The Role of Language in Learning Physics

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THE ROLE OF LANGUAGE IN LEARNING PHYSICS

BY DAVID T. BROOKES

A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy Graduate Program in Physics and Astronomy

Written under the direction of Prof. Eugenia Etkina and approved by

New Brunswick, New Jersey October, 2006 c 2006

David T. Brookes

The Role of Language in Learning Physics

by David T. Brookes

Dissertation Director: Prof. Eugenia Etkina

Many studies in PER suggest that language poses a serious difficulty for students learning physics. These difficulties are mostly attributed to misunderstanding of specialized terminology. This terminology often assigns new meanings to everyday terms used to describe physical models and phenomena. In this dissertation I present a novel approach to analyzing of the role of language in learning physics. This approach is based on the analysis of the historical development of physics ideas, the language of modern physicists, and students’ difficulties in the areas of quantum mechanics, classical mechanics, and thermodynamics. These data are analyzed using linguistic tools borrowed from cognitive linguistics and systemic functional grammar. Specifically, I combine the idea of conceptual metaphor and grammar to build a theoretical framework that accounts for

• the role and function that language serves for physicists when they speak and reason about physical ideas and phenomena,

• speciﬁc features of students’ reasoning and difﬁculties that may be related to or derived from language that students read or hear.

ii The theoretical framework is developed using the methodology of a grounded theoretical approach. The theoretical framework allows us to make predictions about the relationship between student discourse and their conceptual and problem solving dif- ﬁculties. Tests of the theoretical framework are presented in the context of “heat” in thermodynamics and “force” in dynamics. In each case the language that students use to reason about the concepts of “heat” and “force” is analyzed using the theoretical framework. The results of this analysis show that language is very important in students learning. In particular, students are

• using features of physicists’ conceptual metaphors to reason about physical phenomena, often overextending and misapplying these features,

• drawing cues from the grammar of physicists’ speech and writing to categorize physics concepts; this categorization of physics concepts plays a key role in students’ ability to solve physics problems.

In summary, I present a theoretical framework that provides a possible explanation of the role that language plays in learning physics. The framework also attempts to account for how and why physicists’ language inﬂuences students in the way that it does.

iii Preface

The dissertation is organized in the following way:

In Chapter 1 I will provide a brief introduction to the broad issues of learning physics, with some theoretical background on the approaches I will take. I will explain why language is important in the broader context of representations of physics knowledge. I will also try to recreate for the reader, some of my thought process that led me to think that language might be more important than generally thought.

In Chapter 2, I will present a broad survey of related literature. This survey will cover relevant research on the topics of analogy, linguistics, cognitive linguistics, metaphor, systemic functional grammar, and student learning of physics. This chapter will also serve as an introduction to some of the linguistic analysis techniques that will be im- plemented in later chapters.

The goal of Chapter 3 will be to try and draw the literature of Chapter 2 together, present a theoretical framework of how language in physics works, and how students interpret that language. This chapter will also cover all the methodological approaches applied to later chapters.

Testing the theoretical framework and presenting the data on which it was constructed, is the goal of Chapter 4. This will be done entirely in the ﬁeld of quantum mechanics. I will show how the linguistic framework presented in Chapter 3 successfully describes the way physicists talk about quantum mechanical ideas. I will then use the analysis to show how the theoretical framework is applicable to student reasoning in quantum mechanics. I will present two case studies that can be explained with the theoretical framework.

iv Chapter 5 will be the largest chapter of my dissertation. Here I will take the theoretical framework developed in the context of quantum mechanics and show how can be applied to the ﬁelds of thermodynamics and Newtonian mechanics. I will show how physicists’ language can be analyzed with the tools I have developed. I will make predictions about features of students’ reasoning and test these predictions in an interview study. I will also apply the linguistic analysis to reexamine the old “misconceptions” literature about force in Newtonian mechanics and show that many so-called student “misconceptions” may be reinterpreted as linguistic difﬁculties. Finally, I will extend the linguistic perspective beyond the theoretical framework I developed. I will attempt a second hypothesis that really connects cognition and language together as one act rather than two separate ones. This means that we can understand our students’ thought processes by listening more carefully to what they say. Listening to them through the linguistic framework can profoundly change what we hear students saying and asking.

Chapter 6 is the concluding chapter. Here I will show how the theoretical framework may help us to understand physics better. I will summarize what the thesis has accomplished, discuss future directions of research (particularly testing experiments) and make some general suggestions about the role of language in teaching.

v Acknowledgements

Although a dissertation can list only one author, I cannot claim that such a difficult intellectual endeavor is the work of one individual person. The first group of people I wish to thank are those who made a significant intellectual contribution in the form of ideas, suggestions or challenges to my ideas. Without my advisor, Prof. Eugenia Etk- ina, none of these pages would exist. You have encouraged and supported me without question, offered deep insight, and challenged me more than anyone else. The idea to explore the role of language came from my sister, Heather Brookes. Your two reading recommendations have formed the basis of this thesis. I wish to thank Prof. George Horton for questioning my work. Without those challenges, I would never have started exploring the role of grammar in the language of physics. Leslie Atkins’ quiet suggestions and prodding helped me to understand the cognitive mechanism of metaphor comprehension in terms of ad hoc categorization. Prof. Alan Van Heuvelen has tire- lessly supported and advocated my research, and was one of the people who inspired me to focus on representation. Finally, Prof. “Joe” Redish has listened to my ideas, encouraged and supported me and offered many helpful suggestions.

Sometimes I have felt that every idea that has ever been thought has already been thought of by someone else before me. I wish, therefore, to acknowledge my “intellectual parents”, the people who, through their writing and ideas, transformed how I saw the world. They are: Michelene Chi, Eugenia Etkina, Michael Halliday, Mark Johnson, George Lakoff, Jay Lemke, Nancy Nersessian, Michael Reddy, Clive Sutton.

I want to thank Andi and Michael for help with translating German to English.

vi My thanks also go out to:

All my dear friends for your friendship, for your insight, help and support and for great conversations. Alain, Brian, Brigitte, Bronwen, Indranil, Isabel, Retha, Sahana. Yuhfen for supporting me, reading my thesis, and caring for me. My family for their support. In particular, my father has had a profound inﬂuence on my views about education. The entire physics education research group at Rutgers University, especially Aaron, Alan, Anna, David R., Eugenia, Marina, Maria, Michael, Sahana, and Suzanne. Work- ing with all of you was a lot of fun and made it all worth while. Members of the PER community with whom I have had the most wonderful and con- structive discussions. My thesis committe for advice and support during my research. The staff of the Center for International Faculty and Student Services for their support in the daily struggle with the U.S. bureaucracy.

vii Dedication

To Eugenia.

‘You are sad,’ the Knight said in an anxious tone: ‘let me sing you a song to comfort you.’

‘Is it very long?’ Alice asked, for she had heard a good deal of poetry that day.

‘It’s long,’ said the Knight, ‘but very, VERY beautiful. Everybody that hears me sing it–either it brings the TEARS into their eyes, or else–’

‘Or else what?’ said Alice, for the Knight had made a sudden pause.

‘Or else it doesn’t, you know. The name of the song is called “HAD- DOCKS’ EYES.”’

‘Oh, that’s the name of the song, is it?’ Alice said, trying to feel interested.

‘No, you don’t understand,’ the Knight said, looking a little vexed. ‘That’s what the name is CALLED. The name really IS “THE AGED AGED MAN.”’

‘Then I ought to have said “That’s what the SONG is called”?’ Alice corrected herself.

viii ‘No, you oughtn’t: that’s quite another thing! The SONG is called “WAYS AND MEANS”: but that’s only what it’s CALLED, you know!’

‘Well, what IS the song, then?’ said Alice, who was by this time completely bewildered.

‘I was coming to that,’ the Knight said. ‘The song really IS “A-SITTING ON A GATE”: and the tune’s my own invention.’

(Lewis Carroll)

“A serious difficulty in the study of the development of a scientific concept lies in the necessarily inherent vagueness of its definition. This complica- tion arises from the fact that the concept in question finds its strict specifi- cation only through its exact definition in science. This definition, however, historically viewed, is a rather late and advanced stage in its development. To limit the discussion to the concept thus defined means to ignore a major part of its life history.”

(Max Jammer)

