from a sternly positivist position acquired during my own trainingas a physicist towards a more careful view by recognising that even in physics, a multitude of accountsof a phenomenon may be valid and beneficial (Buhmann 2013). On the other hand, I have come torecognise that learning is a highly individual and active process whereby knowledge is constructed. Yeta deep conflict remains and shall form the underlying question to be addressed in this study: If thereare objective truths about physics on the one hand and if learners construct knowledge very individuallyand subjectively on the other hand, then in what sense can they ever be said to arrive at a commonunderstanding of these truths?

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To contain the investigation to the learning-theoretical side of the issue, I will treat physics as givenand unproblematic body of knowledge understood within a positivist paradigm. To facilitate this view, it isadvantageous to study learning in an established area of physics whose basic facts and interpretations areuncontroversial. Classical electromagnetism is an ideal candidate, because it constitutes a core subjectof any undergraduate physics curriculum whose basic laws have been known since Maxwell’s (1865)discipline-founding work.

Electromagnetism has recently been facing external and internal challenges which further motivateits choice for this study. Externally, the place of electromagnetism within the physics core curriculumworldwide faces competition from subjects such as nano-electronics, photonics and computer science(Chew et al. 2012) which has already lead to a reduction from full-year to one-semester courses atsome US institutions (Bansal 2012). Internally, electromagnetism suffers from its students’ perception asdifficult and unexciting (Hänninen 2012), creating a need to incorporate topical issues to raise interestand motivation (Bansal 2012).

The conflicting requirements to include new material, but spend less time in doing so lead to a ten-sion which can only be resolved by investigating which elements of the electromagnetism curriculumare indispensable and how they are interlinked. A snapshot answer can be generated by studying stu-dent understanding of a single core concept that brings together many of these elements. Arguably themost important concept in electromagnetism is ‘light’. It requires an integration of many aspects of thesubject and its many facets point far beyond to other disciplines such as (quantum) optics, photonics,atomic physics, quantum field theory or relativity where it takes diverse roles as object of study, designor measurement tool, exemplar or even primary postulate. Nobel prize laureate Weinberg (1975) hasargued that ‘light’ serves a privileged function in the generation of fundamental physical theories. Thetopical relevance of ‘light’ to both physics and society has been stressed by the United Nations’ recentannouncement of an International Year of Light 2015 (European Physical Society 2013).

Understanding the concept of ‘light’ is a challenging process where increasing levels of complexity areintroduced throughout a student’s learning trajectory in the spirit of Bruner’s (1966) spiral curriculum.A thorough analysis must therefore pay attention to the temporal dimension by studying student under-standing at different stages of their university education. The learning dimension will be probed from aconstructivist perspective (Woolfolk, Hughes & Walkup 2012, Anderson 2010) where understanding is anidiosyncratic, continually growing and evolving entity.

The original very broad issue of the conflict between natural- and social-science paradigms apparentin physics learning has thus been funnelled into the following concrete research question: What is theconceptual understanding of ‘light’ of physics students at different stages of their education? Studyingthis question will serve a multitude of purposes on different levels. Learning-theoretically, it will shedlight on the complex learning process during which conceptual understanding develops in a hard-puresubject (Neumann & Becher 2002). On a disciplinary level, it will explore the internal structure of elec-tromagnetism as a very abstract and interrelated body of knowledge and indicate obstacles that studentsmay face trying to master it. In this way, the study may pave the way for compacting and restructuringthe subject to allow for the inclusion of topical new aspects and hence help electromagnetism evolve andprosper within the physics curriculum of the 21st century. Furthermore, the focus on a particular conceptresonates with Gardner’s (2009) idea of a core topic and could serve as an antidote for the threat ofknowledge fragmentation intrinsic to the expository teaching mode (Gülpinar & Yeğen 2005) dominatingphysics education.

Linking back to the two cultures conflict, the ambitious underlying theme of the endeavour will be toexplore the boundaries between objective knowledge and subjective understanding in the natural sciencesand physics in particular. To rephrase Jackson’s (1982) knowledge argument: Do different physicists reallysee the same thing when they ‘see light’?

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Chapter 2

Literature review

Before even attempting to tackle the issues raised in the introduction, three preliminary questions cometo mind: How is it possible to access a student’s understanding of a complex concept? Which learningtheories may provide a framework for studying conceptual understanding? What previous attempts havebeen made to investigate conceptual understanding in physics and of ‘light’ in particular? These questionswill guide the following three sections in order to arrive at an informed position for addressing my researchquestion.

2.1 Visualising understanding: concept maps

A concept is ‘a category used to group similar events, ideas, objects or people’ (Woolfolk, Hughes &Walkup 2012, p. 332). Alternative theories in cognitive psychology hold that concepts are representedin our minds as prototypes (idealised representatives), exemplars (specific examples) or schemas (listsof attributes) (Anderson 2010). A physics concept such as ‘light’ is most adequately described by thelatter view which suggests that its various attributes such as properties, examples and relations to otherconcepts form an intricate structure.

A concept map as developed by Novak & Gowin (1984) is a ‘very powerful and concise knowledgerepresentation tool’ (Novak & Cañas 2006, p. 332) for visualising such a structure. It is a hierarchicalnetwork of concepts and their relations. In close similarity to the above notion from cognitive psychology,Novak (1998) defines a concept as ‘a perceived regularity in events or objects or records of events or objects,designated by a label’ (p. 22); it is represented by a box containing the label. Darmofal, Soderholm& Brodeur (2002) distinguish such records from the objects or events themselves by using the termsabstract vs. concrete concepts. To capture the multiple levels of abstraction which permeate physics, Iwill interpret the above concept definition rather openly by admitting abstract concepts about abstractconcepts (records of records of events or objects). In addition, while Novak & Cañas (2008) suggest that‘the label for most concepts is a word’ (p. 1), Tifi, Lombardi & Villamor (2008) have proposed to permit‘concept labels to be formed by an indefinite number of words’ (p. 412). The prevalent use of mathematicalformulas and diagrams in physics has prompted me to go even beyond this call for more flexible conceptmapping and allow for symbolic and graphical formats for concept labels. This is in resonance withmodern computer-assisted mapping tools (Cañas et al. 2004) and their use in knowledge management(Novak & Cañas 2010), where concepts are enhanced by graphics, videos, texts etc.

Originally linked by simple lines (Moreira 1977), the relation between two concepts is now commonlyvisualised by a line with a linking word such that the two concepts and their linking word form aproposition (Novak & Cañas 2008). While studies aiming at identifying expertise have argued in favour ofa very accurate and restricted use of linking words (Kharatmal & Nagarjuna 2006, 2008), my investigationis aimed at capturing learning in progress by learners which are mostly not yet experts. To allow for a freeexpression of their growing knowledge, I will allow for arbitrary linking words or phrases or even the use ofunlabelled lines. Further possible elements of a concept map include cross-links, which are ‘relationshipsbetween concepts in different segments or domains of the concept map’ (Novak & Cañas 2008, p. 2) andexamples, ‘that help to clarify the meaning of a given concept’ (Novak & Cañas 2008, p. 2). The latterare usually not surrounded by a box.

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Adopted in this form, concept mapping is an ideal tool for studying the interface between knowledgeand learner. With the inclusion of physical and mathematical quantities, laws, theories, experiments,measurements and units as concepts and both immediate relations between physical quantities and in-tricate cross-connections as links, concept maps are ideal for capturing physics knowledge. On the sideof the learner, their ‘bas[is] on an explicit cognitive psychology of learning, and constructivist epistemol-ogy’ (Novak & Cañas 2006, p. 177) firmly justifies their use for studying conceptual understanding. Thistwofold alignment with the structures of both the subject and the cognitive processes involved in learningit have motivated my choice of concept mapping over alternative graphical tools such as mind mappingor argument mapping (Davies 2011).

A look at the origin of concept maps may help sharpen its use for my study. Concepts maps wereoriginally developed during an investigation of science teaching for young school children (Novak &Musonda 1991) as a means to visualise information gained from interviews in a compact way. While theinterpretive and creative power of concept mapping in this approach rests with the researcher (Kinchin,Streatfield & Hay 2010), it is now common practice to allow learners to express themselves directly viaconcept maps. The resulting shift of ambiguity from interview data to concept maps and accompanyinguncertainty regarding the varying mapping abilities of the learners will be alleviated for my study bysupporting the concept-map analysis with interviews. The very early use of concept maps in studyingand facilitating physics learning (Novak, Gowin & Johansen 1983) stresses their applicability to thesubject. However, the early attention on young learners has led to a focus on very small units of scienceinstruction and an atomistic ‘[o]veremphasis on correctness [which] would fit an objectivist philosophy of‘transmission teaching’ (Kinchin 2001, p. 1260). Hay et al. (2009) support this critical conclusion:

‘In this view, meanings are not synthetic, but emerge absolutely as a product of the partsthey subsume. The approach leaves no room for learners to experiment as they try to un-derstand. Personal ‘knowing’ is either right or wrong because facts do (or do not) comprisecertain sub-sets of information at a lower level.’ (p. 25)

By contrast, a higher-education context with a focus on a subject-spanning topic such as ‘light’ calls fora much more holistic and open use of concept mapping where whole sub-disciplines may be referred toand linked and the correctness of relations is a matter of interpretation and creativity.

A range of possible uses of concept-mapping in a higher-education setting has been discussed by Hay,Kinchin & Lygo-Baker (2008), which can be roughly classified as teacher vs. learner-centred. Comple-menting expository teaching, concept maps have been invoked as an aid for educational design (Czarnocha& Prabhu 2008, Darmofal, Soderholm & Brodeur 2002), instruction (Czarnocha & Prabhu 2008), diag-nostic (Taber 1994, Treagust 1988) and formative (Austin & Shore 1995) assessment. In this context, theuse of pre-constructed concept maps with an emphasis of accuracy and correctness dominates. A moreopen-ended format is needed in learner-centred approaches, where concept maps have been employed tofacilitate active (Hay & Kinchin 2008), collaborative (Kinchin, De-Leij & Hay 2005) or dialogic learning(Hay et al. 2009) and to foster reflective practice (McAleese 1994). Here, concept mapping is an active,free, and associative exercise with an emphasis on process rather than product. A recent meta-analysisindicates that ‘in comparison with activities such as reading text passages, attending lectures, and partici-pating in class discussions, concept mapping activities are more effective for attaining knowledge retentionand transfer’ (Nesbit & Adesope 2006, p. 434). Despite being situated in a physics environment with itspredominant expository teaching, this study’s focus on learning rather than teaching or assessment is infavour of this latter, student-centred approach.

In parallel with these two traditions, techniques for analysing concept maps are typically eitherquantitative or qualitative. As seen from Strautmane’s (2012) review, quantitative measures can be furtherdivided into purely structural attributes such as the number of links (Conradty & Bogner 2008, Austin &Shore 1995, Novak & Gowin 1984), cross-links (Miller & Cañas 2008, Prosser et al. 2000, Novak & Gowin1984) or hierarchical levels (Novak & Gowin 1984) on the one hand and content-related criteria such asthe correctness (Conradty & Bogner 2008, Miller & Cañas 2008, Prosser et al. 2000, Novak & Gowin1984) and quality (Austin & Shore 1995) of propositions and completeness (Miller & Cañas 2008) on theother. Often, scoring schemes or criterion maps are used to aid assessment (Novak & Gowin 1984).

While being easy to implement and relatively free of ambiguities, quantitative assessment schemesfor concept maps fail to grasp important holistic aspects which the more interpretative, qualitative

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Stefan Yoshi Buhmann 2 Literature review

approaches can provide. The starting point for such approaches is a morphological classification of conceptmaps in terms of their global structures. Kinchin, Hay & Adams (2000) have identified spoke, chain andnetwork as distinct morphological classes. Their scheme was extended by Yin et al. (2005) who addedcircular and tree classes. Based on a topological analysis, Koponen & Pehkonen (2008) have proposed analternative classification as chains, loose and connected webs.

Using the very graphic and suggestive morphological classifications as point of departure, a range ofinterpretations have been put forward. Hay & Kinchin (2006) have developed thinking typologies, suggest-ing that spoke structures are ‘indicative of superficial and undeveloped knowledge’ (p. 139) or, in a morepositive view, of ‘learning readiness’ (Hay & Kinchin 2006, p. 139). By contrast, chains are ‘indicativeof achievement, drive and goal-directed behaviour’ (Hay & Kinchin 2006, p. 138), while networks repre-sent ‘a rich body of knowledge in which complex understanding is demonstrated’ (Hay & Kinchin 2006,p. 138). Originally associating this latter type with expert knowledge, Kinchin (2008) has later locatedexpertise in the ability to dynamically transform between ‘chains of practice ... [and] the underlyingnetworks of understanding’ (p. 3). This notion relates back to Novak & Gowin’s (1984) observation thatlearning involves a transition between ‘written or spoken messages [that] are necessarily linear sequencesof concepts and propositions’ (p. 53) and ‘knowledge [which] is stored in our minds in a kind of hierarchi-cal or holographic structure’ (p. 53). Thinking typologies provide an accessible and powerful frameworkfor interpreting and comparing concept maps produced by students at different stages of their learningtrajectories, as required for my study. However, one needs to bear in mind that such an interpretationneglects influences on morphology other than the learner’s knowledge structure, such as the knowledgestructure of the subject or the graphical abilities of the learner.

Alternative interpretative frameworks such as Hay’s (2007) and Kinchin’s (2011) diagnosis of surfacevs. deep learning, Kim’s (2008) and Czarnocha & Prabhu’s (2008) conceptualisations based on Vygotsky’s(1986) zone of proximal development or Prosser et al.’s (2000) and McPhan’s (2008) classifications interms of Biggs & Collis’ (1982) Structure of the Observed Learning Outcome (SOLO) taxonomy are ofless relevance to this study. The former two require a more fine-grained analysis of the learning dynamicswhile the latter focusses too narrowly on the end point of learning.

As argued, quantitative concept-map analyses fail to provide a holistic perspective, while qualitativemethods alone run the danger of limited reliability. In order to benefit from the advantages of bothapproaches, I will build on Kinchin’s (2000) suggestion to combine them and construct an analysis wherequalitative interpretation rests on a basis on quantitative data.

2.2 Theoretical frameworks

I will draw on a general and a discipline-specific theoretical framework to underpin my analysis. Ausubel,Novak & Hanesian’s (1978) theory of meaningful learning has informed the development of concept-mapping from the very beginning, while Nespor’s (1994) actor-network theory of university-physics edu-cation provides a unique perspective on the discipline.

2.2.1 Ausubel’s theory of meaningful learning

Rooted in educational psychology, Ausubel, Novak & Hanesian’s (1978) constructivist learning theory isconcerned with

‘meaningful learning, [which] takes place if the learning task can be related in a nonarbi-trary, substantive (nonverbatim) fashion to what the learner already knows, and if the learneradopts a corresponding learning set to do so.’ (p. 27)

Whether meaningful learning can take place thus depends ‘both on the nature of the material to be learnedand on the nature of the particular learners cognitive structure’ (Ausubel, Novak & Hanesian 1978, p. 43).Of Schwab’s (1973) four educational ‘commonplaces of equal rank: the learner, the teacher, the milieu,and the subject matter’ (pp. 508-509), Ausubel singles out learner and subject matter and assigns themequal responsibility for facilitating meaningful learning. This balance is in line with my study’s interest inthe interface between knowledge and learner. It distinguishes meaningful learning from Marton & Säljö’s

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(1976) notion of deep learning, where the stress is on the learner’s intention. Ausubel, Novak & Hanesian(1978) hold that their distinction of meaningful versus rote learning is independent from that of receptionversus discovery learning, a view which has been contested by the advocates of discovery learning (Lee& Anderson 2013).