ix Table of Contents

Abstract ...... ii

Preface ...... iv

Acknowledgements ...... vi

Dedication ...... viii

List of Tables ...... xvii

List of Figures ...... xx

1. Introduction ...... 1

1.1. The Role of Representation ...... 1

1.2. Orientation and Theoretical Standpoint ...... 3

1.2.1. Cognitive Apprenticeship ...... 4

1.2.2. Representations and Legitimate Peripheral Participation . . . 6

2. Literature Review ...... 10

2.1. Introduction ...... 10

2.2. Language, Thought, and Communication ...... 10

2.2.1. The Interaction of Language and Behavior: The Sapir-Whorf Hypothesis ...... 11

2.2.2. Communication as Construction ...... 12

2.3. Analogy ...... 14

2.3.1. Introduction ...... 14

2.3.2. What is an Analogy? ...... 14

x 2.3.3. The Role of Analogy in the Development of Models in Science 16

2.3.4. Student Difﬁculties in Analogical Reasoning ...... 18

2.4. Metaphors ...... 19

2.4.1. What is a Metaphor? ...... 20

2.4.2. Metaphorical Language, Metaphorical Conceptual System . . 22

2.4.3. More on the Conceptual and Cognitive Role of Metaphor . . . 24

2.4.4. Metaphors and Analogies ...... 26

2.4.5. Summary ...... 27

2.5. Grammar and Meaning ...... 28

2.5.1. The Ideational Function of Language ...... 29

2.5.2. The Ergative Model and Material Processes ...... 30

2.5.3. Summary ...... 31

2.6. Language in Physics ...... 32

2.6.1. A Linguistic View of the Difﬁculties of the Language of Physics ...... 32

2.6.2. Language and Students’ Difﬁculties: Perspective From Physi- cists and Physics Education Researchers ...... 35

2.7. Research on Students’ Cognition and Difﬁculties ...... 41

2.7.1. Robust Misconceptions ...... 41

2.7.2. Knowledge in Pieces ...... 42

2.7.3. Conceptual Change ...... 43

2.8. Summary ...... 44

3. Hypotheses and Methodology ...... 46

3.1. Introduction ...... 46

3.2. Methodology ...... 48

3.2.1. Grounded Theory ...... 48

xi 3.2.2. Stages in Developing a Grounded Theory ...... 51

3.3. Language as Representation ...... 56

3.3.1. Analogical Models Encoded as Metaphors ...... 56

3.3.2. Ontological Underpinnings ...... 61

3.3.3. Summary ...... 68

3.4. Student Difﬁculties, Student Learning ...... 69

3.4.1. Student Difﬁculties Interpreting Metaphors ...... 69

3.4.2. Student’s Ontological Confusion ...... 71

3.4.3. Students’ Ontological Groping ...... 72

3.5. Summary ...... 73

4. Analogies, Metaphors, and Students’ Difﬁculties in Quantum Mechanics 75

4.1. Introduction ...... 75

4.2. The POTENTIAL WELL Metaphor ...... 77

4.2.1. Original Descriptive Analogy ...... 77

4.2.2. Analysis of Modern Language ...... 77

4.2.3. Productive Modes ...... 84

4.2.4. Summary ...... 86

4.2.5. Student Difﬁculties ...... 86

4.2.6. Discussion ...... 89

4.2.7. “Robust Misconceptions” Related to the POTENTIAL WELL Metaphor ...... 90

4.3. The BOHMIAN Metaphor ...... 93

4.3.1. Introduction ...... 93

4.3.2. Modern Language ...... 94

4.3.3. The Original Analogy ...... 95

4.3.4. Productive modes ...... 97

xii 4.3.5. Student Confusion ...... 98

4.3.6. Discussion ...... 100

4.3.7. Robust Misconception? ...... 100

4.4. Additional Examples ...... 101

4.4.1. ELECTRONISA PARTICLE ...... 101

4.4.2. ELECTRONISA WAVE ...... 101

4.5. Summary ...... 101

5. Heat and Force ...... 105

5.1. Introduction ...... 105

5.1.1. Issues Concerning Heat in Thermodynamics ...... 106

5.1.2. Issues Concerning Force in Newtonian Mechanics ...... 109

5.2. Summary of PER Literature on Language About “Heat” ...... 110

5.3. Historical development ...... 111

5.3.1. History of caloric ...... 111

5.3.2. Reiﬁcation of Heat from Substance to Process and Substance Again ...... 112

5.4. Modern Language about Heat ...... 113

5.4.1. Grammar and Ontology ...... 113

5.4.2. Subtle Ontological Distinctions in Heat Deﬁnitions ...... 116

5.4.3. Prior Research on Students’ Ontological Commitments . . . . 117

5.4.4. Metaphors ...... 120

5.4.5. Summary ...... 122

5.5. Students’ Difﬁculties: Results and Analysis of the Interview Study . . 122

5.5.1. Description of the Study ...... 122

5.5.2. Students’ Deﬁnitions of Heat ...... 123

5.5.3. Responses to Question 6 (ii) ...... 124

xiii 5.5.4. Analysis ...... 128

5.5.5. Discussion ...... 130

5.6. Introduction to Force ...... 132

5.7. Historical Analogies ...... 133

5.7.1. The Four Force Analogies ...... 134

5.7.2. Historical Development ...... 135

5.7.3. There is No Force ...... 136

5.7.4. Summary ...... 136

5.8. Modern Language: Grammar and Metaphors ...... 136

5.8.1. Results of Coding ...... 138

5.8.2. Summary ...... 138

5.9. Students’ Difﬁculties With Force and Motion: Speciﬁc Examples . . . 139

5.9.1. There is No Force in the Vacuum ...... 140

5.9.2. Passive Objects Don’t Exert Forces ...... 141

5.9.3. Force Causes Motion ...... 144

5.9.4. Ontological Groping: A Reinterpretation of the “Force Causes Motion” Misconception ...... 148

5.10. Summary and Implications ...... 155

6. Summaries, Conclusions and Future Research ...... 158

6.1. Introduction ...... 158

6.2. Niels Bohr, Complementarity, and the Paradigm Shift in Physics . . . 158

6.2.1. Introduction ...... 158

6.2.2. Two Metaphors From Two Different Ontological Categories . 159

6.2.3. Revisiting the Partial Nature of Metaphor ...... 160

6.2.4. Understanding Bohr’s Complementarity ...... 161

6.3. Testing the Grounded Theory ...... 162

xiv 6.3.1. Summary of the Grounded Theory ...... 162

6.4. Coördinating All Representations of Physics: The Ultimate Testing Experiment ...... 164

6.5. Different Language, Different Concepts, Different Understanding . . 169

6.5.1. Kaper’s Study about Thermodynamics ...... 170

6.5.2. Hewson’s Study of Sotho Speakers ...... 173

6.6. Implications For Teaching ...... 175

6.6.1. Can We Speak Better? ...... 176

6.6.2. Language and the Road to Understanding ...... 177

6.6.3. The contextual dependence of students’ knowledge ...... 178

6.7. Conclusion ...... 178

Appendix A. A Glossary of Terms ...... 180

Appendix B. Interview Questions About QM for Physics Professors .... 183

B.1. Questions About Professor’s Own Research ...... 183

B.2. Questions About the Heisenberg Uncertainty Principle ...... 183

B.3. Questions About Visualization or Mental Images ...... 185

B.4. Questions About the Schrödinger Equation ...... 185

B.5. Questions About the Probabilistic Interpretation of the Wave Function 185

B.6. Questions About Wave-Particle Duality ...... 186

B.7. Questions About Measurement ...... 187

B.8. Unclassiﬁed Questions ...... 187

Appendix C. Analysis of Historical Accounts of Force ...... 188

C.1. Origins of the Four Analogies ...... 188

C.1.1. Origins of the “force is like an animate external entity/agent” analogy ...... 188

xv C.1.2. Origins of the “force is like an passive external medium of interaction” analogy ...... 188

C.1.3. Origins of the “force is like an animating spirit” analogy . . . 189

C.1.4. Origins of “force is like an internal passive tendency” analogy 189

C.2. Historical Development of the Analogies ...... 190

C.2.1. Scientiﬁc development of the “force is like an animate external entity or agent” analogy ...... 190

C.2.2. Scientiﬁc development of the “force is like an passive external medium of interaction” analogy ...... 191

C.2.3. Scientiﬁc development of the “force is like an animating spirit” analogy ...... 193

C.2.4. Scientiﬁc development of “force is like an internal passive tendency” analogy ...... 193

C.3. Development of the Modern Language about Force ...... 194

C.3.1. Newton ...... 194

C.3.2. Alternative conceptions of force ...... 195

C.4. The Fifth Theme: There is No Force ...... 198

C.5. Summary ...... 199

References ...... 201

Vita ...... 208

xvi List of Tables

3.1. Table illustrating how Etkina’s model taxonomy ﬁts sucessfully maps into Chi’s (modiﬁed) ontology ...... 63

3.2. Examples illustrating how Etkina’s model taxonomy ﬁts into Chi’s (modiﬁed) ontology ...... 63

3.3. Summary of the mapping between grammar and ontological category 67

3.4. Comparison of lexical and grammatical ontology of thermodynamics 68

4.1. Samples of physicists’ speech and writing for grammatical analysis. . 78

4.2. Ontology ...... 79

4.3. Summary of the metaphorical mapping between the base domain of physical/geographical features and the target domain of interacting QM systems ...... 83

4.4. Misconception: A particle tunneling through a potential barrier loses energy as it goes through or comes out the other side with less energy than it had previously. Summary from three studies...... 91

4.5. Summary of the metaphorical mapping between the base domain of containers and the target domain of the QM wave function ...... 94

4.6. The STATES ARE LOCATIONS metaphor [1] ...... 95

4.7. Electron is a particle metaphor. The examples show how physicists use the particle metaphor as a productive way of talking about what is happening in a QM process...... 101

xvii 4.8. Examples of the ELECTRON IS A WAVE metaphor. All are examples of how the view of the QM particle as a wave is productively used to explain the Heisenberg uncertainty principle...... 102

5.1. A representative sample of clauses from three popular introductory college level physics textbooks containing the word “heat”, grammatical analysis of each clause, and ontological classiﬁcation of “heat” according to grammatical function...... 114

5.2. Heat deﬁnitions from six popular introductory college level physics textbooks...... 116

5.3. Comparison of modern ontology of thermodynamics against the model ontology encoded in the grammar of physicists’ language...... 117

5.4. Comparison of sample expert-novice data from Slotta et al. [2] . . . . 119

5.5. A list of words that cue certain metaphors ...... 121

5.6. Metaphorical classiﬁcation of heat clauses ...... 121

5.7. Summary of the metaphorical mapping between the base domain of material substances and containers, and the target domain of heat in thermodynamic systems ...... 122

5.8. Students’ deﬁnitions of heat ...... 124

5.9. Rankings of students’ ability to solve Q.6 (ii)...... 128

5.10. Rankings of students’ ability to solve the problem (Q.6 part (ii)) compared with their ability to define the meaning of heat. (Process definition (Proc.), operational definition (Op.), or caloric definition (Cal.).)

Spearman rank order correlation coefﬁcient: DTB: rs = 0.9364, p =

0.00007 (2-tailed t test), Rater 2: rs = 0.9053, p = 0.0003 (2-tailed t test). Pearson product-moment correlation coefﬁcient: DTB: r = 0.936, p = 0.0001 (2-tailed t test), Rater 2: r = 0.905, p = 0.0003 (2-tailed t test). 129

xviii 5.11. Comparison of students usage of the HEAT IS A SUBSTANCE, SYS-

TEM IS A CONTAINER metaphor in their responses to Q.6 (ii), against their ability to solve Q.6 (ii). Spearman rank order correlation coef-

ﬁcient: DTB: rs = 0.8165, p < 0.05 (non-directional test), Rater 2:

rs = 0.8656, p < 0.05 (non-directional test). Pearson product-moment correlation coefﬁcient: DTB: r = 0.817, p = 0.0072 (2-tailed t test), Rater 2: r = 0.866, p = 0.0026 (2-tailed t test)...... 130

5.12. Examples of phrases and clauses and how they were coded...... 138

5.13. A list of studies and quoted student responses ...... 143

5.14. Variety of different studies with students who talk about force as a property of the object...... 145

5.15. Study: McCloskey 1983 in Gentner and Stevens: Student Reponses [3] 146

5.16. Study: McCloskey 1983 in Gentner and Stevens: Student Reponses [3] 147

5.17. Comparison between inventors of mechanics and modern students . . 148

6.1. A comparison of students taught with traditional force representations and reformed force representations...... 168

6.2. Examples of common sentences that do not reﬂect the underlying lexical ontology of a particular model and suggestions for their improvement.177

xix List of Figures

1.1. Investigative Science Learning Environment (ISLE) Cycle [4] . . . . 5

2.1. Summary of the ergative model ...... 30

2.2. Chi et al.’s ontology tree that separates all “entities” into matter, processes and mental states (reproduced from [5])...... 44

3.1. Summary of the method of comparative analysis [6] ...... 50

3.2. A revised ontology tree ...... 61

3.3. Summary of the role of analogy, metaphor, ontology and grammar . . 68

3.4. A summary of my thesis ...... 73

4.1. Summary of the metaphorical system and its usage by physicists. . . . 87

4.2. Wave function of the electron in the sudden approximation ...... 99

5.1. Classiﬁcation of heat clauses into ontological categories...... 115

5.2. Summary of the four analogies and examples of modern metaphorical language that uses each analogy as a base ...... 137

5.3. Results of coding...... 139

6.1. Question from second mid-term, physics 193 ...... 167

6.2. Pendulum question from Alan Van Heuvelen’s paper [7] ...... 168

xx 1

Chapter 1

Introduction

1.1 The Role of Representation

In order to improve education, one needs to understand what is happening in the classroom. In particular, what are students doing and learning when they learn physics? One answer, supported by research done in the physics education research community, is that students are engaged in learning or constructing the concepts of physics and changing their prior conceptions about how the world works. (See, for example, [8,9].) Physics education research has devoted a great deal of energy and attention to the question of how to help students learn/construct the concepts of physics faster/better and the difﬁculties that students have in changing their prior conceptions.

In this thesis, I will examine a different approach to what students are doing in the physics classroom. Jay Lemke has suggested that the primary activity that students encounter and participate in, in a physics course, is representing [10]. They encounter many different representations of physics ideas: graphs, equations, tables, pictures, diagrams, and words. These representations of physics ideas are each by themselves incomplete. It takes an act of assimilating, coördinating, and moving easily between many different representations in order to create understanding. Therefore the ﬁrst ability students have to develop is the ability to represent ideas and physical processes in different ways and move between representations.

Some representations may be familiar. Physicists are conscious of the role of equations and graphs in their reasoning. Physics education researchers have extensively 2

studied students’ difﬁculties with these representations and how to help students master activities such as reading and interpreting graphs, or connecting equations to physical reality. Less attention, however, has been paid to language as a representation of knowledge and ideas in physics. The act of speaking and writing is something not often consciously thought about. Physicists are aware that some student difﬁculties may be caused by confusing language (see for example, [11, 12, 13, 14, 15, 16, 17]), but only a small amount of research has been done in this area [18, 19, 20].

The primary starting point of this thesis will be to treat language as a legitimate representation of ideas and processes in physics. I will accord language the same status as other representations in physics such as Feynman diagrams or space-time graphs etc.

From this starting point, there are two questions I will address ﬁrst:

1. What is language representing?

2. How does language work to represent what it represents?

Neither of these questions are easily answerable. One might think, for example, that language represents the concepts of physics. However, the data I have gathered from interviews with physicists, shows that this is not a complete picture. Likewise the way language works in producing a representation, is the subject of more than 50 years of linguistic research and there is no single answer upon which all linguists will agree. There are, however, approaches to answering the second question that are more appropriate than others in the context of physics and these are the approaches that I will use in this thesis.

It is important to answer these two initial questions ﬁrst because, without a theoretical framework, any ideas I may describe about the role and use of language in the physics classroom will be speculation based on intuition. Chapters 2 and 3 are primarily devoted to building up this theoretical framework. After this, I will ask questions and test hypotheses about how students are interacting with linguistic representations and how this interaction affects their learning of physics. In subsequent chapters I will 3

show some of the data upon which the theoretical framework is based, present and analyze data from student interviews, using the theoretical framework, and ﬁnally will present some general ideas about how physics teachers can become conscious of the language they are using in the classroom and how it might be affecting their students’ understanding.

1.2 Orientation and Theoretical Standpoint

In this section I will describe the theoretical underpinnings of my work. My starting point will be the idea that meaning is constructed from a representation rather than conveyed by a representation. One cannot pass information, knowledge or ideas from a sender to a receiver like a slice of bread. All the recipient can receive is a signal (be it photons of light or longitudinal oscillations of the air) from which he/she constructs meaning based on his or her prior experience and understanding of the world [21].

There are two classes of learning theory based on this standpoint. The first is called cognitive apprenticeship [22]. In this view we ask students to reconstruct the knowledge of a particular field by using the same research and reasoning techniques as the experts do or did at the time a particular idea was invented. Students are seen as becoming “apprenticed” in the cognitive strategies of the practitioners of a particular field and employing those strategies in order to learn. In reality this reinvention is impossible without coaching and scaffolding from the instructor, and this becomes his or her role in the class.

In the second view, students are considered to be “legitimate peripheral participants” in a social system of knowledge, its practitioners and their practices [23]. Stu- dents are viewed as moving from the periphery to the centre of this system as they engage in the ﬁeld. They engage in the ideas and the “tools of the trade”, learning to use them more effectively until they become masters. Both approaches closely mirror an emerging view in neuroscience that human memory and cognition is not an act of 4

pattern matching and adding to/recall from a store of knowledge, but rather a never- ending process of active construction/reconstruction of meaning [24].