The theory of meaningful learning displays a strong resemblance in structure with the method ofconcept mapping on two levels. On a surface level, the ‘three basic kinds of meaningful learning: rep-resentational learning, concept learning and propositional learning’ (Ausubel, Novak & Hanesian 1978,p. 47) correspond to the three basic building blocks of a concept map: examples, concepts and links.On a deeper level, Ausubel, Novak & Hanesian (1978) name five mechanisms through which meaningfullearning takes place. As pointed out by Novak (1998), indications for these can be recovered from thestructure and temporal development of concept maps:

• Subsumption denotes ‘the process of linking new information to pre-existing segments of cognitivestructure’ (Ausubel, Novak & Hanesian 1978, p. 58). It corresponds to the most elementary processby which a coherent concept map can grow: A new concept or example is added by linking itto existing concepts. It is important to note that this ‘results in a modification of both the newlyacquired information and the specifically relevant aspect of cognitive structure to which the newinformation is linked’ (Ausubel, Novak & Hanesian 1978, p. 57), as represented primarily by theneighbourhood of the new concept on the map.

• Obliterative subsumption is a subtle variant of subsumption whereby

‘[t]he less stable (and more specific) meaning of a subordinate idea is gradually incor-porated within or reduced to the more stable (and inclusive) meaning of the specificallyrelevant idea in cognitive structure that assimilates it.’ (Ausubel, Novak & Hanesian 1978,p. 131)

It is one of the strengths of meaningful learning that rather than being forgotten, the new conceptfades from memory in such a way that it leaves a trace in the learner’s cognitive structure. Thisprocess is much harder to represent on a concept map; it might correspond to processes whereexplicit details on a map gradually lose importance or disappear, being replaced by more generalconcepts.

• Superordinate learning takes place ‘when one learns an inclusive new proposition under which sev-eral established ideas may be subsumed’ (Ausubel, Novak & Hanesian 1978, p. 59). Within thehierarchical dimension of a concept map, a superordinate concept is placed above, and immediatelylinked to, its subsumed concepts.

• Progressive differentiation of an existing concept is achieved when ‘new information is learned andthe subsuming concept ... is modified’ (Ausubel, Novak & Hanesian 1978, p. 124). This gradualprocess is reflected by the addition of more and more concepts and examples to an existing concepton a map, specifying and refining its meaning to the learner.

• Integrative reconciliation is a ‘recombination of existing elements of cognitive structure’ (Ausubel,Novak & Hanesian 1978, p. 124) which may ‘take on new organization and hence new mean-ing’ (p. 124). The impact of this very advanced process of meaningful learning can range fromthe creation of new cross-links between remote areas of a concept map to an entire reorganisationof its branches.

With its detailed account of a range of cognitive processes involved in meaningful learning, Ausubel’stheory is a rich framework for the interpretation of concept maps and their change over time.

2.2.2 Nespor’s chain of mobilisations in physics

Nespor’s (1994) theoretical framework for interpreting physics education is the result of ’ethnographicfieldwork [he] did from June, 1986 to July, 1987 at a large public university ... in the United States’(p. 1). The extensive data collected includes interviews with academic staff and students, observations oflectures, tutorials and informal student study and academic transcripts. Interpreting this data by means

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of Callon’s (1986) theory, Nespor (1994) has conceptualised physics as an actor-network which ‘constitutes[itself ] in part through educational practices that shape and sort would-be participants and organise theirparticipation in disciplinary productions of space and time.’ (p. 13). He states this network is formedand stabilised by means of four mechanisms: problematisation, the ‘definition of allowable identities andinterests’ (p. 13) for physicists; interessement, the ‘creation of barriers’ (p. 14) between physicists andoutsiders; enrolment, the ‘weaving [of ] students into the actor-network’ (p. 14); and mobilisation, the

‘shap[ing of] a space of practice by mobilizing—in the form of textbooks, cases, problems,equations and so forth— ... physically distant disciplinary spaces ... so that they can be trans-ported into the educational space of the physics ... program.’ (p. 19)

Nespor’s (1994) theory as a whole forms a social-constructivist framework for a comprehensive analysisof ‘undergraduate education in the field of physics’ (p. 6). Within the context of this study, it can providedetailed insights into the context and background against which physics learning takes place. The campus-based Anglo-American institution which informed his theory is roughly comparable to this study’s UKuniversity.

However, my study can only tap into the full interpretative power of Nespor’s (1994) theory by aradical shift of perspective away from the social-constructivist and macroscopic account of physics as adisciplinary actor-network to a cognitivist and much more microscopic focus on individual learning. Toachieve this, one needs to extract the impact that the actor-network has on the individual physics learner.Nespor (1994) has described this impact as a ‘chain of mobilizations’ (p. 53). Mediated by textbooks,lectures, class notes and problem solving, ‘the product of these mobilizations [is] a progressive strippingaway of the ‘everyday world’ and its replacement with a mathematized world’ (p. 55). The accompanyingtransformation of the student’s understanding is achieved via the following three steps: ‘The spacesof the physics textworld [are] initially described by analogy or contact with the ‘everyday’, non-physicsworld’ (p. 57). Next, diagrams, or idealised images, ‘[bundle] together a set of representations’ (p. 63).Finally, students ‘dispense with such diagrams and operate primarily in mathematical terms’ (p. 58). Bythis sequence, everyday world—idealisation—mathematisation, the students manage to ‘strip away allexcept the equations and the diagrams: the basic building blocks of ‘reality’ in the representational spaceof physics’ (p. 73).

Nespor’s (1994) chain of mobilisations, seen as a movement towards a ‘representation of inanimatephenomena in increasingly ‘abstract’ ... forms’ (p. 110) whereby ‘abstract formulations became physicalintuitions’ (p. 77), presents a tailor-made framework for thinking about the development of understand-ing in physics. However, one needs to bear in mind that it was obtained from a strict outsider perspectivewhere Nespor (1994) ‘wasn’t trying to ‘understand’ physics ... the way the students did’ (p. 52). Whileenabling him to raise his point of view above disciplinary details, his complete lack of understanding ofthe physics content also raises questions, some of which shall be discussed in my study: Is the chain ofmobilisations an appropriate description from the point of view of a physicist? What role does mathe-matics play in this process? Is the chain of mobilisations a one-way street or can students find their wayback by applying their abstract physics knowledge to real-world problems?

2.3 Discipline-specific studies: conceptual understanding in physics

Conceptual knowledge takes very specific forms in physics. This begins with the meaning of conceptsthemselves: Osborne & Gilbert (1979) have pointed out that ‘certain everyday words ... are used in asubtly different way in physics’ (p. 85). The process of acquiring these meanings corresponds to the firststep in Nespor’s (1994) chain of mobilisations:

‘Students tend to have a much more personalized view of physics, and their explanationsare related to human experience and feelings, far more frequently than would be expected froma mature physicist.’ (Osborne & Gilbert 1979, p. 89)

Apart from this very distinct assignment of concept meanings, conceptual knowledge in physics is alsodistinguished by its ‘organised and coherent ... structures’ (Nousiainen 2013, p. 505). As revealed byan epistemic evaluation of concept-map links in terms of ontology, facts, methodology and justification,

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Nousiainen (2013) has found that ‘[t]he highest criterion set for [coherence] seems to be very demandingto fulfil and even in the advanced level of studies only a fraction of students manage to reach it’ (p. 505).Conceptual coherence is a feature which is largely missed by Nespor’s (1994) theory due to limitationsof his outsider perspective.

While the above characteristics hold for conceptual knowledge in physics in general, a number ofprevious studies have been concerned with the understanding of ‘light’ in particular. A range of methodshave been used to identify misconceptions, which can be located on various points of Nespor’s (1994)chain of mobilisations. Ravanis, Papamichael & Koulaidis’ (2002) interview-based study on primary-school children has investigated their grasping of the concept of ‘light’ as a travelling, space-filling entity.This demonstrates that Nespor’s (1994) movement away from everyday, sensory experiences towardsabstract idealisations begins long before university education. Saxena (1991) has constructed intricatephysics problems complemented by interviews to diagnose high-school and undergraduate universitystudents’ understanding of various aspects of light such as shadow formation, vision, reflection, refractionand colour. His findings indicate that learners are often unable to apply general principles to concreteproblems, hinting that Nespor’s (1994) chain of mobilisations might indeed essentially be a one-way-street. Yalcin et al. (2009) used a similar method to test the understanding of first-year university studentsregarding light sources, speed, perception and superposition. Their questions revolved around relativelyabstract aspects of ‘light’ where the accepted physics concept interferes with intuitions from everydaynotions. Yalcin et al.’s (2009) conclusion that ‘most of the misconceptions are originated from students’experiences of daily life’ (p. 1091) stresses the difficulties that students face during the first step ofNespor’s (1994) chain of mobilisations when making the transition from everyday to the discipline-specificmeanings of physics concepts.

The later stages of the chain as governed by the use of diagrams and equations have been investigatedby Ambrose et al. (1999):

‘[T]he diagrammatic representation of a plane [electromagnetic] wave commonly used inintroductory textbooks is often incomprehensible to students ... Students also have difficultieswith the mathematical expressions for oscillating electric and magnetic fields and with theequations that relate each field to changes of the other.’ (p. 891)

Using test questions and interviews, Ambrose et al. (1999) identified an incomplete understanding of thementioned diagram as a main cause for misconceptions about ‘light as a wave’. They showed that thisproblem can be overcome by combining ‘diagrammatic, graphical, algebraic, and vector representationsthat are ... intended to help students to account for physical situations in terms of the concepts of physicaloptics’ (Ambrose et al. 1999, p. 897). This demonstrates that the diagrams and equations do not neces-sarily follow a fixed sequence as suggested by Nespor’s (1994) chain of mobilisations, but go hand in handin aiding the understanding of abstract physics concepts. In this example, the more abstract equationsinform the interpretation of the supposedly more concrete diagram.

Cleverly designed physics problems augmented by interviews have thus been established as a sen-sitive probe for students’ misconceptions of specific aspects of ‘light’. Studies based on concept mapshave largely shared this focus on the correctness of individual propositions. This is most evident fromHimangshu’s (2012) fill-in-the-blank concept maps used to diagnose university-students’ understandingof the ‘wave energy of light’. In her efforts to use concept maps to facilitate learning, she also introduceda more open-ended format where maps were co-constructed by the students. However, even the under-standing expressed in these was assessed with an exclusive focus on proposition-correctness. Aiming toimprove both understanding and attitude towards physics, Broggy & McClelland (2008) used a similarlyopen-ended concept-mapping format where first-year university science students freely constructed mapsindividually and in groups. Their concept-map analysis allows for varying degrees of correctness, butagain with a focus on individual propositions.

A first step towards a more holistic perspective was made by Roth & Roychoudhury (1993) who usedconcept-mapping activities to foster and assess the understanding of high-school students of ‘light as awave’. Although the mapping-task was somewhat restrictive by providing a set of pre-given concepts,the students were allowed to add their own concepts in the course of the exercise. The assessment of themaps moved beyond an atomistic focus on the correctness of propositions by also taking into account

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the hierarchical organisation of the maps. Roth & Roychoudhury’s (1993) investigation may be seen ascomplementary to my study in several respects: The former places an emphasis on the map-constructionprocess in a group-learning exercise of senior high-school students before entering university, whereas thelatter is primarily concerned with the products of individual map construction by students who have justentered university.

Previous studies concerning the conceptual understanding of ‘light’ have thus largely been informed bythe discipline-specific confines of individual propositions and misconceptions. To unleash the true potentialof the concept-mapping tool, I will in this study adopt a more global, learning-theoretic perspective onthe formation and evolution of coherent patterns of understanding.

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Chapter 3

Methods

On the basis of the literature as discussed in the previous chapter, the central research question of thisstudy can now be refined: Can concept maps be used to reveal conceptual understanding of ‘light’ ofphysics students at different stages of their education? The question can be unpacked by breaking itdown into its methodological (How . . . ?), essential (What . . . ?) and developmental (Whither . . . ?)aspects:

• How can concept maps be analysed in order to reveal conceptual understanding?

• What is the essence of conceptual understanding of ‘light’ that emerges from such an analysis?

• Whither does this conceptual understanding develop over time?

As my research question focusses on individual learning, I will attempt to answer it from a construc-tivist point of view (Woolfolk, Hughes & Walkup 2012) using a phenomenological research approach. Inorder to obtain a rich, multi-faceted picture in line with the phenomenological philosophy, I make use oftriangulation (Kuper, Reeves & Levinson 2008) of data (students at different stages, experts), method(concept maps, semi-structured interviews) and analysis (quantitative, morphological, content).

As central part of this study, BSc and MSc students at Imperial College London have been asked toconstruct concept maps of ‘light’ in order to provide snapshots of their understanding at different stagesof their learning trajectories. Quantitative, morphological and content analyses of the maps have beenfollowed up by semi-structured interviews with some of the students to verify and extend the emerginginterpretations of the maps. As a complement and point of reference, an expert concept map of ‘light’ hasbeen generated during a semi-structured interview with the two leaders of the Electromagnetism courseat Imperial College. The experts were able to provide vital background information on the teachingactivities that had informed the student understanding as expressed in the maps.

Discussions of the research approach, site, participants, researcher context and perspective, datageneration, analysis and quality and ethics are given in the following sections.

3.1 Research approach: phenomenology

Phenomenology (Savin-Baden & Major 2013, Reeves et al. 2008) descends from Husserl’s (1968) philoso-phy bearing the same name. Drawing on Macann’s (1993) summary of Husserl’s oeuvre, one may identifythree key ideas which have informed the phenomenological research approach and which I in turn willadopt for this project.

• Consciousness and experience: The central tenet of Husserl’s phenomenology is that knowledgerests on the lived experience of the intentional individual consciousness. As a consequence, phe-nomenological studies aim to describe the essence of participant’s perceptions of a phenomenonwith a focus on meaning. I have used concept maps and interviews (Kvale & Brinkmann 2009) togain access to this experience. In accordance with the phenomenological approach, I will initiallyperform a descriptive analysis of the data and only later complement it with a discussion on thebasis of the theoretical frameworks presented in Sect. 2.2.

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• Adumbrations: Husserl claims that the essence of phenomenon can only be grasped approximatelyby ‘runn[ing] through all the adumbrations specifying every possible aspect’ (Macann 1993, p. 40).Accordingly, phenomenological researchers consider a range of facets of their object of study. I, forinstance, am aiming for insights about the participants’ conceptual understanding, the process ofexpressing it and their learning which has formed it. Analysing the participants’ concept maps, Iwill use triangulation to support a meaning-focused analysis of content with quantitative and mor-phological aspects. Yoshimi (2007) has argued that mathematical tools can indeed be incorporatedinto a phenomenological study, provided that ‘we first trace these concepts back to their sources inintuition’ (p. 280).