Both approaches to learning are important to my thesis. Cognitive apprenticeship approaches are informed by what physicists are doing and thinking when they construct their ideas. If I can understand what physicists do, it will give me an idea of what language is encoding. Legitimate peripheral participation asks what students are doing in a physics classroom. If students are engaged in representing, as Lemke has suggested, it is possible that language plays a critical role in the construction of ideas and understanding. It is thus important to understand how language works as a representation. Likewise the idea that students are engaged in an active construction of meaning implies that the meaning they may construct from language may differ from the intended meaning of the speaker/author. This point is crucial for understanding how students interact with the linguistic representations they read and hear.

1.2.1 Cognitive Apprenticeship

When I started my thesis, I wanted to understand more about how physicists came up with their ideas and how we could help students to use those reasoning techniques so that they could reinvent those ideas themselves. If we want students to become cogni- tively apprenticed in physics, the ﬁrst priority is to establish what it is that physicists do when they do physics. A study of the history of physics reveals some common patterns in physicists’ activities [25]. Figure 1.1 below is an obviously simpliﬁed, linearised model of a highly non-linear thought process, but some common elements of physicists’ activities are there [4]. Can we get students to learn physics by engaging in this cyclic reasoning process?

In this model, students learn about the physical world by conducting observation experiments, identifying patterns in the data, constructing explanatory models and/or descriptive models of what is observed, using those models to make new predictions about the physical system under investigation, and then conducting testing experiments 5

Observation Experiments, data collection More Patterns Did not collect data carefully in Qualitative explanation or experiments? Or, made some Check experiment, revise assumptionsDifferent quantitative rule relating incorrect assumptions? Or, physics quantities proposed a faulty explanation/rule

No Prediction

Testing Experiment: Does the outcome match the prediction based on the explanation/rule?

Yes

More testing experiments

Practical applications

Figure 1.1: Investigative Science Learning Environment (ISLE) Cycle [4] to test those predictions. Models which are not rejected by testing are kept until the limits of their validity are discovered and new models are constructed to account for the discrepant data in some manner.

I began my search for a thesis topic looking at Figure 1.1. In the context of quantum mechanics, I wondered how does a physicist or a student make the leap from observation to construction of a model? How does he/she know what is being seen, how is that information processed and organised, and upon what foundation is the model built? For example, how did Schrödinger come up with his famous equation? He certainly was not looking directly at experimental data, although there was certainly a general awareness that the “old” quantum mechanics of Planck, Einstein and Bohr was an extremely incomplete and unsatisfactory model of what was going on at the quantum level. Examination of Schrödinger’s papers revealed that analogical reasoning built on prior knowledge played an essential role in his reasoning process. A review of research 6

on analogy and how students fail to use it successfully, could ﬁll an entire dissertation by itself. But it appears that analogical transfer is very difﬁcult without prior domain expertise [26].

Other researchers have suggested the importance of analogies for scientists. Kevin Dunbar conducted extensive in vivo studies of biologists [27,28]. Nancy Nersessian focussed on historical records relating to the development of touchstone ideas in physics (for example, Maxwell’s equations) [29]. Both approaches revealed that analogy was one of the most essential elements in scientists’ thought processes. Likewise Mary Hesse [30] has argued from a philosophical point of view, that analogy plays a central role in the development of science.

1.2.2 Representations and Legitimate Peripheral Participation

As mentioned before, it is critical to my thesis to understand what physicists do and think when they do physics. Understanding how physicists construct models of the world is key to answering what physicists’ language encodes. I will show that one major function of physicists’ language is to encode the analogies used to develop new ideas in physics. When analogies are developed, physicists are very careful about the limitations of a particular analogy. However, gradually the limitations of a particular analogy disappear from the language and what is left is a metaphor. Clive Sutton suggested that when physicists or physics teachers use analogies to explain ideas to the students they seldom make explicit, the analogies on which the models are based. Rather they speak about these model analogies metaphorically as statements of fact [31, 32]. Max Jammer has argued that:

“It is no exaggeration to contend that the role of analogy is very important for the progress of knowledge, reducing the unknown and the strange to the terms of the familiar and the known. In this sense all cognition is recognition. When studying the historical formation of a concept, one should remember that the metaphor is a powerful agent in the evolution of language as well as of science; it is instrumental in the transference of a word from its ordinary meaning to the 7

designation of a speciﬁc concept as a deﬁned construct within the conceptual scheme of science.” [33] (p. 16).

I began to think that is it was more appropriate to see students as participating in already established knowledge and ideas (legitimate peripheral participation), as well as building ideas up from scratch (cognitive apprenticeship). These educational approaches, combined with the idea that metaphor may be the means by which physicists’ analogies were encoded in language, led me to consider the role of different representations in students’ learning. In particular, I started investigating the role language as a representation and the role of metaphor as the primary representational mode in physicists’ language.

By compiling observations of professors talking about their understanding of quantum mechanics, students struggling with quantum mechanics problems and a study of original papers about quantum mechanics, I arrived at a topic for my thesis. Below is a brief summary of what I will present:

• I will argue in this thesis that language and other forms of communication and representation are part of a knowledge system in which students participate.

• I will approach students as active participants in a system called physics. On the one hand they may be engaged in constructing the concepts of physics. But they are also peripheral participants in already established language conventions of physics. Students have to interpret and assimilate already established ways of speaking and writing. Often the meanings associated with terms in physics is specialized and different from every-day usage of the same terms.

• In the context of physics, I will treat language as one of many representations of the physical models that physicists construct to describe and explain the physical world they observe. I will show that physicists use language to describe the objects in their models and the processes by which objects interact and how 8

physical systems evolve. I will show how language is used in physics to express these ideas.

• I will assume that language is a legitimate way to represent some physics knowledge and will question the role of language as a part of learning physics. I will show that written and spoken language in the physics classroom matters a great deal in the learning process.

If we see students as participants in representing physics knowledge and focus on language as one possible representation, then we can ask more detailed questions: For example, if students interact with language to construct knowledge, can we see their struggles and confusion as attempts to make sense of, and assimilate this representation? I will show that studying linguistic representations can go a long way to help us make sense of what students are thinking and saying in the classroom.

I hypothesize that students are struggling with a disconnect between what physicists say and what they really mean. The key to this disconnect is the difference between the lexical meaning of physics terms, and their actual usage. I hypothesize that students construct meaning not only from the deﬁnitions of terms, but also from the usage of these terms. If the meaning and the usage of a term contradict each other, then the resulting ambiguity may be connected to certain student difﬁculties. Moreover, it will turn out that linguistic representations of physical models are not particularly isomorphic to other representations. At best, much information is lost, at worst, the language can completely contradict the ideas of the model. (Think about how physicists talk about “heat” for example.) Physicists may not even be consciously aware of some of the models encoded in their language! (For example, see the Bohmian metaphor in Chapter 4.)

Oftentimes, language is simply insufﬁcient to express the desired meaning. Physi- cists are aware of the limitations of language: They know that the mathematical representations tell them a whole lot more about how things work in a physical model. 9

For example, one professor I interviewed responded in the following manner to the question of “what is oscillating” in a quantum mechanical wave:

Prof B: “The problem is you’re trying to shoehorn a phenomenon into ordinary everyday English language, and I think the problem is with the language, not with the phenomenon. So, if you ask me to explain it in English, I think En- glish has limitations which make it impossible to give a satisfactory explanation in English. But, I don’t have to understand it in English. I mean, I think I sort of know what’s going on. At least I have realized the limitations in English and, um, uh, it doesn’t bother me.”

Students, on the other hand, may not be aware of these limitations and may be mislead by the language they encounter.

By the end of this thesis I will not be able to say exactly how much language matters, but I will

1. introduce systematic methods which may enable us to quantify ways in which language inﬂuences understanding and decode the representations of physics knowledge that are encoded in language,

2. suggest ways in which language may matter in learning,

3. present data which suggests that there are some clear connections between what is said in the classroom, what is written in textbooks and exam questions, and how students reason,

4. suggest how a linguistic view can help us understand and facilitate some of the struggles our students undergo in trying to make sense of what they are learning, and

5. show how language and linguistic issues might be addressed successfully in the physics classroom. 10

Chapter 2 Literature Review

2.1 Introduction

In Chapter 1 I have reviewed the ideas of cognitive apprenticeship and legitimate peripheral participation, and how they relate to student learning and the practice of physicists. What physicists do (generate analogies make models etc.) lies at the center of the knowledge system. On the surface or periphery, students interact with representations of that knowledge. Thus the representations must be in some way connected to the activities that lie at the center. In this chapter I will present a theoretical perspective on linguistics and communication, and review some of the research on analogy (part of the center of the knowledge system). Then I will review existing research on the role of language in physics and science education as well as certain aspects of linguistic models of meaning that will be relevant to the theoretical framework that I will develop in Chapter 3. I will also brieﬂy cover some of the literature on students’ difﬁculties in learning physics.

2.2 Language, Thought, and Communication

In this section, I will lay out the theoretical standpoint for language as a representation of knowledge and the relationship between language and cognition. Language and thought is a two way interaction. Language can shape our reality. It can deﬁne and constrain how we think and reason. It can give us an insight into how we think. I am going to use all of these ideas as a starting point for the role of language in physics. 11

2.2.1 The Interaction of Language and Behavior: The Sapir-Whorf

Hypothesis

In this section I will review the hypothesis that there is a bidirectional interaction between language and thought [34, 35, 36, 37] and introduce the notion of linguistic relativism, upon which cognitive linguistics bases its aims, beliefs and methods.

Benjamin Whorf [35] elaborates the connection between communal behavior and language by suggesting that certain social behaviors may be cued by the way a group speaks about a certain situation. In addition, these “language habits” are largely unconscious. In talking about how habitual thought and behavior are related to language, Whorf notes for example how cyclic concepts such as “summer” and “time” are turned into objects as they are treated as nouns in the English language. Objectification of “time” as a substance makes it quantifiable, ordered and spatially orientable. Thus we can talk about “a second of time” in the same sense as “a slice of bread”. We can orient and order time in a linear fashion, as revealed in the phrase “this week, next week, the following week”. In the Hopi language such words as “summer” and “time” do not exist as nouns. Whorf argues that the way the Hopi speak about “time” interacts with their cultural behaviour. The Hopi count events in ordinals. This means that the focus is not on a collection of events, but rather on the reappearance of the same event (the first, the second, the third. . . ) This in turn reflects and is reflected in, a focus on elaborate preparation for the future. Consider for example time as reckoned in successive visits of the same man. (The first visit, the second etc. . . ) If desired, the focus will be on influencing that one man through elaborate preparation for future visits rather than trying to influence him only during his present visit.

Whorf, and later George Lakoff and Mark Johnson [38], argue that our western conception of time as a quantifiable object affects and/or reflects our behavior too. For example, many people are paid, or charge for services, by the hour, reflecting the conception of time as a quantifiable and limited resource. We break diary pages 12

in to weeks or days and calendars into months. Art history is broken into periods: romanticism, impressionism, modernism, and so on. Such conceptions are so ingrained that it is almost impossible to imagine that humans might conceive of time in a different manner.

Communication of theories and concepts in physics follow similar patterns. Dis- course in physics is replete with examples where abstract physical concepts such as “heat” and “force” are spoken about as if they were physical objects or substances. It may be argued that these quantities should be viewed more accurately as processes or interactions. As, for example, physicist Robert H. Romer [15] vehemently argues: “Let’s strike a blow for clear thinking by ridding the English language of the word heat as a noun.”

The contrast between the Hopi and English conceptions of time is one of numerous examples which suggest that there is an inescapable subjectiveness in language [34]. Whorf [36] makes this notion explicit by introducing the idea of “linguistic relativism”:

“It was found that the background linguistic system (in other words, the grammar) of each language is not merely a reproducing instrument for voic- ing ideas, but rather is itself the shaper of ideas. . . We dissect nature along the lines laid down by our native languages [my italics]. . . We are thus introduced to a new principle of relativity, which holds that all observers are not led by the same physical evidence to the same picture of the universe, unless their linguistic backgrounds are similar, or can in some way be calibrated.” (pp. 212-214)

This statement is referred to as the “Sapir-Whorf hypothesis”. It suggests, that we can gain insight into how people organize their ideas and understand the world by examining their language.

2.2.2 Communication as Construction

In this section I will apply ideas of information theory to student learning in physics. Michael Reddy [21] suggested that people construct meaning from the words that they hear, based on their prior knowledge and experience. For example, If you ask me: 13

“Are you sad?” And I respond with a “1”: What would this response mean? By itself it means nothing, it is simply a signal. Imagine that we have a list of possible responses: either 1, 2 or 3. This is called a “repertoire” of responses. After we have established a repertoire of responses we need to assign a meaning to them. Say we agree before hand that 1 = “yes”, 2 = “no”, and 3 = “unable to give a deﬁnite answer”. Now we have established a shared repertoire of acceptable signals, plus a shared code. We have the means to communicate. This example shows that by itself the signal is meaningless. A recipient has to construct the meaning using a commonly understood repertoire and a previously shared code. The code has to be shared a priori between sender and receiver.

We can apply these ideas to teaching physics, as a starting point for models describing how students understand what they hear or read in a physics class. Often the teacher and the students share neither repertoire nor coding scheme. This leads to a possible miscommunication. Most often the common code is the missing piece. Physi- cists often talk about the language of physics involving specialized meanings which are not commonly understood.

For example, when a physicist says “the electron is in the ground state,” she means that the electron has a particular energy. However, if the students do not share the code for the word “state” as the energy state, they may construct a spatial interpretation from the same statement. I have observed exactly this problem and will cover it in greater detail in Chapter 4.

From the above discussion, it follows that meaning cannot be directly passed, conveyed or in any way transported from the instructor to the student. The teacher has to help the student construct meaning by elaborating the code. Students can then use this code to decode the words that the instructor uses.

Based on this, I hypothesize that different students should come up with different meanings for what they hear in the classroom. The hypothesis will be supported if I can predict a future student response based on a systematic interpretation of the language used by the instructor. In Chapter 4 I will present an example. 14

Another implication of this model of communication is that it gives us an opportunity to understand better what students are saying. Sometimes what sounds as “wrong physics” or misconceptions may simply be students’ attempts to add to or reﬁne their coding scheme. For example, students often ask instructors, when analyzing the forces exerted on a projectile in the air, “should I put the force my hand exerted on the projectile into the free body diagram?” Traditionally such question is considered to be a sign of a misconception, however from the “reﬁnement” point of view it can be considered as an attempt to better understand the meaning of the term “force”. I will elaborate on this example in Chapter 5.

2.3 Analogy

2.3.1 Introduction

In this section I will review how physicists and scientists in general use analogy to generate new knowledge. First, I will examine research in cognitive science related to analogy. Then I will analyze the role that analogy plays in physics research. And ﬁnally I will describe studies related to the use of analogies by students.

2.3.2 What is an Analogy?

The study of analogy belongs to the domain of cognitive science and philosphy. Lin- guists have not focussed much attention on analogy because analogy is used as a cognitive device, not as a ﬁgure of speech. However, there is little agreement between different authors on what analogy is. Below I will describe two deﬁnitions of analogy and focus on the latter since it is considered to be the most productive.

Amos Tversky suggested the simplest model of analogy [39]. For him, analogy consists of a set of similar features from the two systems being compared. The greater the overlap of similar features, the stronger the analogy. Most cognitive scientists 15

consider this definition to be inadequate because many analogies involve comparisons between domains that have no obvious similarity. For example a traditional physics analogy between the solar system and the atom does not fit Tversky’s definition. In this analogy relationships between objects are similar rather than any surface features.

Mary Gick, Keith Holyoak and Dedre Gentner [26, 40, 41] suggested a different deﬁnition of analogy that emphasizes a mapping between similar relations rather than similar features of the two systems. Their model of analogy is formalized as “Structure- mapping theory” [26, 41].

In this model, an analogy consists of a mapping between two domains: a target (a new domain that we are trying to understand) and a base (the familiar domain). (Note that I will use the terms base and target, as they are deﬁned here, throughout this thesis, to refer to both analogical and metaphorical comparisons.) A domain may be loosely deﬁned as a system of objects and predicates. Predicates can either be attributes (e.g., the Sun is yellow), or relations between objects (e.g., the Sun is more massive than the

Earth). Given a set of objects in the base: {bi} and a set of objects in the target: {tj}, we want to form a mapping M between them: M : bi −→ tj. A mapping between base objects and target objects with similar attributes, is called a “literal similarity”. To generate an analogy, Genter suggests mapping objects in the base to objects in the target that preserve similar relational predicates. Often, this requires ignoring attributional predicates.

For example, an analogy between a model of a “mother” and “the Earth” is not based on any similar attributional predicates (physical similarities) between a mother and the Earth. We map mother to Earth based on the similarities of the relational structures between mother and child on the one hand, and the Earth and the creatures that live in her protection on the other. (Namely: nurture, provision of sustenance, protection, and so on.) The same analysis applies to physics. Schrödinger based his derivation of the wave equation for matter on a similar relational structure between (1) how the wave equation of wave optics reduces to the Eikonal equation of geometrical 16

optics in the small wavelength limit, and (2) how a matter wave equation should reduce to the standard Hamilton-Jacobi equation for massive particles in the small deBroglie wavelength limit.

It is difﬁcult to draw a clear line between a literal similarity and an analogy based on a relational mapping. Hesse [30] pointed out that there is a continuum of possible types of comparison. These range from a comparison involving similar surface features (Gentner’s “literal similarity”), to a comparison where the base and target domains share no physical similarities and the mapping is entirely relationally based.

Hesse also showed that when scientists use an analogy to construct a model of a target domain, they divide the analogy into a positive, negative and neutral analogy. The positive analogy denotes those aspects of the base and target systems which may be mapped to each other. (In a planetary model of the atom the Sun is mapped to the nucleus, the planets are mapped to the electrons.) The negative analogy consists of those aspects which are not mapped. (For example, there is no physical aspect of the atomic system which is analogically related to the color of the Sun.) The neutral analogy consists of those aspects of the base system whose analogical equivalents in the target system have yet to be examined. These are the predictions of the analogical model.

2.3.3 The Role of Analogy in the Development of Models in Science

There is considerable debate between physicists, historians, cognitive scientists and philosophers as to whether analogy is central and indispensable to scientiﬁc reasoning, or whether it is peripheral and unnecessary. There is not likely to be a resolution to this debate. However, analysis of the history of physics and the practice of science shows that scientists tend to use analogy frequently. Examples from history of physics, philosophy and modern scientiﬁc practice are given below. 17

• History: Nancy Nersessian has analyzed the historical writings of several physicists. For example, she showed that Maxwell used a combination of two base domains (ﬂuid mechanics and machine mechanics) to construct a hybrid vortex- idle wheel model of electromagnetic processes. He used this model to derive the differential forms of the equations of electrodynamics [29].

When scientists produce analogies they do it in a similar manner to lay-people who use analogies in everyday life. Nersessian called this a continuum hypothesis which means that scientiﬁc reasoning is a natural extension of everyday reasoning [29].

• Philosophy: For a long time, philosophers considered analogy an example of “weak” reasoning methods. Other examples of weak reasoning methods are simile and metaphor. Hesse [30] suggested that these weak reasoning methods are, central to the development of ideas in science. Her views are in opposition to those who consider propositional logic (axioms and logic rules) to be sufﬁcient for generating knowledge.

• Modern science: Kevin Dunbar and colleagues have observed members of suc- cessful biology laboratories during discussions and group meetings. They found that scientists use analogies extensively to communicate ideas and explain unfamiliar biological systems. Dunbar divided analogies generated by scientists into three broad classes: (1) local analogies, (2) regional analogies, and (3) long- distance analogies [27]. Each type of analogy served a speciﬁc purpose in the reasoning of the biologists. Local analogies typically involved using an old experiment to guide the design of a new experiment. Regional analogies involved comparisons between different domains within biology and were essential in building new theoretical models. Long-distance analogies were those that used concepts outside the domain of biology to describe and elaborate a biological concept. These analogies served an instructional/teaching role in the research 18

groups that were studied.

2.3.4 Student Difﬁculties in Analogical Reasoning

As shown above, analogies play an important role in generating ideas in science. How- ever, research on students’ learning indicates that students have difﬁculties with analogical reasoning. The explanations of the differences between scientists’ and students’ analogical reasoning to explain new phenomena lies in:

1. Students lack basic understanding of the base domain [26].

2. Students do not see the negative and neutral aspects of an analogy, they tend to overextend the analogy and they focus on surface similarities rather than relational similarities.

3. Students struggle to make the analogical mapping if the two domains don’t look alike. Gick and Holyoak hypothesize that students need to develop abstract levels of representation in order to see an analogy [40, 42].

Various attempts have been made to overcome these problems. The teaching with analogies (TWA) model contains six steps [43]:

1. Introduce the target concept.

2. Recall the analog concept.

3. Identify similar features of target and analog.

4. Map similar features.

5. Draw conclusions about concepts.

6. Indicate where the analogy breaks down. 19

Shawn Glynn notes that: “An analogy can lead students down the wrong path,” and, “. . . analogies are double edged swords.” [In other words, they break down.] The point of using the TWA model is to avoid these and other pitfalls. Glynn’s observations serve to illustrate, again, one of the difﬁculties that students have with analogy. Students tend to over-extend an analogy or apply it inappropriately.

To overcome the analogical transfer problem, John Clement [44] proposed a system of “bridging analogies”. From a distant anchoring example [base domain of the analogy], students were given an intermediate domain more closely related to the target domain and asked to explain the similarities and differences. For example, students were initially unable to construct an analogy between an example of an object com- pressing a spring, and the normal force exerted by a table on a book. To make a connection, students were asked to consider an intermediate example of a book resting on a table made out of two supports and a thin ﬂexible board. Clement observed that:

1. “Students appear to readily understand the anchoring cases.”

2. “However, many students indeed do not initially believe that the anchor and the target cases are analogous.”

3. “Some of the bridging cases sparked an unusual amount of argument and con- structive thinking in class discussions.”

4. “Students were observed generating several types of interesting arguments . . . even spontaneous generation of bridging analogies.”

2.4 Metaphors

As we discussed in the previous section, analogies may be an essential part of scientiﬁc reasoning. However, they are not part of our language because they are not ﬁgures of speech. When we speak, instead of analogies, we use metaphors. In this section I 20

will discuss the relationship between the two and the difﬁculties that this relationship creates for students learning physics.

2.4.1 What is a Metaphor?

There is very little consensus about what a metaphor is, and it is not possible to present a single deﬁnition of what a metaphor is that will be nice and succinct.

Traditionally, metaphor has been regarded as a superfluous literary figure. In other words, a metaphorical expression may be replaced by a literal expression which may be more cumbersome, but equivalent in meaning. For example, “the clouds are a blanket” might be replaced with something like “the clouds form an insulating layer which traps heat” [45]. The study of metaphor has however moved beyond the notion of metaphor as a figure of speech and into the domain of cognition. Cognitive scientists and linguists are starting to see metaphor as an indispensable part of human cognition [38, 41, 46, 47, 48], and it is from this point of view that I want to pursue the role of metaphor in learning physics.

On a cognitive level, a metaphor is considered to be some sort of comparison between domains. Traditionally, a metaphor is often seen as a condensed simile. (Namely a metaphor is a simile with the word “like” omitted.) In some modern views of metaphor, the inclusion or exclusion of “like” in a comparison is largely irrelevant. There is however some experimental evidence that simile and metaphor promote different cognitive processes [46, 48]. I will elaborate this point later in Section 2.4.3.

Hesse pointed out that analogical comparisons run a gamut from comparisons of surface features to a comparisons of deep relational structures [30]. Likewise the linguistic representations of those comparisons can range from a literal similarity (“hawks are like eagles”) to a metaphor (“the Earth is like a mother”). In the latter case the comparison is entirely a mapping of relations between mother and child, and the Earth and the creatures that live in her protection. (See Section 2.3.2 for a discussion of structure mapping). In fact, Gentner has suggested that “many (perhaps most) metaphors are 21

predominantly relational comparisons, and are thus essentially analogies” [41].

There is, however, a problem with all of the views of metaphor presented so far. Max Black and David Ritchie pointed out that thinking of a metaphor as a comparison of any sort, results in a circular deﬁnition of metaphor [45, 49]. For example, how can one make a comparison between “money” (base) and “time” (target), (the metaphor

TIMEISMONEY) without ﬁrst invoking a metaphorical interpretation upon which to guide the mapping? Before you can make any comparison between time and money, you have to see time as money. Thus metaphor itself must precede any sort of comparison [32] where the base and target are devoid of surface similarities. Even when it appears that there are some obvious similarities upon which to base the comparison, one is generally on dangerous ground since, in most cases, similarities in novel comparisons are a matter of cultural perception, rather than objective similarity. For example, the planetary model of the atom is so ubiquitous that it is sometimes difﬁcult for a physicist to step back and remember that no one has ever directly seen a proton or an electron and therefore there is little basis for picturing them as hard spherical balls, or even as particles at all.

One of the solutions to the circularity problem was proposed ﬁrst by Black [45]. It is called the interaction view of metaphor. Many comparisons are culturally situated,

(the metaphor TIMEISMONEY is based on a culture which interprets time as a valuable commodity or substance which can be broken into units and bought and sold), so the essential ingredient of a novel comparison or metaphor is a set of “commonplaces” associated with the notions of time and money. Then, in Black’s interaction view, a metaphor does not function as a comparison, but as a filter (in the “time is money” example, the filter is “money”) through which the target concept (time) is viewed and interpretted. Thus a metaphor functions to create similarity between base and target, rather than find it. The idea of metaphor as a filter through which concepts could be seen and understood, was later taken over by Lakoff and Johnson with their idea of the conceptual metaphor [38]. (See Section 2.4.2 for a detailed account of conceptual 22

metaphor.)