• Bracketing: Husserl decides to not base his philosophy on potentially problematic assumptions suchas the existence of an objective reality. Such assumptions are suspended or bracketed so that thecentral argument does not rest on them. Translated to educational research, bracketing meansthe suspension of “previous knowledge in order to take a fresh view of something” (Savin-Baden &Major 2013, p. 216). To achieve this, I will try to cast aside my own past experiences and personalpreferences of physics learning in order to focus on the learning as perceived by my participants.In particular, I have not created my own concept map of ‘light’.

The compatibility of the phenomenological approach with a constructivist perspective is documentedby their common links to cognitive psychology (Anderson 2010) and neuroscience (Hall 2005): Despite re-maining tensions between cognitive science and phenomenology (Zahavi 2004), their close correspondencehas been demonstrated by Arvidson’s (2003) translation of neuroscientific terms into the phenomenolog-ical framework.

The phenomenological approach promises deep insights into learning as the construction of knowledgein an individual mind experiencing objects and events. In my study of learning, potential philosophicalissues regarding the ontology of the latter will be bracketed. Even within learning theory, phenomenologycarries its own intrinsic limitations: Husserl’s phenomenology struggles to account for any ‘subjectivityother than my own’ (Macann 1993, p. 48), let alone inter-subjectivity. Accordingly, my adopted construc-tivist view of learning exhibits a strong asymmetry between learner and learned, where the former is the(sole) recipient and processor of the latter. Any collective learning phenomena or potential impact thelearner might have on the disciplinary body of knowledge is beyond the scope of this study.

3.2 Research site: Imperial College London

I have chosen Imperial College London as my research site, because my personal history of researchingand teaching at the Department of Physics offered unique access to gatekeepers and potential partici-pants. Imperial College is a large, research-led, campus-based university with a mission to ‘embod[y] anddeliver world class scholarship, education and research in science, engineering, medicine and business,with particular regard to their application’ (Imperial College London 2013d). The Department of Physicsis one of the largest in the UK with 148 permanent staff members employed and 890 students enrolledin six undergraduate programmes in 2011 (Imperial College London 2011).

The students are recruited from both within the UK and overseas via a strict selection process(Imperial College London 2013c). This implies that, while including a broad range of cultural back-grounds, study participants recruited from Imperial College are likely to exhibit a bias towards high-achieving students. In terms of curriculum and teaching methods, the selected research site may beregarded as similar to other higher education institutions within the UK and internationally, with thedistinguishing feature of a particularly broad range of different physics courses offered (Imperial CollegeLondon 2013a).

3.3 Participants: BSc and MSc students and experts

The envisaged quantitative component in the concept-map analysis warrants a sample size of at least10-20 participants in each group, where a much larger size would go beyond the modest scope of thisstudy.

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Stefan Yoshi Buhmann 3 Methods

First degree in Physics ParticipantsImperial College London, UK MSc 6, MSc 9, MSc 12Other UK university MSc 3, MSc 4, MSc 7, MSc 10, MSc 13, MSc 16University outside the UK MSc 1, MSc 2, MSc 5, MSc 8, MSc 11, MSc 14, MSc 15

Table 3.1: Educational backgrounds of the MSc participants.

BSc participants (n = 19: effective random sampling; n = 3: typical-, unique-case sampling). Out ofa total of 240 students enrolled in the first year of any of the Physics BSc (or MSci) degree programmesat Imperial College in 2013, the 21 members of one tutorial group within the course Professional Skillsfor Physicists were asked to participate by contributing a concept map and 19 agreed. They have beenrandomly assigned labels BSc participant 1-19 by order of their submitted maps. With the first-yearstudents being randomly distributed into tutorial groups (effective random sampling), the participatinggroup was chosen for convenience: It was one of the two groups tutored by the course administratorhimself who had provided access to the participants (the other group contained a member of a vulnerablegroup and was avoided for this reason).

Based on the analysis of the submitted maps, 3 students were selected for interviews. Aims of theselection were to represent both typical and unique maps and to obtain information on a range of aspectsemerging from the concept-map analysis (see Sect. 4.1). Initially, a sample of 3 students matching thesecriteria were invited to an interview, 1 of them agreed. In a second round, all remaining students wereinvited. Of the 7 students responding, 2 were able to participate within the prescribed time frame, andthe resulting sample matched the above criteria.

The BSc participants are students who have displayed a strong interest and talent for physics dur-ing their secondary education, but have not yet been exposed to any university-level knowledge aboutelectromagnetism or light.

MSc participants (n = 16: comprehensive sampling, n = 3: typical-, unique-case sampling). All ofthe 29 students enrolled in the MSc in Physics programme at Imperial College in 2013 were invited toattend a teaching session entitled Writing a Literature Review and Using Concept Maps. 17 studentsattended and 16 of these agreed to contribute a concept map produced during the session. They havebeen randomly assigned the labels MSc participant 1-16. Following the same sampling strategy as above,3 students were chosen for interviews. None of these students of the first recruitment round responded,but 3 of the 8 students responding to the second, open round were able to take part, fulfilling the abovecriteria.

The MSc participants have obtained a first university degree in Physics at a variety of institutions:3 have previously studied at Imperial College, 6 have attended another university within the UK and7 have previously been enrolled at universities abroad (cf. Table 3.1). They have all attended a mainElectromagnetism course and other lectures and courses which have made reference to ‘light’. In addition,they have previously worked on a research project.

Experts (n = 2: critical-case sampling). Electromagnetism and ‘light’, in particular, are primarilytaught in the second-year Electromagnetism course which is a core course of all undergraduate Physicscurricula at Imperial College. From 2011 to 2014, the course has been taught by two course leaderswho deliver the lectures in the first and second half of the second term, respectively. Owing to theirunique expertise and experience in electromagnetism teaching, both course leaders have been invited toa joint interview and both agreed (critical-case sampling). They have been randomly assigned the labelsExperts 1-2, by order of their first response during the interview.

The experts are able provide privileged insights into electromagnetism and light, providing a pointof reference for the desired conceptual understanding. In addition, they contribute their knowledge onthe teaching of electromagnetism which has informed the understanding of the MSc participants. Itcan be assumed that they have taught those MSc participants 6, 9 and 12 who have completed theirundergraduate studies at Imperial College.

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3.4 Researcher context and perspective

I am myself a physicist by training, having studied the subject at the Universities of Cologne (Germany),New South Wales (Australia) and Jena (Germany). In addition, I have worked as a researcher and teacherof physics at Imperial College for six years, including a role as tutor for Electromagnetism for two years.My resulting insider perspective is both beneficial and potentially problematic. On the positive side, myknowledge of the subject matter enables me to analyse developing understanding of conceptual knowledgein physics on a level inaccessible to outsider ethnographers like Nespor (1994).

On the other hand, my inside expert knowledge on the subject matter brings the risk of narrowing myview to intricate details of the subject matter as opposed to potentially discipline-transcendent processesof conceptual learning. To avoid this problem, I have purposefully directed my attention away from thecorrectness of individual physics propositions which often lie at the heart of discipline-specific studies. Inaddition, there is a risk of confusing the views of the participants with my own personal experience as aphysics learner. I attempted to reduce the interference via bracketing, i.e., by providing the participantswith room to express their view without interruptions and by focussing on their experience. To reducepossible interference further, I chose not to create my own concept map of ‘light’ at any stage of theresearch.

3.5 Data generation

A concept map of ‘light’ was generated by each of the BSc and MSc participants after instructive sessionson concept-mapping. About two months later, 3 participants from each of these two groups attendedindividual semi-structured interviews. Independently, the Experts 1 and 2 gave a joint semi-structuredinterview which included the construction of an expert concept map of ‘light’.

3.5.1 Student concept maps

Concept maps were generated as part of the regular curricula. All first-year Physics BSc students atImperial College attend a weekly Professional Skills for Physicists course in groups of roughly 20 students.During the first three weeks of this course, they research a topic of their own choice in groups of 5-6students. As support, I prepared a 10-minute presentation Tips on Researching Topics and StructuringYour Talk to be used in week 2 by all session tutors, which I personally delivered to the group chosenas participants for the study (recall Sect. 3.3). The presentation included rules for generating conceptmaps and an incomplete example map on ‘the universe’ (see App. A); the latter was inspired by Novak& Cañas’ (2008) expert skeleton map for the focus question ‘What is the structure of the universe?’. TheBSc participants were asked to individually draw a concept map of ‘light’ on a sheet of A3 paper. Theywere told to work without time limit and indicate when they felt they had completed the task, whichfor this group was the case after 15-20 minutes. This time frame is at the lower end of Hay, Kinchin &Lygo-Baker’s (2008) suggestions that ‘[t]he method can be taught in 10-20 minutes, and most studentswill find another 20-30 minutes sufficient to construct a reasonable map’ (p. 302).

MSc Physics students in their first year at Imperial College undertake self-study projects of their ownchoice. I offered a session entitled Writing a Literature Review and Using Concept Maps to guide themduring this project. The session followed the same protocol as described above where the MSc participantsagain took about 15-20 minutes to produce the concept maps. The original session was missed by morethan half of the students due to short notice, so I offered a repeat session.

Kuntz (2010) urges ’qualitative scholars to critically recognize the impact of the spatial on their re-search’ (p. 152). The settings used for the concept-mapping activities were standard seminar rooms asavailable. The respective rooms were well filled by the 21 students taking part in the BSc session, but onlyvery sparsely populated during the MSc sessions. They provided an environment that the participantswere comfortable and familiar with. The BSc participants were seated at group tables while the MScparticipants were seated in rows. The former arrangement provided some distractions for this intendedindividual exercise, but also seemed to trigger an atmosphere of high motivation.

During the concept-mapping activity, the participants worked in quiet concentration at various paces.For some students, phases of reflection preceded intense map-drawing, some discarded their first draft

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Participant Recording Location

BSc participant 7 49:04 minutes Office + Optics Reading RoomBSc participant 8 30:50 minutes Optics Reading RoomBSc participant 17 34:59 minutes Office

MSc participant 5 37:08 minutes OfficeMSc participant 10 51:33 minutes OfficeMSc participant 12 50:39 minutes OfficeExperts 73:21 minutes Office

Table 3.2: Lengths and locations of recorded semi-structured interviews.

map or used scrap paper before drawing a final map. In each session, about 1-3 participants used mobilephones, possibly to look up facts. The BSc participants were mildly distracted by the regular sessioninstructor who individually asked them about their group projects.

I made efforts to obtain fresh, unbiased data by providing the students with as little prompts aboutthe content of the maps as possible and by not mentioning the topic ‘light’ until immediately prior to themapping exercise. The provided example map ‘the universe’ was chosen to have similarities in structure,but no overlap in content with the concept maps on ‘light’ to be generated. Their observed heterogeneityin content and structure demonstrates the success of this strategy with one exception: The Einsteinequation E = mc2 appeared on an unusually high number of maps, which may have been prompted byits inclusion on the example map.

The mapping task itself with specified root concept ‘light’ but without suggested or prescribed con-cepts offers a high degree of freedom in content and structure (Cañas & Novak 2012). Providing a rootconcept was preferred over the comparably open-ended alternative of using a focus-question which wouldhave been likely to prompt a bias towards a specific aspect of light.

In order to achieve rich, high-quality maps, I did indicate possible content types (mathematical orphysical concepts, properties, formulas, examples, procedures) during my instructions and reminded theparticipants to label their links. To minimise the impact of my instructions on concept-map variation, Itook care to convey exactly the same information about concept-mapping in each of the three sessions.

My two different roles as teacher and researcher as assumed during the concept-mapping sessionsshared common interests: The teacher aimed at providing the students with good concept-mapping abil-ities, which allowed the participants to generate high-quality maps to the benefit of the researcher.

3.5.2 Student interviews

The individual semi-structured phenomenological research interviews with the BSc and MSc participantswere partly informed by the preceding concept-map analysis. They were led by myself on the basis ofthe interview protocol given in App. B.1. This protocol included a definition of ‘light’, a discussion ofthe concept-map creation as well as the physics content of the participant’s map, some specific physicsquestions related to the map and a broader conversation about learning in physics. Finally, each partic-ipant was presented with a preliminary analysis of their map as obtained during this study and givenan opportunity to comment. The interviews were audio-recorded in full with a digital recording device;these recordings varied between 30 and 50 minutes in length, see Table 3.2. The interviews took place inan office, or if available, in the slightly more spacious Optics Reading Room. Both locations constitutednatural university environments with which the participants were familiar.

The interviewees all seemed interested in the topic and keen to contribute. They were generally veryarticulate, took time to reflect and answered with care and consideration. Some of the participants seemedto partially perceive the specific physics questions as an assessment.

I as the interviewer had acquainted myself with the recording device in a test run. Having no priorexperience in interviewing, my interviewing skills were mainly based on Kvale & Brinkmann’s (2009)in-depth advice. To facilitate an improvement of these skills from reflected experience, I took care toconduct only one interview on the first day. In particular, I learned from this first interview to leave

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time for the participants to reflect on my questions, to allow them to elaborate these answers withoutinterruptions and to stress that the interview does not constitute a test.

During the first interview (BSc participant 07 ), the audio recording failed. The entire interviewwas repeated immediately afterwards. As a result, the answers of this interview may be slightly lessspontaneous. On the other hand, the effective rehearsal of the interview may have counter-balanced myinitial lack of interviewing skills.

3.5.3 Expert interview and concept map

The joint semi-structured interview with the Experts was informed by an analysis of the second-yearElectromagnetism syllabus as well as the student concept maps and interviews. It was conducted bymyself in an office and took slightly longer than the student interviews. Again, the entire interview wasaudio-recorded, see Table 3.2.

The interview was loosely based on the interview protocol given in App. B.2 which included twoinitial questions about ‘light’, the collaborative generation and discussion of an expert concept map, adiscussion of the specific physics questions also presented to the students, and some questions aboutteaching and learning in Electromagnetism. The concept-map generation was supported by 24 suggestedconcepts which I had extracted from the syllabus of the Electromagnetism course. In this way, the expertswere prompted to focus their map on the understanding of the ‘light’ concept taught in their course asopposed to their personal views informed by their research experience. The interviewees added 8 newconcepts during the exercise.

The interview took place in a friendly, collegial atmosphere which benefited from the interviewees’excellent rapport with each other. They were both very open and willing to share their expertise inelectromagnetism and teaching, creating a balanced discussion. To maximally profit from their uniqueinput, I deviated substantially from the interview protocol, allowing the discussion to progress freelywith occasional prompts towards my original protocol. In particular, an alternative, very open concept-mapping format evolved during the interview where a strong focus is placed on vertical hierarchy andlinks are indicated by proximity rather than labelled lines.

3.6 Data analysis

The concept maps were analysed in three stages: quantitatively, morphologically and with respect tocontent. The findings from each stage informed the subsequent stages which in turn substantiated theirpredecessors. The student interviews were conducted and analysed based on meaning condensation andcoding after completion of the concept-map analysis. The expert interview took place after the studentinterviews, but prior to their analysis. Its analysis and an investigation of the emerging expert conceptmap was performed last.