In summary, I am most interested in the conceptual role of metaphor. Metaphor undoubtedly involves some sort of comparison yet, as a cognitive and conceptual device I have shown that metaphor is necessarily antecedent to any novel comparison. This is, however, not so important to my thesis. In the sections that follow I will show that conceptual metaphor assumes a role subsequent to any analogical mapping (see Section 2.4.4). in other words, to explain a new phenomenon, event or system, one ﬁrst creates an analogy. After an analogy becomes well established in the community that uses it, it becomes labeled with conceptual metaphors.

2.4.2 Metaphorical Language, Metaphorical Conceptual System

Cognitive linguistics is a branch of linguistics which aims to understand the human conceptual system by studying how we speak. Cognitive linguists’ hypothesize that language and thought are metaphorical. “Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature. . . The essence of metaphor is understanding and experiencing one kind of thing in terms of another. [Their italics] [38] pp. 1,3”

This passage describes what has been termed the conceptual metaphor, or in Brian Bowdle and Dedre Gentner’s language, a conventional figurative [46]. Conceptual metaphors are often unconscious metaphors and seldom made explicit. They have become quite literal, losing their figurative origin through their unconscious and frequent use. George Lakoff and Mark Johnson argue that we function by using these common- place metaphors in our everyday living. An important aspect of metaphors is that they highlight and hide. For example, the metaphor ARGUMENT IS WAR highlights combat- ive aspects and hides the coöperative aspects. Thus in a particular situation an argument may (for all intents and purposes) become a conflict.

In Lakoff and Johnson’s model, human concepts and conceptual systems are themselves metaphorical in nature. Metaphors are organized together into systems. One 23

could think of these systems as analogical models of the world. Just like analogies, metaphorical systems are grounded in terms of objects and experiences that are familiar to us. More abstract and ethereal concepts (emotions, ideas, time) are described in terms of more concrete and clearly delineated concepts (containers, spatial orientations etc. . . ). And like an analogical model, a metaphorical system has an underlying ontology of substances or things that undergo processes of movement or transformation.

For example, the metaphor TIMEISMONEY is based on another ontological metaphor which may be stated roughly: TIME IS A VALUABLE SUBSTANCE WHICH MAY BE

BOUGHTANDSOLD. However there is a big difference between a metaphorical system and an analogical model. Unlike an analogical model, a metaphorical system has no underlying basis of objective comparison. An attempt to deﬁne the metaphorical system this way would lead to circularity and inﬁnite regress. As I will show later, this is not the only difference between a metaphor and an analogy

For example, the metaphorical system, TIMEISMONEY, entails TIMEISALIM-

ITEDRESOURCE (based on an ontological metaphor in which TIME IS A SUBSTANCE/OBJECT

THATCANBEBOUGHTANDSOLD) In turn, this implies TIME IS A VALUABLE COM-

MODITY. All these together form a conceptual system which may be labelled as TIME

ISMONEY. This is evidenced by common phrases such as:

“ You are wasting my time This gadget will save you hours. How do you spend your time these days? That ﬂat tire cost me an hour. I’ve invested a lot of time in her. He’s living on borrowed time. You don’t use your time proﬁtably. Thank you for your time.” [38] (pp. 7-8)

The most important thing to observe from these examples is how to identify the presence of the metaphors in the language. The italicized words are all words we associate with the conceptual idea of money, and are now being attached to the concept of time. It is this use of words from the base conceptual system and associated with the target that permits these metaphorical systems to be identiﬁed. 24

My goal is to apply the idea of conceptual metaphor to communication in a physics classroom. (Remember that communication is about constructing meaning.) Notice how different a dictionary meaning of the word “time” may be from its subtle meanings suggested by its metaphorical use in language. Rather than an objective meaning for time, humans interpret time in terms of several different systems of concepts. In

addition to the metaphorical system TIMEISMONEY, we use TIMEISAMOVINGOB-

JECT. (We speak about time as if it was moving past us, as suggested by the phrase, “this week, next week, the following week.”) We choose a metaphorical system depending on the context of a particular situation.

If we assume that communication through language in physics happens in a similar way, similar structures can be observed. For example, I have observed in interviews with physicists how they switch quickly and easily between two metaphorical systems

to describe an electron (namely THE ELECTRON IS A WAVE and THEELECTRONISA

PARTICLE) depending on the type of question they are asked and the direction of the discussion. If we see meaning-making as driven fundamentally by metaphor, it is possible to ask questions about (1) what unconscious metaphorical systems are encoded in physicists’ language, and (2) how are students interpreting those metaphorical systems? It will turn out that this is a very productive way of thinking about the role of language in learning physics. I will use it as one of the major theoretical underpinnings of how I approach language in later chapters.

2.4.3 More on the Conceptual and Cognitive Role of Metaphor

It has been noted and experimentally observed that the insertion or omission of “like” in a comparison triggers different cognitive processes. Bowdle and Gentner found that subjects preferred the insertion of “like” (“X is like Y”) when the comparison was novel or unfamiliar (a novel ﬁgurative), while they preferred the omission of “like” (“X is Y”) when the comparison was a familiar conceptual metaphor (conventional 25

ﬁgurative) [46]. The authors note that a statement such as “time is money” is grammatically equivalent to a statement of category membership (for example, “water is a ﬂuid”). A statement such as “time is like money” is grammatically equivalent to a literal similarity (for example, “a hawk is like an eagle”). The insertion of “like” invites comparison.

In everyday experience, it has been found that we classify things according to how well they match a “physiologically situated” or “culturally relative” prototype [50]. Thus in colloquial use, “water” might be considered a prototypical example of the category “fluid”. To a physicist however, asserting water is a fluid means that water must satisfy all the pre-defined properties of a “fluid” (whatever those properties have been chosen to be). It need not satisfy those properties exactly, but must be close enough so that differences are negligible in the context of the problem in which you are assuming that water is indeed a fluid. This is what I will term a physicist’s classification scheme.

There are then two possibilities for a metaphor

1. If the metaphor is surprising to the listener (a novel metaphor), a statement of the form “X is Y” has to be interpreted through the formation of a new shared category (an ad hoc category) of which Y is a prototypical member [48]. For

example, to comprehend THEELECTRONISASMEAREDPASTE, the reader has to come up with an ad hoc category shared by both. A physicist who understands the quantum mechanical behavior of an electron, might suggest a category of “things that don’t have a well-deﬁned location”. There is no guarantee that a student will come up with the same classiﬁcation unless it is made explicit.

2. If a metaphor is an accepted part of everyday language it is a conceptual metaphor or conventional ﬁgurative. Conceptual metaphors are those that are well understood by a community of speakers through pervasive use. For physicists, a 26

conceptual metaphor, such as THE ELECTRON IS A WAVE, is often used unconsciously and is more akin to a statement of category membership. But such an assertion is known not to be literally true even though it is asserted as such.

Rather, the ELECTRON IS A WAVE metaphor highlights certain wave-like aspects of electron behavior and ﬁlters out other properties and behaviors which are de- cidedly unwave-like. What is highlighted and hidden is well understood by the community that uses the metaphor.

2.4.4 Metaphors and Analogies

To understand the second role of metaphor, I need to draw together most of the ideas covered so far.

In the context of science education, Clive Sutton [31, 32] has suggested that as an analogical model becomes well established, the role of language in articulating that model changes. Tentative or interpretative terms (any novel comparison), evolve into a system of labels. These labels appear as metaphors — they are statements of fact and their interpretative origins are lost. (Since Sutton wrote these ideas before much of the modern research on metaphor, one could say with hindsight that he is referring to the evolution of novel ﬁguratives into conceptual metaphors.) For example, physicists seldom make explicit the analogy between classical waves and quantum systems. It is now sufﬁcient to say that the electron simply is a wave.

Bowdle and Gentner have suggested the same idea with their the career of metaphor hypothesis [46]. They suggest that a novel ﬁgurative (essentially an analogy with little or no surface similarities between base and target) evolves to a conventional ﬁgurative that has become unquestioned through continual usage.

A physicist’s conceptual metaphor (e.g., THE ELECTRON IS A WAVE) may well be a physics student’s novel metaphor. The student will have to come up with an ad hoc category to comprehend the metaphor. The student interpreting the comparison must decide in what respect an electron and a wave are similar. What common category 27

could he/she put them both in? A physicist knows electrons and waves both belong to the category of things that can be polarized and diffracted. Shen’s idea of ad hoc category formation, permits the possibility that the listener may place the electron and a wave in a completely different category as compared to the accepted expert view. For example, a student may categorize electrons and waves as things that need a medium to travel in.

In another example, David Meltzer [51] observed that 69% of physics students interviewed, (incorrectly) viewed heat as a state-function of the thermodynamic system. I can explain the students’ reasoning as mis-categorization of heat into the category of real substances (suggested by the ubiquitous conceptual metaphor HEAT IS A FLUID) rather than into the category of things that obey a diffusion equation, which would be a more acceptable “expert” view. Meltzer observed that those students who viewed heat as a function of state frequently mentioned the volume and temperature of the system as an indicator of the amount of heat in the system, in justifying their reasoning.

2.4.5 Summary

In Section 2.4 I have presented partial answers to both of the questions asked in Chap- ter 1. (Namely, what is language encoding and how is it encoding it?) The literature surveyed in this section, if applicable to physics, suggests that physicists’ language encodes analogical models and it achieves this through the evolution of language from novel metaphors to conceptual metaphors that are well understood by the community who uses them. Most importantly however, the literature suggests a mechanism behind students’ potential confusion with language, namely students form incorrect ad hoc categories in the process of metaphor comprehension. 28

2.5 Grammar and Meaning

According to functional grammarians, meaning is more than a dictionary deﬁnition assigned to each word. When someone reads or hears a sentence, meaning is made by how the words work together in the sentence. In other words, grammar is an essential part of meaning construction, and thus essential to any thesis involving a study of how meaning is constructed from language.

Consider the sentence “his foot accelerated the ball.” The subject (“his foot” — noun group) and object (“the ball” — noun group) are considered grammatical participants participating in a grammatical process (“accelerated” — usually a verb or verb group) [52]. In another language the same sentence might be structured: “his foot, the ball accelerated.” (I.e., subject, object, process.) Yet the idea of participants and process is still there. The ordering gives the native speaker an idea of what is a participant and what is the process. Michael Halliday argues that although different languages have different grammar, the underlying notion of process and participant seems to be universal [52]. An English speaker can ﬁgure out the strangely ordered “his foot, the ball accelerated,” because we know the dictionary meaning of the words, but the following extract from Lewis Carroll’s nonsense rhyme, “Jabberwocky”, illustrates the importance of grammar in making meaning: “the slithy toves did gyre and gimble in the wabe”. The subject (the slithy toves), object (the wabe) and process (did gyre and gimble) are immediately obvious from the ordering (as well as additional cues such as the judicious use of “the” and “did”).

In this section I will discuss one particular view of language (the ideational view) that is relevant to physics (see Section 2.5.1). There are many models of how grammar produces meaning in the ideational view of language. In Section 2.5.2 I will describe one such grammatical model (the ergative model) that is well suited to understand the language that physicists use. 29

2.5.1 The Ideational Function of Language

According to grammarians, language has three meta-functions. I will only consider one, namely how language functions “as a means of representing patterns of experience” [52]. This is referred to as the ideational function of language. The ideational perspective is uniquely suited to analyzing language in physics. Physics is all about describing and modeling the world. Language performs exactly the same purpose from the ideational perspective.

In the ideational view, grammatically, each sentence is made up of at least one process. (A process is usually denoted by a verb or verb group.) Within the grammatical process category there are a number of different types of process. The one I will be most concerned with is the material process. (For example, “Heat ﬂows from the hot reservoir into the engine.”) In addition there are relational processes (for example, “the electron is a wave” or “a force is a push or a pull,” or “the system is like a water well”), and mental processes (for example, “John likes the gift”). There are other categories of process but are not important to my analysis. The grammatical process refers, not to the whole sentence, but to the verb or verb group. In the examples above these are, “ﬂows”, “is”, “is like” and “likes”.

Surrounding the verb or verb group are a selection of one or more grammatical participants that participate in the process. A participant is generally denoted by a noun or noun group. From the above examples, the participants are, “heat”, “the electron”, “a wave”, “a force”, “a push”, “a pull”, “the system”, “a water well”, “John”, and “the gift”.

Surrounding the participants are an optional set of one or more grammatical circumstances. These are extent/location, manner, cause, accompaniment, matter, and role. Each of these circumstances serves to elaborate something about either the process or one of the participants. A circumstance is easily identiﬁed since it is either an adverb, adverbial phrase or prepositional phrase. In the examples above (“Heat ﬂows 30

from the hot reservoir into the engine,”) there were two circumstances of location; “from the hot reservoir”, and “into the system”. The adverbs “from” and “into” are the keys to spotting the circumstantial elements.

I have shown in Section 2.5.1 that the ideational view is, in some sense, the language of physics. In Section 2.5.2 below, I will show in more detail how the ideational function of language can be modeled grammatically.

2.5.2 The Ergative Model and Material Processes

The part of grammar that is most important in physics is that of material processes. Grammarians separate material processes two categories. The first is action, which can be identified as the answer to the question: “What did X do?” (For example: “The environment did work on the gas”, or “A force of 50 N acts on the box.”) The second is event, which can be identified as the answer to the question: “What happened to X?” (For example: “Heat flows from the hot reservoir into the system”, or “A beam of electrons scatters off a delta function potential.”) Note that “heat” and “a beam of electrons” did not do anything. There are other external agencies, not mentioned in the sentences, that are causing the events.

Halliday [52] presents the following model to help understand how material processes are realised grammatically (see Fig. 2.1).

Figure 2.1: Summary of the ergative model 31

In this model1 there are two core constituents of a sentence, the process and the medium which are present in every sentence. For example, “heat [medium] flows [process]”, or “A force [medium] acts [process]”. The medium is a “special” participant because it is the one noun/noun group that is indispensable to the meaning of the sentence. Without it a complete sentence cannot be formed. Surrounding the central core is a layer of secondary participants that serve to elaborate the process further. These are agent, beneficiary, and range. The agent is the “doer of the deed”. For example “Object A [agent] exerts [process] a force [medium] on object B.” Beneficiary and range could be thought of as sentient/active recipient and passive/non-living recipient as I will illustrate in the following examples: “I [agent] sent [process] a gift [medium] to my friend [beneficiary],” which may be turned around and restated, “I [agent] sent [process] my friend [beneficiary] a gift [medium].” In the case of range you cannot do this: One can say “I sent my proofs to the printer [range]”, but “I sent the printer [range?] my proofs” makes no sense unless “the printer” refers to a human who op- erates a printing machine of some sort. Notice how the very subtle ordering of the words in the sentence interacts with the lexical knowledge (dictionary knowledge) of the reader to make meaning! Around the participants is a tertiary layer of grammatical circumstances that function to elaborate something either about the process itself or one of the participants in the process. These are explained in Section 2.5.1 above.

2.5.3 Summary

The purpose of reviewing grammar here is that it is part of the apparatus humans use to make meaning out of language. In subsequent chapters I will use the grammatical analysis described above to analyze physicists’ language and come to a better understanding of how that language is interpreted by a physics student. I will also show how

1The ergative/nonergative model of grammar is an alternative to the transitive/intransitive model. For example, “The boy kicked the ball”/“the ball rolled” would be considered an ergative (agency present)/nonergative (agency implied) pair. 32

grammar and metaphor function together to make meaning. Although the two topics appear quite separate at the moment, I will combine them together into one theoretical approach to understand how physicists describe their ideas in language and how students interpret that language.

2.6 Language in Physics

Physicists are aware that when they speak and write, some students have a great deal of difﬁculty understanding what is said or written. Linguists have also studied the language of physics. In Section 2.6.1 I will provide a sample of this analysis. In Section 2.6.2 I will review the research of physicists and physics education researchers on the role of language in learning physics.

2.6.1 A Linguistic View of the Difﬁculties of the Language of Physics

Linguists have identified patterns of grammar peculiar to scientific fields. They suggest that these patterns have evolved in the last 500 years in order to support the process of scientific reasoning [53]. Scientists do not employ these grammatical patterns consciously and it is therefore easy to misuse them. Such misuse can make scientific talk and writing extremely difficult to comprehend.

Scientists employ a “given-new” grammatical structure to support a logical progression of ideas. For example: “x happens” → “Given that x happens, [given] then y is true [new].” Using this structure, ideas can build naturally upon each other to make up a complex chain of reasoning.

The given-new structure of scientiﬁc argument is facilitated by a grammatical device called grammatical metaphor2. Grammatical metaphor is a device whereby one

2Grammatical metaphor is different from conceptual metaphor. Conceptual metaphor involves a comparison between two different conceptual domains, grammatical metaphor involves the substitution 33

grammatical class is substituted for another. For example, replacing a verb with a noun. The most common grammatical metaphor in science is nominalization. Nominaliza- tion refers to a process whereby a noun or noun group is substituted for a verb or verb group. For example, instead of writing “x happens [verb] and y happens [verb],” one may encounter “happening x [noun group] causes/is related to [verb group] happening y [noun group].” In this example the two processes “x/y happens” have been replaced with two noun groups: “happening x” and “happening y”, that serve as names for the two processes. Using this grammatical device, scientists can express a causal relationship or correlation between different events or processes.

Used carefully, this given-new structure with the use of nominalization forms the backbone of scientiﬁc writing and speech. The ﬁrst page I turned to in a popular introductory college level physics textbook [54] yielded the following example: “As the ice skater. . . pushes herself away from the railing, there is a force. . . on her from the railing. . . ” This was followed two sentences later by: “Here energy is transferred internally. . . via the external force.” This was followed another sentence later by “We want to relate the external force. . . to the internal energy transfer.” [54] (p.184).

In the above sequence the authors start out by describing an (already familiar) process whereby a skater pushes off a railing which means that the railing exerts a force on her. In the second sentence this process is replaced by “external force”, a noun group that refers to the process by which the railing exerts a force on her. The second sentence also includes a new process in which energy is transferred internally. In the third sentence, the authors want to relate the two processes to each other and replace both processes with noun groups that refer to them: “external force” and “internal energy transfer”.

Used inappropriately (outside of the given-new structure), the use of nominalization can result in grammar that can be confusing. For example, the following sentence was found in the same textbook: “The vibration amplitudes of the atoms and electrons

of one grammatical class for another. 34

of the metal at the ﬁre end of the poker become relatively large because of the high temperature of their environment.” [54] (p.443).

The above sentence illustrates the three most important possible problems due to the use of extremely large noun groups. According to Halliday, in this sentence we can ﬁnd examples of (1) lexical density, (2) lexical ambiguity and (3) lexical discontinuity.

1. Lexical density refers to the number of technical terms (with potentially specialized deﬁnitions) per sentence. When using a noun group, this density invariably rises. For example, “. . . vibration amplitudes of the atoms and electrons of the metal at the ﬁre end of the poker. . . ” contains seven words that may have specialized meanings and implications, which the reader may or may not be aware of. Halliday estimates an average lexical density of around four per sentence in regular speech and writing [53].

2. Lexical ambiguity is a rather poorly named term that refers to ambiguity in the causal structure of the argument resulting from the grammar of the sentence. The separation of the sentence into “[a large noun group] becomes relatively large [verb group] because of [another large noun group]” contains a number of ambiguities for a naïve reader. For example, what is the environment of what? Does the environment comprise of the surrounding atoms in the poker or the ﬁre itself? Is “the temperature” the cause or “their environment”?

3. An extension of lexical ambiguity is lexical discontinuity. Often when a noun group is substituted for a complicated process, the reader has to fill in missing details that are omitted by the substitution. These are details that only someone who is a domain specialist could fill in. For example, in referring to “. . . the high temperature of their environment.” as a cause, the authors omit the complex process by which faster moving air molecules and ions in the fire collide with atoms of the metal, thereby increasing the vibration amplitudes of the atoms and electrons in the metal. 35

2.6.2 Language and Students’ Difﬁculties: Perspective From Physi-

cists and Physics Education Researchers

There are many different ways of examining why language can cause difficulty. Sec- tion 2.6.1 provides a brief flavor of the issues as linguists view them. It also covers a fairly limited area of linguistics. Halliday and Martin are concerned mainly with relatively concrete issue to do with the odd grammatical features of scientific writing and how the specialized grammar increases the language processing load when used carelessly.

Physicists and physics education researchers have written a great deal about language and teaching physics. The first issue that physicists are concerned about is lexical precision. I use this term to refer to the desire to define terms as precisely as possible to avoid confusion and the effort to remove unnecessary figurative structures (for example, metaphors) from the language. Many argue that physics (and other scientific fields) employ a large vocabulary of words whose technical meanings and everyday meanings may be quite different. Students have to assimilate these numerous new technical meanings. If the technical meanings contradict everyday usage, the task can become extremely difficult or demanding. If physicists disagree on a definition for a term, this may cause more confusion.

Linguistics is evolving beyond the simple lexical view of meaning. That is, the meaning of a term is more than an association of a term with a dictionary deﬁnition. Meaning is also denoted by how a word functions grammatically in a sentence and also the type of metaphorical images that are created around that word. For example, the word “force” acquires meaning based on its grammatical function in the sentence and based on its place in schemes of metaphors. Physicists associate special meanings with the word “force” yet at the same time their usage of the word does not always reﬂect these technical meanings. Nor are these technical meanings often made explicit amongst the physicists themselves. 36

Linguists would argue that it is impossible to remove all vagueness from language. Likewise, language cannot have every ﬁgurative form expunged from it. Thus I want to move beyond lexical precision to investigate what images are present in the language that physicists use and also how lexical meaning interacts with grammatical function to create meaning, both literal and ﬁgurative.

Physicists and physics education researchers have considered all three aspects of meaning ((1) lexical precision, (2) imagery, and (3) grammatical function) in their writing. I will cover a selection of this in the section below to illustrate the current thinking on the subject of language.

Most has been said about lexical precision and the vagueness of ﬁgurative language. For example:

• In a 1994 American Journal of Physics paper Lillian McDermott et al. suggest that students may have linguistic difﬁculties with the concept of “tension in a string”. They write:

“However, the term tension usually appears in the problem statement and often evokes confusion in the minds of students. Most have only a vague, undifferentiated sense of tension as both internal and external to the string. . . Regardless of the care taken in deﬁning tension properly, conceptual difﬁculties that have a strong linguistic element still remain.” [55]

They argue that “In daily life, the word is used to convey an idea of ‘tautness’ in a string.” Their issue is with the confusion between the colloquial meaning associated with tension and the specialized usage in physics. The authors are not so focussed on the image of a container suggested by the use of the preposition “in”, but they may be conveying some discomfort with this way of speaking when they talk about the “vague, undifferentiated sense of tension as both internal and external to the string.”

• Arnold Arons presents a similar view when he says:

“Many presentations start in by ignoring the fact that the words ‘force’ and ‘mass’, which, in everyday speech, are heavily loaded metaphors, are 37

being taken out of everyday context and given a very sophisticated technical meaning, completely unfamiliar to the learner. . . Students have, in general, not been made self-conscious about, or sensitive to, such semantic shifts, and they continue to endow the terms with the diffuse metaphorical meanings previously absorbed or encountered.” [11] (p.57-58)

• H. Thomas Williams [16] is concerned with physics getting its “linguistic house in order.” His stance on the matter is best summarised with the following quotation from his paper:

“Physics is often called an ‘exact science,’ and for good reason. At our best, we are precise in our measurements, equations, and claims. We do not seem to be at our best, however, when we write and talk about physics to introductory students. Language usage which presents few problems when used among ourselves because of shared assumptions, is potentially misleading or uninformative when used with the uninitiated. This problem can be solved by a self-conscious striving for precision in language.”

In his survey of textbooks he identiﬁes three types of problem with the language in those textbooks:

“(1) a common word used with a specific definition which differs from that of its everyday use; (2) a technical word or phrase defined differently from author to author, or in different contexts; or (3) a principle or important notion which is defined imprecisely.”

Williams focuses on semantics and lexicography. He argues that a lot of confusion would be alleviated if physicists agreed on a precise set of definitions for terms. He suggests that part of physics must involve some engagement in learning a new vocabulary. The difficulty in learning this vocabulary is compounded by the fact that many of the terms have everyday meanings which are quite different from their scientific meanings, physicists disagree fundamentally on how to define things, and sometimes definitions are vague and/or tautological.

Many physicists are aware that the use of language in physics goes beyond a set of precise deﬁnitions. Physicists and physics education researchers have already started 38

talking about imagery and grammatical function as I will show in the following examples:

• Arnold Arons beautifully summarizes the view of language as more than dictionary deﬁnitions when he says:

“Few students, even at college level, have had direct experience, making them self-conscious about examining how words acquire meaning through shared experience. They tend to think that words are deﬁned by synonyms found in a dictionary and, when it comes to concepts such as velocity and acceleration or force and mass, are completely unaware of the necessity of describing the actions and operations on executes, at least in principle, to give these terms scientiﬁc meaning. Since the words, to begin with, are metaphors, drawn from everyday speech, the students remain unaware of the alteration unless it is pointed to explicitly many times—not just once.” [11] (p.18)

• John McClain suggest that students’ view force as a property of an object rather than an interaction between two objects may be encouraged by certain images invoked in physicists’ language about force:

“It is unfortunate that attempts to correct this misconception [students’ belief that force is a property of an object] are thwarted by the phraseology used by some textbooks, where one may read “force of bat on ball” or even “bat’s force on ball,” expressions that imply that the force in some sense belongs to the bat.” [14]

Arons echos the exact same sentiment when he says:

“‘Force’ is interpreted by many students as something given to, being a property of, or resident in a moving body or one being accelerated. (How much is this reinforced by our tendency to talk about forces ‘imparted’ to a body? I myself ﬁnd the latter locution difﬁcult to avoid.)” [11] (p.74)

• Jerold Touger criticises textbook sentences that suggest a force is the doer of the action or the agent. Two examples he gives are sentences with the locution: “the force acts. . . ” or “the force pulls. . . ”. 39

“. . . from its use as a noun subject, a beginning student might infer that force is a concrete noun (a thing or a person), and thus an agent, that is, the doer of the action rather than the action itself. The student, I am suggesting, may be drawing misleading inferences about the meaning of the language from the form that the language takes.” [20]

This is one of few examples where a physics education researcher has explicitly addressed the connection between grammatical function (using force as noun subject) and imagery (force could be interpretted as “the doer of the action rather than the action itself.”)