3.6.1 Concept maps

Following the phenomenological idea of adumbrations, the concept-map analysis was aimed at separatingand then recombining different aspects: quantitative measures, morphological features and content. As apreparation for both the quantitative and morphological investigations, the original concept maps weretransformed into normalised and comparable forms in two steps: First, the concept maps were redrawnwith all concept- and link-labels removed. This step led to a content-free map which was faithful instructure and geometrical layout to the original, see Fig. 3.1. Here, concepts are represented by opencircles and the root concept is indicated by a shaded square. To ensure greater comparability among theparticipants, boxes with multiple content were occasionally split into two or more boxes; and multi-linkswith a single common label were elevated to concepts (Fig. 3.2).

In a second step, the content-free maps were geometrically rearranged to facilitate an easier comparisonof their structure (Fig. 3.3). During this topological normalisation, the root concept was placed at thetop of the map and the other concepts were arranged on levels corresponding to their distance fromthe root concept. Such a procedure had been applied previously by Koponen & Pehkonen (2008) withthe aid of an automated graph-theoretical software. In the manually implemented normalisation of this

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(i) Original map

(ii) Content-free form

Figure 3.1: Concept- and link-label removal for the concept map of MSc participant 12.

study, branches were further ordered from left to right according to their depth while striving for greatestpossible balance among the branches in cases of ambiguous sub-branch assignments. The exact rulesleading to unique normalised maps are shown in App. C.

The normalised maps provided a convenient starting point for the quantitative analysis. This stepfocussed on the pure structure of the maps without referring to any content or even specific geometricallayout. From a mathematical point of view, the normalised maps are graphs: collections of verticesand edges (Gould 1988). Drawing on ideas of graph theory, some basic structural parameters of theparticipants’ concept maps were extracted by a simple counting procedure. These were used to calculateadditional higher structural parameters. For details, see Sect. 4.1.1.

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(i) Concept splitting

(ii) Multi-link elevation

Reflection

Reflection

TransmissionTransmission

ApplicationsApplications

Figure 3.2: Normalising procedures applied during concept- and link-label removal.

Figure 3.3: Topological normalisation of the concept map of MSc participant 12.

The morphological analysis was carried out by simultaneous comparative visual inspection of allstudent concept maps in their content-free, normalised forms. The maps were grouped into 6 classesof similar morphological appearance and the maps in each class were ordered by the number of theirconcepts. In a second step, a note containing the parameters from the quantitative analysis was attachedto each map. The parameters of the maps within each class were compared to identify typical values orranges for a given class. These ranges were subsequently used to re-asses the class assignment of each map.This last step refined the morphological classification and ensured its consistency with the quantitativeanalysis.

The content analysis was performed separately for the BSc vs. MSc participants. For each group, alist was composed that contained all concepts in alphabetical order together with the number of mapsmentioning this concept. Next, the concepts were re-ordered by the number of maps mentioning them.Finally, concepts were classified according to their content type. In addition, mixed structure–contentinformation was extracted from the maps. This included creating ordered lists of concepts most stronglylinked to other concepts, misconceptions and the content of cross-links.

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3.6.2 Interviews

The interviews were transcribed by a professional transcription service in intelligent verbatim style (fordetails, see App. D). The first few minutes of the interview with BSc participant 7 were transcribed by theresearcher himself to provide the transcription service with a style sample. The full transcript as receivedfrom the service was verified by comparing it with the recording. During this process, phrases markedas inaudible or uncertain by the provider were completed and long sentences were broken up in order toenhance readability. The service was asked to apply this last step to all subsequent interviews. For these,passages marked as inaudible by the service were completed and a proofreading was carried out. Thisincluded a referral to the recording for problematic passages such as incorrect uses of physics-specificterms by the non-physics transcribers.

The transcripts were subjected to meaning condensation. This “entails an abridgement of the meaningsexpressed by the interviewees into shorter formulations” (Kvale & Brinkmann 2009, p. 205), which wereplaced into in the transcript margins. In a second step, coding was applied by associating these shorterformulations to categories. These categories corresponded to the different stages and questions containedin the interview schedules (App. B).

3.6.3 Quality

A range of procedures was applied to ensure the reliability and consistency of the data analysis (Cohen,Manion & Morrison 2011). Firstly, each step of the data analysis was applied in a compact time windowacross the entire sample. During the quantitative analysis, check-sums were used to spot errors (seeApp. E). The mutual consistency of quantitative and content analysis was ensured by referring backto the content-removal procedure for identification of concepts during concept extraction. In addition,the number of concepts extracted from each map was compared with the number of vertices of thecorresponding normalised maps from the quantitative analysis. The re-ordering and classification duringcontent analysis was performed with the aid of word-counting tools. Interview transcripts were read forlogical consistency and, in cases of inconsistencies, compared with the recordings. In addition, they weresent to the participants for inspection together with a list of all direct and indirect quotes used in thisdissertation. Of the 8 participants contacted, 4 replied within the time-frame necessary for inclusion inthis dissertation and voiced agreement with these quotes (member checking).

3.7 Ethical considerations

Central ethical issues for this study involve maintaining the anonymity of the participants and treatingtheir data with respect. I ensured the former by assigning each participant with a label (see Sect. 3.3)and storing their identities on a separate, encrypted device. Concept maps and interview transcripts werestored and cited with reference to the participant labels only. The transcription service provider workedwith the anonymised recordings and agreed to treat these confidentially (see App. D). Anonymity wasfurther assisted by the facts that the BSc and MSc participants were drawn from rather large poolsof students. This is not the case for the two Experts who could in principle be identified via theirrole at Imperial College. However, as not disclosing the research site would have strongly reduced thetransferability of the study, they consented to this risk. The danger of hurting the feelings of participantsby revealing deficiencies in understanding was addressed by avoiding judgemental statements in the dataanalysis and offering all participants the chance to obtain feedback after reading the report on this study.

Power issues were not present as the researcher was not involved in any assessment of the studentparticipants or indeed in any of their current or future teaching apart from the session on concept mapping.To avoid any educational advantage from this session, it was offered to all students regardless of theirparticipation. No vulnerable groups were involved. Informed consent was sought from all participantswhich they were able to withdraw at any time.

With these considerations, the study was classified as minimal risk and in accordance with the Eth-ical Guidelines for Education Research by the British Educational Research Association (2011). Ethicalapproval was obtained via the Education Ethics Review Process by Imperial College London (2013b).Details on the decisions I made to ensure an ethical conduct of my research can be found in App. F.

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Chapter 4

Findings

As a first step in addressing the central research questions, I will in the following report on the findingsfrom the data analysis described in the previous chapter. In accordance with the phenomenologicalapproach, I will describe them with a focus on the essence of the student experience of their developingconceptual understanding and defer interpretations in terms of theoretical frameworks to the next chapter.The presentation roughly follows the order in which data and results were obtained: Starting from theanalysis of the concept maps, I will subsequently substantiate and extend this analysis with the aid ofthe student interviews. The student views will be compared and contrasted with the teacher perspectivefrom the expert interviews.

4.1 Student concept maps

The analysis of the student concept maps proceeds in three consecutive stages: quantitative, morph-ological, content. Each stage marks the point of departure for the following stage.

4.1.1 Quantitative analysis

Drawing on ideas from graph theory (Gould 1988), the structural complexity of concept maps can bequantified via characteristic parameters. In the following, the parameters and their potential relevance willbe introduced in everyday terms and the observed ranges will be discussed. More precise mathematicaldefinitions of the parameters can be found in App. E. We begin with some basic structural parameterswhich can immediately be read off the topologically normalised concept maps, recall Fig. 3.3. They areillustrated in Fig. 4.1; and their values for each of the participants are shown in Table G.1 in App. G.

Figure 4.1: Basic structural parameters of MSc participant 12.

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Number of concepts. For the present mapping task without pre-given concepts, the most basicstructural parameter is the number of concepts. Mathematically, it corresponds to the order which isdefined as the number of vertices. The maps of the BSc participants feature about 31± 12 concepts withsome exceptionally large maps containing up to 70 concepts. The number of concepts on the maps createdby MSc participants is slightly less, ranging around 28± 13 concepts, with a maximum of 65 concepts.

The observed numbers show that the open-ended mapping task with root concept ‘light’ triggersrich associations within the students, who easily conceive a large number of concepts. The variationbetween the students is relatively large. The slightly reduced concept number for the more experiencedMSc participants suggests that the concept number alone is not a good measure of the extent of studentknowledge, possibly hinting towards a deeper understanding underlying these smaller maps.

Number of links. The number of links, mathematically defined as the size or number of edges, reflectsthe connectedness of the concept map. The observed parameter ranges of 33±12 for the BSc participantsand 30 ± 14 for the MSc participants are indicative of well-connected conceptual understanding. Theobserved variability is slightly larger than for the number of concepts.

The number of links lies at the heart of content-based scoring criteria for concept maps (Novak &Gowin 1984), where it represents the number of correct propositions. In this purely structural analysis,the emphasis is on the perception of a connection by a student, irrespective of whether the student hasfully developed the corresponding precise proposition.

Diameter. The diameter is the greatest distance across the map in any given direction (Sanders et al.2008). The observed relatively large parameters 7±2 are indicative of the presence of chain-like structuresin the concept maps.

Radii. The in-radius and ex-radius measure the minimal and maximal distances from the root conceptto the periphery of the map, respectively. The rather small typical values of 1.5± 0.5 (BSc) and 2.0± 0.4(MSc) for the in-radius signify the presence of under-developed areas of understanding on most conceptmaps, while the larger ex-radii of 4±1 show that very well-developed areas of knowledge are also present.

Degree sequence. Finally, the degree of a given concept of the map is the number of concepts to whichit is connected. Koponen & Pehkonen (2008) refer to degree 1 concepts as outliers, degree 2 concepts asjunctions and higher-degree concepts as hubs. The degree sequences of the individual maps as displayedin Fig. 4.2 exhibit a large amount of variability, in particular for the MSc participants. However, onedoes observe a tendency of the BSc maps to be strongly dominated by outliers and a few hubs of veryhigh degree, signifying spoke-like overall structures. The MSc map on the other hand exhibit enhancednumbers of junctions and low-degree hubs. This points towards the presence of chain-like and web-likestructures. These different trends are confirmed by Fig. 4.3, where the percentages of concepts with givendegrees are displayed as averages across the two groups.

The degree of the root concept determines the gross structural organisation of a concept map. Here,we observe an extreme variability among both BSc and MSc maps with root degrees ranging between2–10. This indicates strong individual differences regarding the basic organisation of the concept maps.

The basic characteristic parameters have been used to calculate higher parameters which convey morecomplex structural information. The observed values are displayed in Table G.2 of App. G.

Cross-linkage. The cross-linkage is the number of excess links (which are not required to hold theconcept map together) relative to the total number of links. Note that this total number is unique whereasthe decision as to which particular link is a cross-link is not. The observed values for both BSc and MScmaps exhibit an extreme variability of 9% ± 7% and 12% ± 10%, respectively. The observed abundanceof maps with almost no cross-links hints towards an untapped potential of knowledge integration.

The number of cross-links is again a central element of Novak & Gowin’s (1984) original scoringscheme. In their content-based scheme, a cross-link has to connect distinct parts of a concept map. Inour purely structural scheme, the ambiguity as to which link is a cross-link is lifted.

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Stefan Yoshi Buhmann 4 Findings

(i) BSc participants

(ii) MSc participants

Degree

Degree

Relative number of concepts

Relative number of concepts

Figure 4.2: Degree sequences for individual concept maps.

Dimension. The dimension is a parameter relating the number of concepts with the diameter of aconcept map. In close analogy to the dimension of fractal sets (Mandelbrot 1967), the dimension has asimple intuitive interpretation where maps of dimension 1 are dominated by linear, chain-like structures,maps of dimension 2 typically exhibit branches with few cross-links and higher-dimensional maps indicate

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(i) BSc participants (ii) MSc participants

Figure 4.3: Average degree sequences across the two groups.

a high degree of inter-connectivity. The observed dimensions of the participants’ concept maps rangearound 1.6 ± 0.1, midway between one- and two-dimensional. The full potential of connectivity offeredby a two-dimensional page has not been explored in full.

Balance. The balance of a concept map is the ratio between its in- and ex-radii. It is a measureof how balanced the generally skew-shaped topologically normalised graphs are. The observed values41% ± 21% (BSc) and 49% ± 12% (MSc) exhibit a large variability ranging from very balanced mapswith balance 100% at one extreme to very imbalanced maps (balance 17%) at the other. The latter mapsexhibit areas with a particular potential for future learning: As indicated by Reichherzer & Leake (2006),concepts close to the root concept are potentially particularly relevant. For very imbalanced maps, manyof these are not sufficiently substantiated by embedding them in a network of related concepts (see alsoSect. 5.1.4).

To summarise, the quantitative analysis of structural parameters has set the stage by providing therange against which to compare individual concept maps. Common features of most maps are a relativelysmall degree of cross-linkage accompanied by a low dimension. Signifiers of individuality are the largelyvariable root degrees and the strongly differing balances. Differences between BSc and MSc participantsare most prominent in the degree sequences, where indications of spoke-like vs. chain- and web-likestructures have been observed. In the next section, I will use the structural indicators to identify globalmorphologies.

4.1.2 Morphological analysis

Visual inspection of the topologically normalised maps reveals that they all roughly resemble (partsof) trees. I therefore chose to classify them on the basis of flora-based analogies. Using the parametersroot degree, balance and cross-linkage which have been seen to provide the strongest contrast across thesample, it is possible to distinguish 6 classes as shown in Fig. 4.4.

Seed (Indicator: Unconnected). Seed maps exhibit several distinct, unconnected areas of knowledge.These areas act as seeding points from which new knowledge can form to eventually result in a connectedmap. Potential for learning lies primarily in the creation of cross-links to connect the different seeds.

Sprout (Indicator: Balance < 20%). Sprout maps display regions of very detailed, deeply developedknowledge, possibly areas in which the learner has a particular interest. These are contrasted by areas

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(i) Seed (BSc participant 5) (ii) Sprout (BSc participant 15)

(iii) Crown (BSc participant 13) (iv) Tree (BSc participant 11)

(v) Root (MSc participant 1)

(vi) Fungus (BSc participant 17)

Figure 4.4: Morphological classes for concept maps.

of partial knowledge. Potential for learning lies particularly in these areas which can easily be identifiedfrom the topologically normalised maps.

Crown (Indicators: Root degree ≥ 5, short chains). Crown maps are very broad, with many linksemerging from the root concept. The many end points or leaves are areas to which new learning caneasily be added. Crown maps indicate a large degree of receptiveness or learning-readiness.

Tree (Indicators: Root degree 2–6, average chains). Tree maps are a balanced intermediate between thevery broad crown maps and the deep root maps. Different areas of knowledge are equally well developed.Potential learning can involve both a deepening or broadening of understanding and in particular, theforming of new connections.

Root (Indicators: Root degree ≤ 4, long chains). Root maps exhibit very deep and long chains, indicativeof detailed, typically procedural knowledge. New learning should perhaps seek to place these long chainsinto a broader context and create cross-links between the different chains.

Fungus (Indicator: Cross-linkage ≥ 13 %). Fungus maps are morphologically distinguished by theirhigh degree of interconnectedness. The many cross-links connect different areas of knowledge to form awell-integrated, holistic understanding. Particular care must be taken when new learning requires therearranging of what might be a stable and relatively inert network.

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Class Participants

Seed BSc 5, MSc 15

Sprout BSc 7, BSc 8, BSc 10, BSc 15

Crown BSc 2, BSc 4, BSc 13, BSc 16, BSc 19, MSc 8, MSc 16

Tree BSc 1, BSc 6, BSc 11, MSc 3, MSc 5, MSc 9, MSc 12

Root BSc 12, MSc 1, MSc 7, MSc 10, MSc 11, MSc 14

Fungus BSc 3, BSc 9, BSc 14, BSc 17, BSc 18, MSc 2, MSc 4, MSc 6, MSc 13

Table 4.1: Morphological classification of the participants’ concept maps.

The assignment of the participants to these partially overlapping classes is shown in Table. 4.1. Theclassification scheme has been implemented by first identifying the seed, sprout and fungus maps andthen placing the remaining maps on the crown–tree–root continuum.

One observes that the three classes seed, tree and fungus are equally populated by BSc and MScparticipants. This has very different implications for those classes and their exponents. The fact thatseed maps occur even for the experienced MSc participants perhaps points towards problems with thegraphical tool of concept mapping alternative to the interpretation in terms of student understandingproposed above. The tree class represents a very typical or average morphology, so it is not surprisingthat exponents of this class can be found within both the BSc and MSc groups. The fact that fungus-classconcept maps are equally prominent amongst the BSc participants as amongst the MSc participants isa surprise. It indicates that connectedness of understanding is subject to a large degree of variabilityamong learners and may be quite well-developed even at high school level. On the other hand, theoverall rather low cross-linkage across the entire sample might indicate that an integration of knowledgecould potentially be emphasised more in physics teaching. For alternative interpretations in terms of themapping process, see Sect. 4.2.1.

Marked differences between the two student groups can be observed in their population of the sprout,crown and root classes. The sprout class is exclusive to BSc participants who also dominate the crownclass. The prevalence of these two classes for learners at the beginning of their education confirms theabove interpretation in terms of receptiveness or learning-readiness. MSc participants, on the other hand,overwhelmingly populate the root class. This class seems best adapted to their detailed and specialisedknowledge. One may conclude that the development of conceptual understanding during undergraduatephysics education manifests itself morphologically via a transition from imbalanced or broad maps towardsdeep maps.

4.1.3 Content analysis

Having analysed the pure structure of the concept maps, we next turn our attention to their content. Asa measure of individuality of content, Fig. 4.5 shows how many distinct concepts are included by 1, 2 etc.participants on their maps. One observes that the content expressed on each map is highly individualwhere a large fraction of concepts is mentioned by one or a few participants only. This is highly surprisingin view of the very homogeneous groups of learners mapping a very central concept of their discipline.Comparing the two participant groups, the MSc maps show a relative enhancement of concepts mentionedby 3–5 members. This may be interpreted as evidence for the formation of sub-disciplinary groups amongthese more advanced students according to their chosen specialisation within physics.

A common inventory of key concepts can still be identified for each of the groups. Concepts men-tioned by at least 7 of the 19 BSc participants are listed in Table. 4.2. To summarise the essence of ‘light’according to these concepts, one notes that the competing ontologies of light as wave or particle/photonare at the focus of attention. They are substantiated by the associated key experiments (double-slitinterference, photoelectric effect). This is followed by the processes that light can undergo (refraction,reflection, diffraction), its basic properties (speed of light, energy, polarisation, phase velocity, frequency,wave length), the electromagnetic spectrum/colour and its various parts (radio, infrared, visible, ultravi-olet, x-ray). In addition, some applications of light are mentioned (fibre optics, laser).

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Fraction of concepts

. . . shared by # Participants

Figure 4.5: Concepts shared among the participant groups.

Concepts mentioned by BSc participants

Electromagnetic wave, Particle [18]

Refraction [16]

Diffraction [15]

Speed of light [14]

Reflection, Speed of light (value) [13]

Photoelectric effect, Wave-particle duality [12]

Photon, Spectrum, Ultraviolet, Visible [11]

Colour, Fibre optics, Gamma, Infrared, Photon energy, Polarization [9]

Laser, Phase velocity, Radio, X-ray [8]

Frequency, Double-slit interference, Wave length [7]

Table 4.2: Concepts most frequently included in the maps of the BSc participants. The numbers ofparticipants mentioning a concept are given in square brackets.

Table. 4.3 lists concepts mentioned by at least 5 of the 16 MSc participants. While the wave vs.particle nature remains a focus also for these more advanced learners, other content included by the BScparticipants has vanished (the spectral components, most basic properties of light) or lost importance(light-related processes). Instead, two new elements capture the participants’ attention: the differenttheoretical frameworks for describing light (Maxwell equations, quantisation, quantum physics, specialrelativity) and aspects of how light can be created (light-matter interaction, sources).

A marked shift in focus between the two participant groups is also manifest in the content type.This can be shown by classifying all concepts included on any concept map into the categories physicsconcepts (electromagnetic wave, reflection, Snell’s law . . . ), formulas (c = 3× 108 m/s, E = h̄ω . . . ),theories (geometric optics, special relativity . . . ), experiments (photoelectric effect, Michelson–Morleyexperiment . . . ), applications (laser, CD, information transfer . . . ), history of science (Planck, corpuscle,aether . . . ), non-physics concepts (photosynthesis, colour, beauty . . . ) and structuring categories (prop-

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Concepts mentioned by MSc participants

Electromagnetic wave [15]

Photon [13]

Laser [9]

Energy, Interference, Maxwell equations, Particle, Photon energy, Wave length [8]

Frequency, Light-matter interaction, Speed of light [7]

Applications, Colour, Diffraction, Photoelectric effect, Quantization,Quantum physics, Reflection, Sources, Special relativity, Spectrum [5]

Table 4.3: Concepts most frequently included in the maps of the MSc participants.

Fraction of concepts

Figure 4.6: Concepts shared among the participant groups.

erties, devices, evidence . . . ). Fig. 4.6 displays the number of distinct concepts mentioned within each ofthese categories. Physics concepts strongly predominate in both participant groups, followed by formulas,applications and, rather surprisingly, non-physics concepts. Comparing the participant groups, one ob-serves that the MSc participants lay an even stronger focus on physics concepts and stress formulas andtheoretical frameworks. On the other hand, the focus on applications, history of science and non-physicsconcepts is markedly reduced in comparison with the BSc participants.

Combining structural and content information, one can identify common misconceptions in the par-ticipants’ understanding. A total of 18 instances of a misconception were observed within the BSc maps,which is a very small number. The dominant misconception concerns polarisation (7 instances): thisproperty of light is often wrongly categorised as a process. This can be addressed in teaching, as canoccasional wrong connections (e.g., classical electromagnetism–particle). Other misconceptions involveincomplete knowledge about topics not yet covered (neutrinos faster than light) and erroneous referencesto history of science (electromagnetic light–corpuscles, aether–phlogiston). The former are an indicatorof learning readiness rather than true misconceptions while the latter show that history of science has tobe used with great care. No misconception was observed on the MSc participants’ maps.

In summary, the content analysis has revealed that the concept maps produced by the participantspossess a common core, but are also highly individual. This is especially true for the MSc participants,where hints towards the formation of specialised sub-communities have been found. The development of

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understanding during university physics education seems to be marked by a motion away from applica-tions and non-physics concepts towards physics concepts, formulas and theories. The analysis has furtherrevealed polarisation as a potentially problematic topic.

4.2 Student interviews

The interviews with the student participants are able to provide additional evidence for or against theinterpretation of the maps put forward in the previous section. They add more detail and provide infor-mation not available from the maps alone. The findings from the interview analysis will be presented inprogression along this continuum from verification (Sects. 4.2.1 and 4.2.2) and enrichment of information(Sect. 4.2.3) to completely new insights (Sect. 4.2.4 and 4.2.5).

4.2.1 External influences on the concept-map creation

The concept-map analysis given in the previous section attributed both morphology and content of themaps entirely to the conceptual understanding of the participants. The interviews have revealed a rangeof possible alternative influences like participant intention and time constraints.

Concept maps with sprout morphologies have been interpreted as an indication of areas of partialunderstanding. As a competing explanation, these maps may simply not have been fully completed dueto time constraints. Of the 6 students interviewed, 3 stated that they would have further expanded theirmap given more time. A BSc participant who produced a sprout map states:

“I ran out of time. I don’t think the map is complete and I was going to expand on thatsection, on the applications of light.”

The section he is referring to is indeed one of the two branches on his map containing only few concepts.On the other hand, an MSc participant whose two main branches wave vs. photon are very differentlydeveloped confirms that sections with fewer concepts are partially due to less knowledge and confidencein the respective areas of physics:

“[Redrawing the map now] I would want to do more on the photon aspect and I think that’spartly because I’ve learnt a lot more since.”

In support of this, the 4 participants who indicated areas where they want to learn more consistentlypointed towards less developed branches of their maps.

Another limiting factor for the full development of the maps is task interpretation. An MSc participantwhose number of concepts (26) is at the lower end of the typical range, states:

“In the context of your lecture [Writing a Literature Review and Using Concept Maps], it wasabout getting the idea of the map down. So I thought, ‘okay, I’ve done it, I get the picture.’. . . I wasn’t actually trying to make a map of ‘light’.”

Another feature of the maps which is potentially influenced by their creators’ intention and modeof working is the number of cross-links. Two participants with high cross-linkages 19% and 14% wereexplicitly aiming to find connections; the latter explains:

“I liked finding ideas that were connected in some way. . . . The ideal scenario was somethingthat was connected like a web.”

A participant with a very crowded map (60 concepts) attributes his low cross-linkage 3% to lack of spaceand aesthetic reasons:

“I thought to myself, ‘I could link [these three concepts]’ . . . but it’s not very important andit will make a mess.”

These spatio-temporal limitations aside, an MSc participant who has split his map into two centralsections associated with the wave and particle pictures confirms that the absence of cross-links can bean indicator of conceptual difficulties:

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“The big thing I did was: I split it into two parts mostly. . . . In terms of crossing over betweenthe two of them, the only [cross-link] I did was between spectra and the frequency of thephoton.”

4.2.2 Polarisation as a common misconception

The interviews have confirmed that the concept of polarisation is a challenge to a considerable fractionof undergraduate students. The 3 BSc participants interviewed had all not included polarisation ontheir maps. When prompted by being asked to name all essential properties of light, only one mentionspolarisation. One of the other 2 states that this topic was covered only at a very basic level at school.An MSc participant supports this:

“I didn’t know polarisation in high school.”

This shows that the observed incomplete understanding of polarisation is probably due to limited teachingand can be addressed by additional instruction at university level.

4.2.3 Variations of conceptual understanding

It is instructive to contrast the observed heterogeneity in content of the submitted maps with the partici-pants’ own perception of individuality. When asked which parts of their maps should be common amongall physicists, they reproduce the central shared elements from Tables 4.2 and 4.3 such as wave, particle,interference, refraction, spectrum and photoelectric effect with remarkable accuracy. They name verydifferent possible areas of individuality: non-physics concepts, special fields of interest within physics,advanced quantum aspects of light. This agrees with the absence of concepts from these areas fromTables 4.2 and 4.3.

The interviews have also provided episodic evidence for the proposed forming of specialised sub-communities among the MSc participants. Two of the 3 interviewed students named photonics/plasmonicsas their common specialisation and identified the application section of their maps as evidence. Thissection contains two shared concepts, laser and communication. The former is so general that it is sharedby 9 of the MSc participants. The latter is indeed shared by a small sub-group of 4 students and may hencepartially explain the raised fraction of concepts shared by 3–5 MSc participants observed in Fig. 4.5. Notethat the third interviewed MSc participant has not yet chosen a specialisation, because he has movedinto physics very recently.

4.2.4 Key elements of the concept ‘light’

When asked to condense the essence of the ‘light’ concept to a short definition, virtually all participantsagree that light is an electromagnetic wave. In addition, 2 of the 6 participants appeal to the fact thatlight is visible and 2 of the 3 MSc participants include the particle nature of light in their definition.

This homogeneity of views level is only observed on a surface level. When asking the participantsto recall a moment when they had understood something really important about light, a whole cosmosof deep and very individual insights opens up. The students’ response sheds some light on possible keyelements of conceptual understanding in physics. A crucial element of understanding seems the perceptionof structure and coherence within the topic. As an instance of perceived structure, BSc participant 7reports:

“We were given a talk by the demonstrator and he was talking about light in a particularway that made the difference between geometric optics and quantum optics much clearer. Hemade me think: ‘Okay, there are two separate ways of thinking about the wave nature of lightand for some phenomena you’d use this way and for some phenomena you’d use this way’.And that was a snap, ‘Oh, there you go, I think my understanding has improved there’.”

Structure and coherence are equally important in BSc participant 8 ’s sudden understanding of wave–particle duality:

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“At that point you’re just doing what you’re told and you’re not really thinking about whatyou’re doing. And then [wave–particle duality] explains what light is and its properties andhow it interacts with matter or different objects and it all comes together.”

Mathematics seems to play a distinguished role in developing this understanding of the essentialstructure of a topic. MSc participant 5 recalls:

“In university, I learned how these two properties are connected. They are all governed by thede Broglie wave function.”

Similarly, mathematics has helped MSc participant 10 perceive unity within the physical laws governinglight:

“People talk about how nice the Maxwell’s equations are but they’re kind of long and unin-tuitive. . . . At the time I could see their uses but they were four separate entities that werebroadly connected with this electromagnetic wave. But then . . . when you take the four vectorpicture, . . . you can end up with a very nice, neat equation. And it feels like, ‘Ah so, and nowI see that’.”

According to MSc participant 12, the light concept itself can unify the fundamental laws of electromag-netism, again mediated by mathematics:

“I think for a point where I understood light a lot better was . . . when you use Maxwell’sequations to make a wave equation. I saw how light was actually linked to electricity andmagnetism as opposed to being told that it’s an electromagnetic wave and it’s caused byoscillating electric fields. . . . It was done on the board: You get a wave equation . . . and thenfrom that you can get this insight that light is caused by the signal sent when you oscillate acharge. And so I could see very clearly how electric and magnetic fields lead to light.”

A very different key element of conceptual understanding seems to be the ability to apply theoreticalknowledge. MSc participant 10 reports on important insights formed during a computing project:

“Things were falling into place. I could see I had all my transition matrix elements and Icould see that certain transitions were allowable and weren’t allowable. And I could go, ‘okay,I remember this from my first year’, where I was learning about electronic transitions in anatom and that had been very abstract.”

4.2.5 Learning history

The interviews have thus helped identify central elements of conceptual understanding which were onlyimplicit on the maps themselves. They also provide information on how the learning of these elements wasfacilitated. All of the 6 students interviewed name expository instruction as a central trigger for learning.The above quote from MSc participant 12 which refers to learning during an Electromagnetism lecturemay serve as a representative example. Interestingly, despite being mediated by expository teaching, notall of the resulting learning is intended. BSc participant 7 ’s understanding of geometric vs. wave opticswas triggered by instructions from a lab demonstrator, but:

“That certainly wasn’t his main point. He was just trying to introduce the experiment, Ithink.”

Alternative learning activities are also named as decisive in forming conceptual understanding. MScparticipant 5 mentions the use of textbooks and experiments, while MSc participant 10 stresses therelevance of computing project and coursework which provide room for intense focus:

“As the end date approaches, I start to really focus in on it and spend hours and hours andhours on the same thing. That’s the point at which I tend to get understanding.”