When it comes to language about heat, work and thermodynamic processes, all three issues (lexical precision, metaphor, and grammatical function) are all addressed in the literature. There is consensus in the literature that physicists’ language can be misleading, but because the issues of metaphor and grammatical function are only addressed on an intuitive level and the authors often contradict each other.

• On the issue of lexical precision Mark Zemansky emphasizes that “heat and work are methods of energy transfer” [17], and he wants them deﬁned as such. He has also considered metaphorical issues and grammar: He is content to accept using heat as a noun as it refers to heat entering a system but explains that you cannot talk about “heat in a body”. The following quotation explains his view:

“When water flows, the hydraulic engineer refers to a current or, more specifically, a ‘water current’. Similarly, when electric charge flows, one refers to an electric current, and when heat flows, one alludes to a heat current. There is, however, a fundamental difference: both water and electricity are matter, which (neglecting relativistic effects) is conserved. Heat, however is not conserved. When water an electricity enter a system, they don’t disappear. When heat enters a system, it has no existence within the system as ‘heat’.”

Zemansky cautions against using the term heat as a verb because “heating a system” could be confused with raising its temperature. Finally, Zemansky cautions against using the term “thermal energy” since it seems to be confused between 40

(non-existent) “heat energy” and the “internal energy” of a system (an issue of lexical precision).

• Robert Bauman [12] is mainly concerned with lexical precision. He explains that physicists’ language “differs very little from the vernacular.” He blames students’ confusion on “the confusion in everyday language,” and makes the following suggestions for careful language use in a physics course:

“a. The property read by a thermometer will be called temperature. b. Energy within a body, responsible for the temperature, will be called thermal energy. c. Transfer of energy, as a consequence of temperature difference, will be called thermal energy transfer... Heat will be retained as a generic term, useful when it is not necessary to distinguish between meanings or processes. For example, the verb heat will loosely designate either the process of adding energy (as Q, as W , or as an arbitrary mixture) or the process of increasing the temperature. . . ”

Note that Bauman seems to be making the mistake that Zemansky cautioned against, namely confusing “thermal energy” with internal energy in one sentence, and with heat in the next sentence.

• On grammar, Ralph Baierlein cautions against using heat as a noun and happy to use it as a verb [56]:

“You may have noticed that the word ‘heat’ was never used as a noun. There was plenty of talk about heating things, but never once was ‘heat added’ or other such locutions used. Students find it difficult to distinguish correctly between the nouns ‘(internal) energy’ and heat. The former is a state function, but the latter is not. I find it simplest to avoid the distinction by using the word heat only as an adjective and in its various verbal forms.”

• Harvey Leff [13] is concerned with imagery that may be inferred from the way physicists talk about heat:

“The point is that the transfer of an entity implies movement of that entity from one storage region to another. . . We conclude that because heat cannot be stored, then the term heat transfer is an oxymoron. . . In particular, one must emphasize and re-emphasize that heat is not a stored energy, in contrast with its former use within the framework of the long defunct caloric theory.” 41

He is however content to use heat as a noun as illustrated in the following sentence from the same paper:

“We have shown how entropy can be a helpful tool for assessing the sign and magnitude of the heat during a reversible process.”

• Finally, Robert Romer connects the Baierlein’s recommendations about grammar with Leff’s awareness of the imagery implied by "heat transfer" and suggests that, because using “heat” as a noun suggests that it is a substance, “heat” as a noun eradicated from the language [15]. Reading between the lines its seems he might feel more comfortable using heat as a verb, but seems ready to discard the term altogether in the light of Zemansky’s objections to using heat as a verb.

2.7 Research on Students’ Cognition and Difﬁculties

Ultimately my thesis is about education and students learning physics. I want to explore the interaction between student learning and language in physics. Thus one major goal is to develop a theoretical framework about the role of language and then use this framework to reexamine previous ideas about how students think and examples of students’ difﬁculties that have already been identiﬁed by researchers.

2.7.1 Robust Misconceptions

A large amount of research in the ﬁeld of physics education has been done on students’ “misconceptions”. Physicists and physics education researchers have suggested that students. . .

• have what has been termed:

– “alternative conceptual frameworks” [57],

– “intuitive beliefs” [58],

– “misconceptions” [59], 42

– “naïve beliefs” [59], and

– “naïve theories” [59].

• These have been described as, or found experimentally to be

– “coherent [self-consistent] theories” [58],

– “difﬁcult to modify” [58],

– “tended to persist through an instructional sequence” [57],

– “highly robust” [9], and

– “obstinate” [60].

• They have been postulated to

– come from “their [students’] experience with the physical world” [59],

– be “derived from extensive personal experience" [61], or

– be “based on years of experience with moving objects. . . ” [62]

In my thesis I will re-examine some of this research. I will show that considering the role of language can provide an alternative explanation for the source of some of students’ difficulties. Likewise, persistent use of confusing language may be able to account for why these difficulties are so robust and resistant to instruction. I will also hypothesize that in some cases student difficulties are in fact part of a process of making sense of language.

2.7.2 Knowledge in Pieces

The ideas about robust misconceptions have been challenged by Jim Minstrell, An- drea diSessa, David Hammer, Andrew Elby and others [63, 64, 65, 66]. They suggest that student knowledge comes in pieces and is only locally coherent. Researchers have found evidence that students’ application of knowledge is highly context dependent [67] and cannot be portrayed as a robust theoretical framework. Hammer and 43

Elby have suggested that students use epistemological beliefs and/or epistemological resources when reasoning about physics problems and that this usage is context dependent. In their view, students difﬁculties stem from the lack of global coherence of their physics knowledge and from bringing ineffective epistemological resources to speciﬁc situations.

I will focus mainly on the view of students’ difﬁculties as robust misconceptions. However, I will show that a linguistic approach allows a possible uniﬁcation of certain aspects of the two competing views (misconceptions versus knowledge in pieces).

2.7.3 Conceptual Change

Much research has been done on how students change their ideas. (See, for example, [8, 9]). It is suggested that learning physics involves some rearrangement/reﬁnement of existing concepts or maybe the elimination of old concepts and the addition of new concepts in the mind of the student. Most of this research is irrelevant to the topic of my dissertation, thus I will not provide an extensive review of this literature.

There is, however, one theory of conceptual change that I want to use in my thesis. Michelene Chi and others have noticed a coherent pattern of incorrect ontology in students’ difﬁculties [2,5,68]. They hypothesize that physics concepts can be arranged into an ontological tree of matter, processes, and mental states 3 (see Fig. 2.2 below).

On the basis of experimental evidence (interviews and textual analysis) they suggest that students misclassify physical concepts into the incorrect ontological category. Students’ difficulties are related to this misclassification. For example, a student might classify “heat” as matter, while a physicist might classify it as a process. This misclassification is the source of their incorrect reasoning. Chi et al. say that conceptual

3When I talk about the ontology of an idea, concept, or model, I am referring to its classiﬁcation into one of the three categories, either matter, processes, or mental states. See Appendix A for a deﬁnition of ontology. 44

Figure 2.2: Chi et al.’s ontology tree that separates all “entities” into matter, processes and mental states (reproduced from [5]). change is a process of ontological recategorization. A student must move the concept of “heat” from the matter category into the process category in order to really understand it and use it effectively to reason.

2.8 Summary

In this chapter I have discussed literature related to analogy and metaphor and also literature on how the two might be connected. The research on how humans categorize their world provides a possible mechanism behind students’ potential confusion with language. It is possible that students mis-categorize ideas based on forming inappropriate ad hoc categories. In Chapter 3 I will show how this idea of mis-categorization can be connected to Chi et al.’s idea that students categorize physical concepts into inappropriate ontological classes. The key to making this connection lies in the grammar that I reviewed in Section 2.5. I have also shown that physicists and physics education researchers have sophisticated intuitive ideas about the role language is playing 45

in learning and understanding physics. In subsequent chapters I will try to show how their ideas can be formalized and systematized within the theoretical framework that I will develop. 46

Chapter 3 Hypotheses and Methodology

3.1 Introduction

The goal of Chapter 3 is to present an overview of the research methodology (Sec- tion 3.2), the theoretical framework that emerged from the data (Section 3.3), and hypotheses that emerged from the theoretical framework (Section 3.4).

In Section 3.2 I will explain the methodology of developing a grounded theory from qualitative data. This method is based on a dialog between the data, the theoretical categories that emerged from the data, and the literature that was covered in Chapter 2.

In Section 3.3 I will describe the theoretical framework that emerged from my initial data collection. It is a framework that explains the emergent category of language as a representation of a physical model. This framework will link together analogies, metaphors, ontology, and grammar in the following way:

• Analogy and metaphor: The initial data show that many models that physicists speak or write about originate as analogies. However, when physicists speak or write, they refer to these analogical models by using systems of conceptual metaphors. They tend to say “X is Y,” rather than “X is like Y in certain respects.” Thus when physicists generate knowledge, they use analogies. But when the knowledge is already established, physicists use metaphorical language without worrying about the limits of the picture.

• Ontology and grammar: I will show that every physical model has an underlying ontology of matter, processes, and states. This means that every component 47

of a physical model can be classiﬁed into one of these ontological categories. This ontology is encoded in the grammar of each sentence that physicists speak or write.

• Grammar and metaphor: Finally, grammar can be used to identify many metaphors. I will show that the conceptual metaphors involving ontology and cause-effect relationships often appear in language as grammatical metaphors.

In Section 3.4, I will use the theoretical framework developed in Section 3.3, to make predictions about the reasoning of physics students in relation to the language that they hear and read. There are three major ideas covered in Section 3.4. These are:

1. Students are known to overextend and misapply analogies that they encounter. (See [26] for example.) I hypothesize that students overextend and misapply metaphorical systems that they read and hear in the physics classroom in a similar manner. These “overextensions” sometimes appear as “robust misconceptions”.

2. Chi et al. have observed that students have trouble classifying concepts into the correct ontological category [5]. This difficulty may be connected with their reasoning about physics problems. I hypothesize that students’ poor ontological classification is related to the underlying ontology of the grammar of physicists’ speech and writing. If this is true, we should be able to identify systematic connections between the ontology encoded in physicists’ discourse and the ontological classification of concepts by students. If students’ ability to categorize concepts correctly really matters, we should be able to observe direct evidence of it in their reasoning. We should be able to identify students using correct/incorrect ontology to justify their reasoning when solving physics problems. Their ontological categorization of concepts should correlate with their ability to solve the problem.

3. My third hypothesis is that students often struggle to deﬁne the meaning of terms in physics. These struggles are analogous to historical difﬁculties that physicists 48

had. I hypothesize that these difﬁculties are crucial for students to develop real understanding. Meaning refers both to lexical deﬁnition and ontological categorization. This struggle for meaning is displayed in certain questions that students ask. These questions have been regarded previously as examples of students’ misconceptions or naïve reasoning.

3.2 Methodology

3.2.1 Grounded Theory

The reader may have noticed that up to now there has been almost no mention of the chronological development of the ideas in my thesis. The only mention was brieﬂy in Section 1.2.1 and Section 1.2.2 in Chapter 1. In this section I will explain the chronological development of my thesis, loosely based on a methodological approach called “grounded theory” [69].

The best way to understand grounded theory is to compare it with “grand theory”. F~ Gm m For example, in physics, two equations, ~a = net and F~ = 1 2 rˆ, aim to account m R2 for all motion between two or more objects caused by gravitational interactions. Gen- eral relativity aims to both account for these two equations and also account for new phenomena that Newtonian gravitation does not explain. When violations of general relativity are found, it too may be replaced by a more complete theory of gravitational interactions. This is the progress of grand theory. As counter-examples are found the theory is rejected and each subsequent theory aims to account for these new phenomena while still being able to explain all of the old phenomena.

In the ﬁeld of human interactions, sociologists realized that grand theories were an unrealistic goal for their ﬁeld. The space of human interactions is too large and complex for a few grand theories that attempt explain everything. So they introduced grounded theory as an alternative. A grounded theory does not attempt to explain all phenomena, but only explain some smaller subset of the given phenomena. As long 49

as a theory is applicable to some general subset of sociological or psychological phenomena, it is considered legitimate. A theory is not rejected by one counterexample. Counterexamples merely serve to delimit the domain of applicability of the grounded theory. Such a theory is only discounted once the domain of applicability of the theory has been reduced to a negligibly small or essentially useless size.

Physicists also develop grounded theories. Models that apply to particular length or energy scales are particularly common in nuclear physics and in condensed matter physics. These models often involve emergent phenomena, analogous to the emergent categories of sociological grounded theory. Edward Redish has suggested that physics education researchers should aim to develop models of limited scope, applicable to particular “length scales” of human interaction and cognition [70].