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Figure 4.7: Expert concept map created during the interview.

To summarise, the interviews have helped identify conditions that may qualify the proposed interpre-tation of concept-map structure and confirmed polarisation as a challenging topic. The participants havestressed the existence of a shared core of understanding and indicated non-physics concepts and fieldsof specialisation as areas of heterogeneity. The interviews have revealed that a range of elements are re-quired to form conceptual understanding in physics, including the perception of structure and coherenceas mediated by mathematics and the ability to apply abstract knowledge. These elements are learned ina similarly broad rage of activities ranging from lectures to projects and coursework.

4.3 Expert contribution

The joint interview with Experts 1 and 2 included the co-construction of a concept map and a discussionof some topics which had emerged from the analysis of the student concept maps.

4.3.1 Concept map

The co-constructed experts concept map is shown in Fig. 4.7. Here, blue and red concepts had beenprovided by the researcher on the basis of the first and second half of the Electromagnetism coursesyllabus, respectively. Yellow concepts were additionally proposed by the Experts during the interview.

Due to constraints, the concept mapping activity did not include the creation of links. However, itwas possible to reconstruct links from the interview transcript. For instance, Expert 2 remarks:

“Dipole radiation, I know where that goes: that goes right here in terms of creation.”

This may be interpreted as a link between the concepts dipole radiation and creation. Figure 4.8 is areproduction of the Experts’ concept map where all links mentioned during the interview have been added.Unfortunately, most link references during the interview where unspecific like the one above, which iswhy the extracted links are unlabelled.

Quantitative analysis. Based on the map of Fig. 4.8, one can determine the relevant structuralparameters. The basic parameters are displayed in Table. 4.4. The number of concepts lies right in themiddle of the range set by the student maps. Note that here the mapping task is less open-ended, because

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Figure 4.8: Expert concept map with added links.

Parti- Con- Links Dia- Radius # Vertices of degreecipant cepts meter In Ex 1 2 3 4 5 6 7 8 9 . . . 15Experts 36 52 5 2 4 16 9 15 5 2 1 0 2 1 0 1

Table 4.4: Basic structural parameters of the expert concept map.

Participant Cross-linkage Dimension Balance

Experts 31% 2.0 50%

Table 4.5: Higher structural parameters of the expert concept maps.

a list of proposed concepts had been prepared by the researcher. The in- and ex-radii are also comparableto those observed in the student maps, showing that the expert map exhibits a similar depth. The degreesequence can be regarded as an extrapolation of the development taking place in the transition from BScto MSc maps. The number of outliers (degree 1) is reduced even further with a corresponding increaseof junctions (degree 2). In contrast to the MSc maps, the Experts’ map exhibits quite a few hubs of veryhigh degree. The most striking feature of the expert map is the large number of links which goes alongwith a slightly reduced diameter.

The large degree of connectivity manifests itself even more prominently in the higher structuralparameters (Table 4.5). The cross-linkage of 31% lies outside the typical ranges 9%± 7% and 12%± 10%observed for the student maps. Correspondingly, the Experts’ map is the only one which is truly two-dimensional (dimension: 2.0). The balance of the Experts’ map (50%) is similar to that of the studentmaps.

Morphological analysis. The topologically normalised concept map of the Experts is displayed inFig. 4.9. Comparison with the morphological classes of Fig. 4.4 clearly identifies the map as a represen-tative of the fungus class. This is in agreement with the observed extraordinarily high cross-linkage andindicative of the fully integrated understanding expected from experts (Hay & Kinchin 2006).

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Figure 4.9: Topological normalisation of the Experts’ concept map.

Content analysis. Of the 36 concepts displayed on the Experts’ concept map, 33 are physics conceptsand the remaining 3 are formulas. Again, this may be interpreted as a continuation of the developmentobserved when comparing BSc and MSc maps. The Experts’ concept map is a stripped-down version ofthe latter maps which has a narrow focus on the physics concepts and formulas relevant to a particularcourse, Electromagnetism, where all references to non-physics concepts, history of science, applications,experiments and even other physics theories have been removed. This finding is partly biased by thesuggestion of concepts by the researcher at the beginning of the concept-mapping activity. However, theseconcepts had been extracted from the course syllabus as originally created by the Experts themselves;and the additional concepts proposed by the Experts also fall into the category of physics concepts.

The physics concepts contained in the Experts’ map exhibit a large degree of overlap with the prop-erties and processes included on the maps of the students, cf. Tables 4.2 and 4.3. Here, these conceptsare included under the headings propagation and interaction, respectively. However, the Experts’ mapcontains a number of concepts not generally included on the student maps. These are primarily conceptsrelated to the creation of light including its theoretical description and examples as well as conceptsilluminating the underlying theory of processes involving light.

The combined structure and content of the Experts’ map is very unique and distinct from any ofthe student maps. Firstly, its fundamental structure is threefold and directly related to the temporallife cycle of light: creation, propagation, interaction. Secondly, the map exhibits a pronounced verticalhierarchy, where the three main categories creation, propagation, interaction are followed by fundamentallaws, derived physics concepts and processes, examples being placed at the bottom.

To summarise, the Experts’ concept map is unique in terms of both structure and content. Thestructural distinction lies in an extraordinary degree of inter-connectivity and strong vertical hierarchy.The content is characterised by a strong focus on course-related physics concepts and formulas and astructuring in terms of the life cycle of light.

4.3.2 Interview

During the interview, the Experts were able to give the teacher’s perspective on the conceptual under-standing of light which is complementary to the learner’s perspective of the BSc and MSc participants.

Polarisation. When confronted with the seeming difficulty of the BSc participants to understandpolarisation as a fundamental property of light, the Experts where able to identify the conceptual coreof the problem and point out a potential remedy. According the Expert 1, the key to understandingpolarisation lies in appreciating the transverse nature of the light wave:

“[Light] is moving, it’s propagating through space in a direction, but it’s going to move theelectron perpendicular to that.”

Expert 2 adds:

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“I think you hit on another real conceptual problem: that electromagnetism is 3-dimensionalin nature.”

The fact that a mathematical description of polarisation requires three-dimensional vector calculus mayexplain why the BSc participants’ understanding of the concept is often incomplete. As suggested byExpert 2, a deep understanding of polarisation can be formed in the students by showing them an imageof light creation by a moving electron:

“For [the students] who get it, there’s a moment of: ‘Oh yeah, okay, so it’s transverse’.”

Their suggestion is in line with Ambrose et al.’s (1999) proposal to use diagrams for teaching this difficulttopic, but adds that a discussion of light creation can deepen the understanding further.

Teaching electromagnetism. The concept map produced by the experts represents their understand-ing of ‘light’ with its complex internal structure of interrelated concepts. However, this structure doesnot necessarily correspond to the way in which the course in organised, as Expert 1 points out:

“Often the best, logical way to do something is not the best way to teach it.”

In particular, he explains that the creation of light, which logically is the first aspect in the life cycle oflight, is only taught at the very end of the course:

“You have to do propagation really before you do the creation, because all these formulas arevery, very hard to derive.”

This incongruence may explain why the MSc participants maps do not always develop a fully intercon-nected structure.

Learning electromagnetism. According to Expert 1, forming an understanding of electromagnetismrequires two central learning activities:

“I certainly emphasise: think about things, but also learn to do.”

The latter aspect of learning seems to be under-represented in the students’ views (Sect. 4.2.4). This maybe due to the fact that learning via problem-solving is more incremental with a slow building of skillsapplied to very particular tasks. Expository teaching, on the other hand, focusses much more on the bigpicture and can thus invoke vivid experiences of sudden understanding.

The Electromagnetism course. According to Expert 1, the Electromagnetism course in its currentform is very condensed in content:

“The problem with our course is really it should be double the length.”

As Expert 2 explains, this is achieved by a strong focus on the essentials for understanding ‘light’:

“The second-year Electromagnetism course is really a course about light in many ways.”

This tight organisation does not seem to leave much room for further compression. Quite the contrary,the Experts mention important aspects that have already been removed from the syllabus. Expert 2 states:

“It’s through time pressure that you end up not really having too many applications.”

Expert 1 makes a similar observation regarding experiments:

“When you really try to construct a good course, you should motivate from experiments whyyou need equations like this. . . . But it’s so long.”

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The removal of applications and experiments from the course due to time constraints may partially beresponsible for the observed tendency of the MSc participants to not include the respective concepts ontheir maps.

The expert interview has thus revealed possible causes of student misconceptions about polarisation.It has hinted that the student account of formative learning activities may be biased towards expositoryinstruction and shown that the student development of a focus on physics concepts is in line with theirteaching. Above all, it has revealed a crucial obstacle for the forming of a well-developed conceptualunderstanding of ‘light’ by the students, namely the unavoidable incongruence between the structure ofthe topic and that of the course.

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Chapter 5

Discussion

As an important second step of the phenomenological analysis, I will next discuss and interpret thefindings described in the previous chapter. The two theoretical frameworks introduced in Sect. 2.2 willeach integrate aspects of them into a meaningful and coherent whole.

5.1 Concept maps as witnesses and facilitators of meaningful learning

Ausubel, Novak & Hanesian’s (1978) theory puts forward five essential processes responsible for mean-ingful learning. Each of them offers an interpretation of an important aspect of conceptual understandingas expressed on the students’ concept maps.

5.1.1 Subsumption

Subsumption is the most basic process of incremental knowledge growth. The incorporation of individualnew concepts into a learner’s understanding cannot be observed at the macroscopic resolution offeredby this study. However, it can contribute to facilitating subsumption in line with Ausubel, Novak &Hanesian’s (1978) famous credo: ‘The most important single factor influencing learning is what the learneralready knows. Ascertain this and teach him accordingly.’ (p. iv). The findings from the BSc participants’concept maps paint a rich picture of their previous knowledge of ‘light’ when entering higher educationin physics. Due to the observed high degree of homogeneity both in existing understanding and in roomfor further development, it is possible to extract a meta-map. This map, which condenses the commonfeatures of the individual BSc maps, is shown in Fig. 5.1. Well-known concepts and links are shown inblack. An extended map includes additional red concepts and links which are less developed on many ofthe students’ maps.

The basic structural parameters of the meta-map as displayed in Table 5.1 confirm that it is repre-sentative of the BSc maps with a slight bias towards a larger number of links. The higher parametersof Table 5.2 show that a very well-integrated map can easily be obtained by adding the red links andconcepts. This can be achieved in a short recapitulation session at the beginning of the course, possiblyaided by showing the meta-map. The meta-map is a multi-faceted tool which can be used by teachers ofelectromagnetism as a tool for course preparation, instruction and diagnostic assessment.

Map Con- Links Dia- Radius # Vertices of degreecepts meter In Ex 1 2 3 4 5 6 7 8 9 15

Meta-map 26 32 6 2 4 8 9 3 3 2 1 0 1 0 0Extended meta-map 28 43 5 2 4 6 8 4 3 4 1 2 0 0 0

Table 5.1: Basic structural parameters of the meta map and its extended version.

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Figure 5.1: Meta-map reflecting the shared understanding of the BSc participants. Concepts in dashedboxes are substituted with alternatives for some participants. Red concepts and links of the extendedmap are only partially known to the students.

Map Cross-linkage Dimension Balance

Meta-map 22% 1.7 50%Extended meta-map 37% 1.9 50%

Table 5.2: Higher structural parameters of the meta maps.

5.1.2 Obliterative subsumption

The elusive phenomenon of obliterative subsumption may help explain two surprising results: the re-duced number of concepts of the MSc participants’ maps compared to those of the less experienced BScparticipants’ and the striking absence of mathematical formulas. The development from the BSc to theMSc maps shows clear evidence of a gradual fading of the kind of detail which is primarily perceived asrelevant when a new concept is first learned. The prime example for this is the deep insight that vastlydifferent phenomena with completely disjoint meanings for our everyday life, radio waves, infrared, visiblelight, ultraviolet, x-rays, are all manifestations for electromagnetic waves. This is very explicit on the BScparticipants’ maps where all these examples are named and linked to spectrum, recall Table. 4.2. The listof examples has thinned out or disappeared on the MSc maps, but the underlying comprehension hasbeen retained and deepened, as shown by the remaining of spectrum and colour and its embedding intothe context of photon energy and frequency.

Although less explicit on the maps, obliterative subsumption may also be at work in the use ofmathematical formulas. These are strikingly absent on the majority of the maps with the exception ofvery simple relations like λν = c.

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Stefan Yoshi Buhmann 5 Discussion

Concept # Mentions Observed degrees

Light 14 10, 10, 10, 8, 7, 7, 5, 5, 5, 5, 4, 4, 4, 4Electromagnetic wave 11 10, 7, 6, 6, 6, 6, 5, 5, 5, 4, 4Spectrum 9 8, 8, 8, 8, 7, 7, 5, 5, 5Particle 6 4, 4, 4, 4, 4, 4

Table 5.3: Hub physics concepts on the BSc participants’ maps.

Concept # Mentions Observed degrees

Light 11 8, 7, 6, 6, 6, 6, 6, 5, 4, 4, 4Electromagnetic wave 5 6, 6, 6, 5, 5Electromagnetism 5 6, 4, 4, 4, 4Photon 4 6, 6, 4, 4Spectrum 4 8, 6, 5, 4

Table 5.4: Hub physics concepts on the MSc participants’ maps.

BSc participant 07 comments:

“There are some things that you can’t understand without mathematics, but I still think youcan talk about physics without mathematics. The main reason I didn’t write many formulaswas simply because . . . the ones I know I’ve largely forgotten. . . . If I do remember a formulait’s because I’ve remembered the physics.”

This description strikingly matches Ausubel, Novak & Hanesian’s (1978) definition of obliterative sub-sumption. The formulas are needed, or at least they were at some point in the past when understandingwas formed. Now their memory has been replaced by that of the more stable meaning of the physicsconcepts.

5.1.3 Superordinate learning

Superordinate learning leaves an explicit trace in the structure of concept maps. Effective superordinateconcepts manifest themselves as hubs, defined by Koponen & Pehkonen (2008) as concepts with a largenumber of links. Tables 5.3 and 5.4 show the physics concepts that most frequently indicate superordinatelearning on the participants’ maps. Firstly, the tables confirm that light itself is a powerful integrativeconcept. The above mentioned spectrum concept features prominently on the BSc maps and looses itsrelevance on the MSc maps as a result of obliterative subsumption. The concepts electromagnetic waveand particle/photon integrate important sub-structures of the maps. Interestingly, the label of the latterhas been turned to the abstract photon for the MSc participants and its integrative power has beenincreased as indicated by the higher observed degrees.

5.1.4 Progressive differentiation

The fleshing out of the photon concept by an increasing number of related concepts can alternatively beread as a case of progressive differentiation of this concept itself. In this way, the particle model of light,intriguing but largely alien to most BSc participants gains depth and meaning by its embedding into arich context of new concepts, including for instance the Casimir effect mentioned by MSc participant 12.Structurally, progressive differentiation is manifest in the observed replacement of degree 1 concepts bydegree 2, 3, 4 concepts during the transition from BSc to MSc maps, recall Fig. 4.3.

5.1.5 Integrative reconciliation

The integrative reconciliation of formerly disjointed areas of understanding is primarily mediated bycross-links. Rather surprisingly, the forming of cross-links seems to be an extremely idiosyncratic process:

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Almost all of the recorded cross-links were unique to the single participant mentioning it. This indicatesa great potential for creating new cross-links from dialogue among the students.

The one exception of a commonly mentioned powerful cross-link is that between the wave and particlepictures of light as mediated by the complex phenomenon of wave–particle duality. The grasping of thiscross-link seems essential for the forming of a coherent understanding of light.

Ausubel, Novak & Hanesian’s (1978) theory of meaningful learning with its five central processes hasthus helped interpret and apply the findings presented in the previous chapter. Obliterative subsumptionmay explain the absence of formulas on the students’ maps and the compression of content observedon the MSc maps, while progressive differentiation has been identified as a major element of maturingunderstanding. Light, wave and particle have been identified as powerful superordinate concepts. On theother hand, we have seen how subsumption can be supported by a meta-map and integrative reconciliationcan be fostered via peer dialogue.

5.2 Concept maps as evidence for a chain of mobilisations?

Nespor (1994) has conceptualised the development of conceptual understanding physics as a chain ofmobilisations. The first step in his chain, the motion away from the everyday world, is compellinglyconfirmed by the observed disappearance of non-physics concepts from the MSc participants’ maps,recall Fig. 4.6.

However, this study casts doubt on the status and sequence of the subsequent links of the chain. Theinterviews paint a rich picture of the essence of conceptual understanding in physics which contradictsNespor’s (1994) rather rigid scheme of abstraction progressing via diagrams towards mathematical equa-tions as the ultimate endpoint. Firstly, the interviewees give a range of nuanced opinions regarding therelevance of mathematics. In contradiction of Nespor (1994), BSc participant 7 states:

“You can talk about physics without mathematics. . . . It’s just a language, really.”

On the other hand, BSc participant 8 points towards an objectifying function of mathematics:

“[Mathematics] backs up your knowledge on light and convinces you that it’s not just yourpersonal interpretation.”

Even further, MSc participant 12 seems to be leaning towards Nespor’s (1994) position:

“From a point of view of understanding I think the equations are very necessary.”

Secondly, an abstract mathematical formulation does not seem to be the ultimate, superior form ofunderstanding in physics. BSc participant 7 proposes an alternative, very different image:

“[The physics that I know] is more of a sensation of change, the sensation of how things areor an image. . . . It’s all dynamic, it’s not so static.”

Similarly, BSc participant 17 states that mathematics is not the end of the line as far as understandingis concerned:

“The interpretation comes after the mathematics.”

Expert 2 agrees:

“I wouldn’t say the mathematics is so important. It’s having a goal in mind: why you’re doingit.”

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Stefan Yoshi Buhmann 5 Discussion

Expert 2 elaborates:

“I want [the students] to be able to translate mathematical symbols into pictures in theirbrain in time.”

In physics, mathematics is hence a tool: a transient means or catalyst of understanding rather than itsend. This conclusion agrees with Greca & Moreira (2001) who state that ‘comprehension in a particularfield of physics is attained when it is possible to predict a physical phenomenon from its physical modelswithout having to previously refer to the mathematical formalism.’ (p. 106).

In view of these findings, it seems appropriate to replace Nespor’s (1994) chain of mobilisations witha more accurate web of abstractions where different vehicles such as diagrams, equations and cause-effectrelations play equally important, complementary roles.

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48

Chapter 6

Conclusion

This study has given methodological (how?), essential (what?) and developmental (whither?) answers toits central research question: can concept maps be used to reveal conceptual understanding of ‘light’ ofphysics students at different stages of their education? Methodologically, an affirmative answer has beengiven: By combining quantitive, morphological and content analyses of student-submitted concept maps,it is indeed possible to obtain a coherent picture of the conceptual understanding of learners in physics.The three elements of the triangulation mutually inform and confirm one another; and the findings aresubstantiated by subsequent interviews and in line with the complementary teacher perspective providedby the course as well as the researcher’s own experience of learning and teaching physics. With respect toessence, the study has shown that the conceptual understanding expressed on the student maps is highlyindividual, but possesses a shared core of physics concepts, theories and formulas which are meaningfullylinked to form a coherent whole. In Ausubel’s language, this understanding can be interpreted as a resultof subsumption of new concepts into a student’s body of previous knowledge; superordinate learning viathe structuring concepts wave and particle; progressive differentiation of originally unintuitive physicsconcepts such as the photon; integrative reconciliation via cross-links; and obliterative subsumption me-diated by mathematics. The latter plays the role of a tool to help students develop an understanding ofphysics concepts; it must not be confused with this understanding itself. With regard to development,the study suggests a three stage-process: The original understanding by BSc students at the beginning oftheir university physics education is marked by a broad interest and receptiveness, including both corecontent and essential structures of the light concept, but garnished with very individual associations ofperipheral non-physics concepts. The transition to the understanding of MSc students as informed bythe course is marked by a focus on physics concepts with a deepening of knowledge structures. The final,expert understanding gained from experience and meta-cognitive reflection requires an introduction ofhierarchy and the forming of a multitude of cross-links.

The limitations of the study are inherent in its primary tool: concept mapping. In particular, conceptmaps are far from one-to-one representations of conceptual understanding. Quite the contrary, they haveto be viewed from a constructivist point of view as fluid entities emerging from the mappers’ negotiationwith their understanding. The interviews have revealed that a range of factors like intent, ability and taskinterpretation influence the resulting product. Interpretations of a concept map hence always require adialogue with their creator to avoid simplistic misinterpretations. When used in this way as a startingpoint for investigating conceptual understanding, the weakness ambiguity can be turned into a strength,resulting in a richness of analysis.

Another weakness of the study, its coarseness, opens a perspective for future work. By following thedeveloping conceptual understanding of individual learners in a longitudinal study involving many conceptmaps, one can hope to obtain further insights into the process of forming understanding. In particular,this could shed further light on the suggested role of mathematics as a transient tool particularly relevantat intermediate stages of learning. In addition, the study has shown that concept mapping is able toreveal deep structural misconceptions such as the failure to grasp polarisation as a property of light.Future investigations can extend this potential of concept mapping.

Some of the findings of this study are certainly limited to physics as a discipline or even morenarrowly electromagnetism as a particular subject: This includes the development of an increasingly

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abstract understanding as mediated by mathematics; the identified common core of previous knowledgeof the BSc students; and the perceived prominent role of instruction in facilitating understanding. Furtherinvestigations are needed to verify whether this latter view is representative of all physics students and towhat degree it is accurate. The fact that most themes have been mentioned by at least two intervieweesseems to indicate saturation in terms of topics, but is far from being able to determine the relevance ofthese topics. The latter could be achieved by means of questionnaires.

On the other hand, two learning-theoretical methodologies have emerged which are likely to be fruitfulwhen applied to other contexts and disciplines. The first is the proposed morphological classification ofconcept maps in terms of six fauna-based types. While comprising a similar number of classes as Yin et al.’s(2005) chain, circular, spoke, tree, network distinction, the interpretative richness offered by the faunaanalogy renders it closer in spirit to Kinchin, Hay & Adams’s (2000) original scheme. The plant/growthmetaphor offers a rich framework for discussing the quality and development of understanding whichstresses the role of individuality and potential for future learning rather than concentrating on uniformlearning targets and the identification of deficiencies. For instance, the vanishing of detail on certainareas on concepts maps in the course of learning is analogous to the process of lignification: the turningof many-leafed sprouts into stable trunks as a solid basis and support for more advanced knowledge.As a second example, the growth of individual areas of special interest and knowledge is nurtured bystimulation in terms of teaching or alternative facilitators of learning just like the asymmetric prosperingof certain parts of a tree is controlled by environmental factors such as sunlight, rain and wind.

The two strengths of the proposed scheme are its objective fundament and its practical consequences.The first is achieved by basing the classification on quantitative structural parameters and a uniquetopologically normalised form. However, this strength can easily be turned into a weakness when applyingthe proposed definitions too narrowly and thus producing a rigid scoring scheme for concept maps. Toavoid this, one needs to bear in mind that the criteria for class assignment are certainly subject- and mostlikely sample-dependent. The parameter ranges used to define the different classes are hence to be seenas suggestions only and need to be adapted to each new context. The true value of the scheme residesnot in the class assignment per se but in the future development and learning suggested for each class.With this focus, morphological classification provides a framework for sharing, discussing and deepeningconceptual understanding.

The other methodology which may possibly transcend the subject-specific and disciplinary boundariesis the construction of a meta-map from the common core of many concept maps. The first and decisivestep for its applicability is the existence and identification of a suitable, course-spanning root concept. Itcan alternatively be found by consulting the syllabus or by asking the question: what is this course reallyabout? A chosen root concept can be used in a second step for diagnostic assessment at the beginningof the course. Given a sufficient degree of homogeneity, concept maps submitted by the participants canthen be used to create a meta-map which captures their common previous knowledge. The meta-mapcan inform the course design and it can be used as an instructional tool to ensure that all student indeedshare this common core of knowledge. As a short-cut, an existing meta-map might possibly be (re-)usedwithout a preceding mapping exercise. As a first step towards a test-in-action, a combined morphologicalclassification and meta-map creation exercise will be implemented in the near future for a different physicssubject and within the different context of a German higher education institution.

Returning to our original point of departure: Do two physicists see the same thing when they ‘seelight’? Most probably not. But for a vast range of specific questions which a physicist typically asksabout light, they will be able to agree on a common answer.

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Appendix A

Teaching concept mapping

In this appendix, the instructional slides used to teach concept mapping to the BSc and MSc participantsare presented. The participants were first shown the example of a beginning of a concept map on ‘theuniverse’ as displayed in Fig. A.1.

Figure A.1: Teaching concept maps: Worked example.

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Figure A.2: Teaching concept maps: Rules for generating concept maps.

The example is a simplified version of a worked example for generating a concept map given by Novak& Cañas (2008). I have chosen a physics example in order to show the participants which types of contentmay be included in a concept map about a physics topic. The example is sufficiently different from thetopic ‘light’ of the study to avoid prompting or biasing specific contents for the maps generated forthis study: ‘the universe’ is a much broader category which encompasses a whole variety of physics sub-disciplines whereas ‘light’ is much more specific and primarily confined to electromagnetism. Furthermore,none of the concepts shown in the example is likely to appear in a concept map of ‘light’, with possibleexception of ‘energy’ and the Einstein equation E = mc2.

Following Novak & Cañas (2008), the generation of the map is presented in two steps where step 1shows only the ‘parking lot’ and step 2 shows the concepts and their relations. The example featuresphysical concepts, properties and formulas to demonstrate a variety of elements which can possibly beincluded in a concept map.

The rules for generating a concept map as shown in Fig. A.2 were displayed during the entire concept-mapping exercise. They are a shortened version of the procedure described by Novak & Cañas (2008).The participants were thus presented with a step-by-step procedure for creating a concept map whichalso indicated the essential properties of the finished product.

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Appendix B

Interview protocols

Very similar protocols were used for the semi-structured interviews with the BSc and MSc participants.They are shown in App. B.1 where questions used exclusively for one of the participant groups areindicated by BSc only and MSc only, respectively. The BSc participants were asked an additional questionabout Newton’s corpuscle theory as an example of exposure to history of science in school teaching,whereas the MSc participants were provided with more specific questions regarding their learning history.The protocol for the joint semi-structured interview with the two Experts evolves around similar themesas the student protocol, but from the teacher perspective. It is given in App. B.2.

B.1 Student Interviews

[Confirm: Interview + recording, consent form, transcript verification]

Header

The following is the interview with [BSc/MSc] participant [number] taking place on [date] at [time] in[Huxley 6M04/the Optics Reading Room].

[Explain structure: Initial question, concept-map generation, content, questions, learning]

Initial question

Let us start by creating a spontaneous snapshot of your knowledge of ‘light’.

• If you had to define ‘light’ in one sentence, what would that sentence be?

[Present concept-map to participant]

Concept-map creation

Let us first focus on how you created your concept map.

• Thinking back to the situation, can you talk me through how you went about this task?

– Did you start with a ‘parking lot’? Did you extend your list of concepts later?– In what order did you generate the elements of the map? Did you place concepts and labels

simultaneously?– What caused you to end the task?

∗ Did you feel that you had enough time?

∗ What would you have done to your map given you had more time?

• What do you think happened while creating the map?

– Did you extract knowledge from your brain?– Did you generate new knowledge?

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– Was there anything that you discovered while drawing the map?

Concept map contents

Let us next turn our attention to the content of your map.

• Can you explain your concept map?

• What are the most important elements of your map?

– What is it that makes these elements important?

• Which parts of the map represent concepts that you know really well?

– Can you summarise your understanding of these parts?

• Where would you like to learn more?

– How would you extend these sections of the map?

• Are there any sections of your map that might be further linked?

– How would you link them?

• BSc only: You mention Newton’s corpuscule theory. How and what did you learn about that?

Questions about ‘light’

I would now like us to use your map to answer some questions about ‘light’.

• What information is needed in order to completely characterise some given light?

– Where does this information appear on your map?

• You mention wave–particle duality. Under which circumstances is light a wave or a particle?

– Can it be both at the same time?– How do you make sense of these two very disjoint descriptions? Are you happy with your

understanding?– How is wave–particle duality reflected in your map?

• Describe how light is created?

– How would this process be expressed in your map?

• Can you explain polarisation?

– Where does it fit into your map?

Learning

Finally, let us focus on your experience of learning about ‘light’.

• BSc only: Can you recall a moment when you understood something really important about ‘light’?

– What was that?– What helped you form this understanding?

• MSc only: At what stage of your university education did you feel that you had understood ‘light’for the first time?

– How did that come about?

• MSc only: Which are the aspects of your ‘light’ concept map that you learned from the Electro-magnetism course?

– Which aspects did you learn from which other courses?

• MSc only: Which part of your map represents your latest understanding?

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Stefan Yoshi Buhmann Appendix

• MSc only: Where did you think you formed most of your understanding, during the electromag-netism lectures, other lectures, solving problems, doing experiments, working on projects or else-where?

• Which parts of your map are strongly linked to mathematics?

• MSc only: What is your main field of interest within physics?

– How is that reflected in your map?

• Which parts of your map represent knowledge that all physicists should agree on?

– Which parts represent your individual understanding?

• How do you think your understanding of ‘light’ differs from that of a non-physicist?

– How is this reflected in your map?

• Are there any changes that you would like to make to your map?

Closing

• Do you have any further comments or anything you want to add?

Thanks for your help!

Debriefing

[Provide concept-map analysis and give feedback, possibly ask for a reaction.]

B.2 Expert interview

[Confirm: Interview + recording, consent form, transcript verification]

Header

The following is the interview with the two leaders of the second-year Electromagnetism course takingplace on 19th December 2013 at 5:00 pm in 736 Blackett.

[Explain structure: Initial questions, concept-map generation, concept-map discussion, physics questions,teaching]

Initial questions

Let us start by capturing a few spontaneous thoughts about ‘light’.

• If you had to define ‘light’ in one sentence, what would that sentence be?

• What is the most important or difficult fact about ‘light’?

[Present list of concepts]

Concept-map creation

I have prepared a list of ‘light’-related concepts from the electromagnetism course.

• Could you agree on a selection of about 20 key concepts which are important for understanding‘light’?

– Please feel free to modify, discard or add new concepts.

• Could you jointly place these concepts onto the sheet of paper to generate a concept map of ‘light’?

• Could you try to connect the concepts on the sheet by adding labelled lines?

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• How well does this map reflect the course?

Concept-map discussion

• Can you explain the resulting concept map?

• What are the most important elements of the map?

– What is it that makes these elements important?

• With which parts do you think the students struggle most?

– What make these aspects hard to understand?– How can you help them understand?

• Are there any sections of the map that might be further linked?

– How would you link them?

Questions about ‘light’

I would now like us to use the map to discuss some questions about ‘light’.

• What information is needed in order to completely characterise some given light?

– Where does this information appear on the map?

• Some students seem to struggle with the concept of polarisation when they enter the course. Howwould you explain polarisation?

– Where does it fit into the map?

• Some students seem to have two very disjoint images about the generation of light—via the os-cillation of charges or via conversion of internal atomic energy. How could they reconcile theseimages?

• The students seem very interested in wave–particle duality. How would you explain to a studentwhether or when light is a wave or a particle or both?

– Do the students seem uncomfortable with these two very disjoint descriptions?– How could the Electromagnetism course prepare the students for wave–particle duality?– How could one express this on the map?

Teaching

Finally, let us focus on teaching and learning about ‘light’.

• What do you think happened while creating the map?

– Did we extract knowledge from our brains?– Did we generate new knowledge?– Was there anything that the map can show or teach particularly well?

• Do you recall episodes when the students seemed to learn something really important about ‘light’?

– How did that come about?

• Which aspects of ‘light’ have to be learned from which other courses?

– How can the Electromagnetism course prepare them for those aspects?– How can we express this on the map?

• How does your own field plasma physics influence the course?

• Where did you think the students form most of their understanding of ‘light’, during the Electro-magnetism lectures, other lectures, solving problems, doing experiments, while working on projectsor elsewhere?

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• Which part of the map do the students understand last?

• Which parts of your map are strongly linked to mathematics?

– What role does mathematics play in understanding light?– Does it matter whether to use component or vector notation?

• The students’ understanding of ‘light’ during university education seemed to become more abstractand further detached from the non-physics world.

– If you think this is a good thing, how do you support this process?– If you think this may be problematic, how would you encourage links to non-physics concepts?

Closing

• Do you have any further comments or anything you want to add?

Thanks for your help!

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Appendix C

Topological normalisation

Topological normalisation is a procedure of transforming a content-free concept map into a unique formwhich preserves the concept vertices and their links. It was developed for this study. Placing the rootconcept at the top, vertices which are once, twice etc. removed from the root are placed on subsequenthierarchical levels and linked as in their original form. Starting from the top, branches emerging fromeach concept vertex are ordered from left to right according to the following rules:

• Place the deepest branch first.

• For branches of equal depth, place the branch with the largest total number of vertices first.

• For branches with an equal number of vertices, place the branch with the largest number of deepestsub-branches first.

• For branches with an equal numbers of such sub-branches, place the branch whose uppermost vertexhas the largest number of sub-branches first.

• For branches with equal numbers of sub-branches of the uppermost vertex, place the branch withthe largest number of cross-links first.

Due to the presence of cross-links, some vertices can alternatively be assigned to two or several branches.To render the procedure unique, the vertex was always assigned to the branch with the smaller numberof vertices.

Topological normalisation results in a unique representation where concept maps appear as maximallybalanced, skew-shaped graphs with deeper, heavier branches on the left and shorter, lighter branches onthe right.

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68

Appendix D

Transcription

Transcription of all interviews was done by

Pro Rata Secretarial Solutions21 St Teilos WayWatford FarmCaerphillyMid GlamCF83 1FAUK

The intelligent verbatim style used is described by the service provider as follows:

“Leave out background noises, ‘um’, ‘eh’, ‘you know’ etc. No repetitions, descriptors or trippingover words included. A tidying up of the transcript but without losing or adding to any ofthe important data.”

The confidentiality agreement with the provider contains the following key passages:

“The Service Provider agrees that they shall not . . . disclose (to a third person, company,firm, business entity or other organisation whatsoever) . . . any . . . Confidential Information(as defined below) relating or belonging to [the researcher] or any of its clients.

Confidential Information includes, but is not limited to, any information relating to clients,. . . intellectual property . . . or any information which the Service Provider has been told isconfidential or which might reasonably be expected to be regarded as confidential, or anyinformation which has been given to [the researcher] in confidence by clients, suppliers orother persons.

In complying with these confidentiality obligations The Service Provider must refrain fromdiscussing, reading or disclosing Confidential Information openly in public areas, such as, ontrains, buses and airplanes, on mobile telephones, or in restaurants. If the Service Provider isin any doubt as to the extent and/or the ambit of these obligations they should, in the firstinstance address any queries to [the researcher].”

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70

Appendix E

Graph theory

In this appendix, we present the mathematical definitions of some basic notions from graph theory(Gould 1988). They agree with the more intuitive notions given in Sect. 4.1.1.

• Graph: A graph G(V,E) is a set of vertices v ∈ V together with a set of edges e = {v,w} ∈ E.

• Order: The order p of a graph G is the number of its vertices, p = #V .

• Size: The size q of a graph G is the number of its edges, q = #E.

• Path: An v − w path is an alternating sequence of vertices and incident edges with end points vand w such that no vertex and no edge is repeated.

• Length: The length of an v − w path is the number of edges in the path.

• Distance: The distance d(v,w) between two vertices is the minimum length of any v − w path inthe graph G.

• Diameter: The diameter diam(G) = maxv∈V

maxw∈V

d(v,w) of a graph G is the maximum distance of any

two vertices.

• Degree: The degree deg v of a vertex is the number of edges incident with it. The number of verticeswith a given degree k is denoted by n(k) = #{v ∈ V |deg v = k}. The degree sequence of a graphobeys the two identities

∑k n(k) = p and

∑k k n(k) = 2q, which can be used as check-sums.

In addition, I have developed the following notion special definitions not common to graph theory,which are useful for the analysis of concept maps. Some of them rely on the fact that a concept map hasa root concept, corresponding to a distinguished root vertex r of the associated graph.

• Cross-linkage: The cross-linkage e/q is the number of excess edges e divided by the size q. Here, e isthe maximum number of edges which can be removed without splitting the graph into disconnectedsub-graphs. It is related to the order p and size q via e = q − p+ 1.

• Dimension: The dimension d of a graph G of order p and diameter diam(G) is defined by theequation [diam(G) + 1]d = p. This definition is based on the relation of diameter and volume inEuclidean space and inspired by the notion of fractal dimension (Mandelbrot 1967). It differs fromErdös, Harary & Tutte’s (1965) standard definition of dimension in graph theory.

• Boundary: The boundary V of a graph is the set of all vertices b ∈ V whose distance d(b, r) fromthe root vertex is equal to or greater than that of their immediate neighbours.

• In-radius: The in-radius Rin of a graph G is the minimum distance of between the root vertex rand any vertex b on the boundary, Rin = min

b∈V

d(b, r). This definition is an adaptation of the in-circleradius of a triangle in geometry.

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• Ex-radius: The ex-radius Rex of a graph G is the maximum distance of between the root vertex rand any vertex b on the boundary, Rex = max

b∈V

d(b, r). This definition corresponds to the ex-circleradius of a triangle.

• Balance: The balance is the ratio Rin/Rex of in-radius and ex-radius.

The following definitions from graph theory will not be used, as they deviate from the intuitive notionswhich are more useful in the context of concept maps.

• Tree: A tree is a connected, acyclic graph.

• Arc: An arc is a = (v,w) is a directed edge.

• Digraph: A digraph is a graph where all edges are arcs.

• Network: A network N = (V,E, s, r, c) is a digraph G(V,E) together with distinguished verticess and r called source and sink and non-negative real numbers c(e) assigned to each edge calledcapacities.

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Appendix F

Ethical conduct of research

Savin-Baden & Major (2013) advocate for ‘moving beyond the typical ethical approval processes’ (p. 319)to ‘ensuring that one has maintained excellence in carrying out research with people’ (p. 336). Reflectingon their suggestions on how two achieve this goal with respect to four different aspects, I have ensured:

• that my study has strong goals and a sound methodological basis. In addition, I acquired the neces-sary skills, learning about qualitative research methods (Savin-Baden & Major 2013), in particularconcept-mapping (Novak & Cañas 2008) and interviewing techniques (Kvale & Brinkmann 2009).

• that my research interests were transparent to the participants and that I protected their anonymityby not discussing their contributions with course administrators. I have treated any submittedmaterial with care, returned it promptly and analysed their contributions with benevolence andrespect. I respected participants’ time by being well-prepared for all data acquisition sessions.

• that I have declared my philosophical position and any preconceptions with respect to the researchquestions; and would admit any possible mistakes that may arise during the study.

• that I have sought the critical opinion of others about my research and portrayed participants’views as honestly as I could. I will strive for a wide dissemination of my results in refereed journalsand conferences. As a first step in this direction, I have submitted a contribution to the Educationfor Electromagnetics session of the annual Progress In Electromagnetics Research Symposium whichhas been accepted for oral presentation (Buhmann 2014).

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Appendix G

Structural parameters of participants’

concept maps

In this appendix, the structural parameters of the participants’ concept maps are shown. They havebeen obtained via the procedure outlined in Sect. 3.6.1, following the definitions given in Sect. 4.1.1and detailed in App. E. Table G.1 displays the basisc structural parameters, while the higher structuralparameters are given in Table G.2.

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Parti- Con- Links Dia- Radius # Vertices of degreecipant cepts meter In Ex 1 2 3 4 5 6 7 8 9 10BSc 1 38 40 9 2 5 16 11 6 2 2 1 0 0 0 0BSc 2 20 20 5 2 3 13 2 1 1 2 1 0 0 0 0BSc 3 29 32 9 2 6 12 7 6 1 2 1 0 0 0 0BSc 4 38 41 7 2 4 19 8 4 5 0 0 1 1 0 0BSc 5 21 20 5 1 1 12 5 2 1 0 0 0 1 0 0BSc 6 48 48 9 2 5 31 5 4 4 1 0 2 1 0 0BSc 7 70 71 9 1 5 37 17 7 1 6 1 0 0 0 1

BSc 8 30 30 10 1 6 14 9 4 2 0 0 0 1 0 0BSc 9 28 33 7 2 4 9 9 4 4 1 1 0 0 0 0BSc 10 33 32 8 1 5 19 6 4 2 1 0 0 1 0 0BSc 11 23 23 7 2 4 11 7 3 0 1 0 1 0 0 0BSc 12 27 30 6 2 4 14 4 2 3 4 0 0 0 0 0BSc 13 25 26 6 1 3 16 3 4 0 0 0 0 1 0 1

BSc 14 22 29 5 2 3 7 9 1 1 2 0 0 2 0 0BSc 15 19 18 9 1 5 8 7 3 0 1 0 0 0 0 0BSc 16 24 26 7 1 4 12 6 1 2 2 0 1 0 0 0BSc 17 22 26 5 2 4 7 7 4 2 1 1 0 0 0 0BSc 18 37 43 7 1 4 17 10 5 0 1 1 1 2 0 0BSc 19 33 36 7 1 4 13 11 5 2 0 0 2 0 0 0Mean 31 33 7 1.5 4 15 8 4 1.4 1.4 0.4 0.4 0.5 0 0.1St. dev. 12 12 2 0.5 1 7 3 2 1.4 1.5 0.5 0.7 0.7 0 0.3MSc 1 27 29 10 2 5 11 10 1 2 2 1 0 0 0 0MSc 2 48 58 8 2 4 21 11 5 5 1 3 1 1 0 0MSc 3 65 66 11 2 6 28 21 9 4 1 1 0 1 0 0MSc 4 27 30 6 2 3 11 8 3 3 1 0 1 0 0 0MSc 5 30 29 7 2 4 17 5 2 5 1 0 0 0 0 0MSc 6 22 28 8 2 5 3 11 5 0 2 1 0 0 0 0MSc 7 22 23 5 2 4 12 5 0 2 2 1 0 0 0 0MSc 8 41 45 9 2 5 17 9 8 5 1 1 0 0 0 0MSc 9 18 18 6 2 4 7 7 2 1 1 0 0 0 0 0MSc 10 26 29 7 2 4 8 10 4 3 0 1 0 0 0 0MSc 11 32 31 9 3 5 14 9 6 3 0 0 0 0 0 0MSc 12 28 29 7 2 4 14 9 1 0 1 3 0 0 0 0MSc 13 16 22 4 2 3 3 4 6 1 1 1 0 0 0 0MSc 14 15 21 7 1 4 2 3 6 4 0 0 0 0 0 0MSc 15 13 12 6 2 4 5 5 3 0 0 0 0 0 0 0MSc 16 16 17 5 2 3 5 8 1 1 0 1 0 0 0 0

Mean 28 30 7 2.0 4 11 8 4 2.4 0.9 0.9 0.1 0.1 0 0St. dev. 13 14 2 0.4 1 7 4 3 1.8 0.7 0.9 0.3 0.3 0 0

Table G.1: Basic structural parameters of the participants’ concept maps. The degree of the root conceptis indicated in bold. For ease of comparison, mean and standard deviation are also shown for each group.

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Participant Cross-linkage Dimension Balance

BSc participant 1 8% 1.6 40%BSc participant 2 5% 1.7 67%BSc participant 3 13% 1.5 33%BSc participant 4 10% 1.7 50%BSc participant 5 0% 1.7 100%BSc participant 6 2% 1.7 40%BSc participant 7 3% 1.8 20%BSc participant 8 3% 1.4 17%BSc participant 9 15% 1.6 50%BSc participant 10 0% 1.6 20%BSc participant 11 4% 1.5 50%BSc participant 12 13% 1.7 50%BSc participant 13 8% 1.7 33%BSc participant 14 28% 1.7 67%BSc participant 15 0% 1.3 20%BSc participant 16 12% 1.5 25%BSc participant 17 19% 1.7 50%BSc participant 18 16% 1.7 25%BSc participant 19 11% 1.7 25%Mean 9% 1.6 41%Standard deviation 7% 0.1 21%MSc participant 1 10% 1.4 40%MSc participant 2 19% 1.8 50%MSc participant 3 3% 1.7 33%MSc participant 4 13% 1.7 67%MSc participant 5 0% 1.6 50%MSc participant 6 25% 1.4 40%MSc participant 7 9% 1.7 50%MSc participant 8 11% 1.6 40%MSc participant 9 6% 1.5 50%MSc participant 10 14% 1.6 50%MSc participant 11 0% 1.5 60%MSc participant 12 7% 1.6 50%MSc participant 13 32% 1.7 67%MSc participant 14 33% 1.3 25%MSc participant 15 0% 1.3 50%MSc participant 16 12% 1.5 67%

Mean 12% 1.6 49%Standard deviation 10% 0.1 12%

Table G.2: Higher structural parameters of the participants’ concept maps.

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When ‘light’ dawns upon them: Mapping the essence of conceptual understanding … · 2014-12-16 · the subjective understanding of its scholars, this study sets out to investigate