The term “grounded” in grounded theory refers to the idea that a grounded theory should be based on data, not on intuitive speculation. While such a point may seem obvious to a physicist, many sociological and psychological theories were not consciously based on data, but only on intelligent guesses by the researchers involved. The resulting hypotheses were then tested against the available data, often with the danger that researchers would force the data to ﬁt the theory, rather than the other way around.

The grounded theory approach does not provide a speciﬁc recipe and set of techniques for generating a grounded theory. Rather, it speciﬁes a set of guidelines and a set of concurrent directions in which the research should progress over time:

1. The primary goal is to develop a substantive grounded theory. Substantive simply means real or related to a specific situation. Thus the research begins with a specific situation. The researcher identifies the players and their roles and takes data (usually interviews or in-vivo observations) in order to develop a theory to explain what is going on. Ultimately the researcher wants to progress to a formal grounded theory. The formal grounded theory is more general and abstracted from the specific (substantive) situation with which the research began. 50

2. The heart of the process of developing the grounded theory is the method of comparative analysis. This is summarized in Fig. 3.1 below. With note taking

Figure 3.1: Summary of the method of comparative analysis [6]

and summary memos, the researcher should try to make repeated comparisons between different data sets and between data and the emergent categories that are discovered. This process of comparison helps to deﬁne the relevant categories, their properties, and to deﬁne the category boundaries (domain of applicability). Relationships between categories should be hypothesized and tested with more comparisons.

3. Theoretical saturation occurs when no more categories or properties of categories can be identiﬁed.

4. Literature should only enter after the ﬁrst attempts at a substantive grounded theory. Literature has the same status as data, and is also emergent. This means that useful literature is identiﬁed by its relevance to the emerging categories and explanations. Literature provides more data and helps to formalize the substantive theory.

5. The process of reﬁnement and growth or decline of the formal grounded theory continues for as long as researchers see the theory as useful applicable or relevant. The grounded theory is a process rather than a ﬁnal product. 51

3.2.2 Stages in Developing a Grounded Theory

The chapters and sections of this thesis attempt to present a neatly packaged product, a reasonably complete formal grounded theory, that was the result of a multi-year dialog between the data, the developing theoretical framework, and the literature. In this section I will provide an overview of all the data I collected and the place of each in the chronological development of the theoretical framework and the hypotheses that followed. I will attempt to recreate an outline of the thought process so that the reader may see an overview of the data and also gain a feeling for the methodology of the grounded approach in action. In subsequent sections and chapters this chronology will be discarded in favor of presenting the more condensed and coherent ﬁnal result.

Initial data and emergent categories: Initial substantive grounded theory. The initial data for my thesis came from the following sources:

1. 1 - 1.5 hour long interviews with 5 physics professors. These interviews consisted of me asking the professors open-ended questions about how they understood or would explain various concepts in QM. (See Appendix B for a full list of questions.) These interviews generated about 7 hours of video, 40 pages of single-spaced transcript and a further 100 or more pages of notes and memos.

2. Original quantum mechanics papers from Born [71], Schrödinger [72] and Bohr [73], as well as an analysis from Goldstein [74] of how Schrödinger developed the wave equation. The paper by Born I translated from German to English myself in order to preserve as closely as possible the original grammatical and metaphorical structure of the German language.

3. Two homework study groups. The ﬁrst group consisted of four junior physics and engineering majors in their ﬁrst QM course. The second group consisted of two senior physics majors in their second QM course. In both cases, I obtained permission to video-tape the students while they worked on their QM homework 52

as a group. These sessions generated about 10 hours of video. Two particularly interesting half hour segments, one from each group, were transcribed and analyzed.

4. A selection of older and more modern, popular introductory quantum mechanics textbooks [75, 76, 77, 78, 79, 80] were analyzed.

5. In the initial stages I also read some of the literature (covered in Chapter 2) on analogy, including Nersessian’s cognitive historical analysis [29] and Sut- ton’s hypothesis about the progression of physicists’ language from analogy to metaphor [31, 32]. This early introduction of literature represents a deviation from the grounded theory approach.

I began my research with open questions about how physics educators could teach QM better. The initial substantive situation was learning and teaching QM. The initial players I identiﬁed were physicists and physics students. The initial data gathering consisted of interviews with physics professors, reading QM textbooks and original papers, and observations of students working on QM homework problems. The initial roles and categories emerged quickly from the initial comparative analysis. For example, I compared the way Schödinger and Born wrote about their ideas (developing the wave equation and the probabilistic interpretation of the wave function) with the way modern textbooks and physics professors wrote and spoke about the same ideas. This led me to deﬁne two separate categories of language used to express ideas in QM:

1. The first category of language was language used by the inventors of QM. They tended to use cautious and figurative language. Ideas were often expressed as figurative comparisons of the form “X is like Y in certain respects”. They made analogies explicit and cautioned against overextending or misinterpreting these analogies. The emergent unit of analysis in the first category was the entire text used to make an analogy explicit. 53

2. The second category that emerged was language used to communicate already established knowledge of QM. (Language used by physics professors and modern QM textbooks.) This language was characterized by statements of fact with little if any reference to the original analogies on which the ideas were based. Here the emergent unit of analysis was an individual sentence.

The roles of the physicists in each of these categories was different. In the ﬁrst category, physicists were engaged in inventing or learning about QM. In the second category, physicists were engaged in teaching QM.

In my observations of students working on their QM homework problems, I saw students engaged in learning about QM. This learning took on many different forms. Students tended to spend a lot of time on procedural tasks such as evaluating integrals or deciding what equation was applicable to a given situation. These events did not captivate me because they did not seem to bear much relation to what the physicists were doing. Occasionally however, students stopped calculating, and engaged in an activity that could loosely be described as sense-making. Of all the video I took, the total time spent on sense-making could be reduced to two half hour episodes, one from each group. What was interesting about these two episodes is that it appeared to me as if students understood the physical ideas, but they were confused about the language used to express the physical ideas. In comparing these observations with the observations of physicists, I realized that the common theme running through all the relevant data was language itself. Thus I decided that the emergent substantive theory would be focussed on language. The theory would attempt to explain how ideas in QM are expressed in language and how students interpret the language that they hear and read in the QM class.

Secondary data, formalizing and extending the substantive theory. The secondary data consisted of most of the literature reviewed in Chapter 2. This literature was used to help formalize the substantive theory. In comparing the initial categories and data with the literature, the formal categories of metaphor, grammar and ontology emerged. 54

The question of language meant that I had to apply and adapt linguistic ideas to understand how ideas are encoded in language and how humans decode language in order to understand what is being said or written. Approaches using both metaphors and grammar seemed appropriate. In this process I hypothesized a formal connection between metaphor and grammar. The connection was a pattern of ontology encoded in the grammar (see Section 3.3.2).

The next step in formalizing the substantive theory involved revisiting all of the data collected in the initial data collection. In particular I returned to the interviews with physics professors and the quantum mechanics textbooks and started to code the language they used, looking for patterns of consistent ontology and patterns of conceptual metaphors used to represent physical ideas. I also looked for evidence that students were overextending the conceptual metaphors they heard or read.

Next, I tried to extend the theory beyond QM to thermodynamics and Newtonian mechanics. Additional data came from three popular introductory college level physics textbooks [54, 81, 82]. I analyzed and coded about 2000 sentences in which the word “heat” or “force” occurred. I also drew on literature about the historical development of the ideas of heat in thermodynamics and force in mechanics [25, 33]. I used several more introductory college level physics textbooks [83,84,85] to broaden the data. The success of these historical and grammatical/metaphorical analyses helped convince me that my theory about language was generalizable to all areas of physics.

Ultimately I concluded that conceptual metaphor and grammar were not reducible to each other. Grammar and ontology seem to be more fundamental than conceptual metaphor, yet at the same time, a metaphorical analysis can illuminate things that a grammatical analysis cannot. Thus the ﬁnal formal methodological component of the theory retains both grammatical and metaphorical analyses.

Tertiary data: Testing the theory in relation to student learning. Previously I had suggested that students were overextending conceptual metaphors they encountered in their learning and that these difﬁculties could manifest themselves as misconceptions. 55

This seemed to be a productive way to frame some student difficulties in QM. In the final stage of building the formal theory, I extended this idea by suggesting that students categorize certain physics concepts into ontological categories, based on the underlying ontology of the language that they hear. This ontological classification drives their reasoning about the related phenomena. If the ontology encoded in the language conflicts with the model ontology, then the students will have specific predictable difficulties. I tested these ideas on the following data:

1. I conducted an interview study of ten students in the honors program at Rutgers who had completed at least two semesters of introductory physics and had gotten a B+ or better in both semesters. I interviewed them using questions about thermodynamics devised by David Meltzer [51]. The interviews lasted about 1 hour each and generated about 100 pages of transcript. The audio recording of one student was lost, so the interview had to be reconstructed from interview notes. I coded parts of these transcripts using the grammatical/ontological/metaphorical analysis I had developed.

2. In the ﬁeld of physics education research, a large amount of work has been devoted to students’ misconceptions. I decided to re-examine the misconceptions literature on thermodynamics, Newtonian mechanics and QM [11, 51,55, 57, 58, 59, 60, 61, 86, 87, 88, 3, 89, 90, 91, 92, 93, 94, 95]. I succeeded in reinterpreting a number of classic misconceptions through the linguistic framework that I had developed. Issues with the underlying ontology were central in many cases.

Summary: The conclusion of this multi-year interaction between the data and the developing theory is a formal grounded theory that will be described in Section 3.3 below. The major hypotheses that emerge from the theory will be covered in Section 3.4. In this section I have covered the methodology of the grounded theory approach and tried to present, in chronological order, how each data set contributed to the construction and formalization of the grounded theory. 56

3.3 Language as Representation

3.3.1 Analogical Models Encoded as Metaphors

Nersessian’s studies have shown, for example, that Maxwell’s notions of the electromagnetic ﬁeld were built on ﬂuid and mechanical analogies [29]. As physicists strive to remove the ambiguities, an analogical model is separated into a negative and positive analogy as the implications of the neutral analogy are explored [30]. These analogical models become, in time, encoded linguistically as conceptual metaphors [31, 32]. The way physicists talk about already established knowledge is different than the way they talk about new ideas they are trying to comprehend themselves.

I will take this idea further. From the primary data (textbooks, original papers, and interviews with physics professors), I have identiﬁed three types of analogical model that metaphors encode. I will classify the types of analogies by their origin and function. The three types of analogical models that metaphors encode are:

1. Current analogical models: For example, Schrödinger based his wave equation

on an analogy to wave optics [72]. The metaphorical system is THEELECTRON

IS A WAVE and is spoken about by modern physicists in terms such as “electron interference”, “electron diffraction”, “wave equation”, and so on.

2. Defunct analogical models: It is often the case in physics that older models, whose limitations have been experimentally exposed and supplanted by better models, live on in the language of physics. The ﬂuid model of electricity lives on

in the following phrases which reflect the ELECTRICITY IS A FLUID metaphor: “Electrical current flows though the circuit”, physicists talk about “sources and sinks of electric field”, “flux of electric field lines”, and so on.

The caloric theory of heat (encoded by the HEAT IS A FLUID metaphor) survives in phrases such as: “heat ﬂows from an object at a higher temperature to an object at a lower temperature” and “waste heat is discharged to the cold reservoir.” 57

These examples are not throwbacks to theories that are better forgotten. Physi- cists use these metaphorical pictures when they reason. I will elaborate this point further below.

3. Descriptive analogies: For example an analogy between a physical valley and

a potential energy graph. The metaphor is POTENTIALENERGYGRAPHSARE

WATER WELLS OR PHYSICAL VALLEYS. Examples of how the metaphor is used in language are: “potential well”, “potential step”, “energy level”, “ground state”, and so on.

Note how analogy types 1 and 2 are equivalent to Dunbar’s regional analogies that biologists used to build new theoretical models, and analogy type 3 is equivalent to Dunbar’s long-distance analogies, used by biologists to instruct each other. (See Chapter 2, Section 2.3.3.)

I will identify metaphors that encode analogies 1 to 3, by identifying the base of the analogy. It is important to identify the analogy on which the metaphorical system is based because the base domain of the analogy will serve as a starting point for identifying words that cue the metaphor. I will use the idea that conceptual metaphors borrow terms from the base of the analogy and apply these words directly to the target concept.

For example, the HEAT IS A FLUID metaphor borrows terms from the base domain of fluids. An example would be “heat flows. . . ” Another example: if we look at the matter – wave analogy we can consider that a water wave or an electro-magnetic wave is the prototypical example which will serve as the “base” of the analogy. Thus words such as “interfere”, “polarize”, “diffract”, and “wave” are used in the context of “an electron”. Such examples will be identified as instances of the ELECTRON IS A WAVE metaphor. Recognizing a metaphorical system is therefore contingent on identifying the base of the original analogy.

I hypothesize that physicists unconsciously prefer to speak and write in metaphors because metaphors have certain features and functions that are advantageous to them. 58

The features and functions of these metaphorical systems are listed below with examples from interview data with physics professors.

1. Feature: Conceptual metaphors encode analogies. They encode a more deep and complex piece of knowledge which is the completely elaborated analogy. That elaboration as an analogical model is, however, tacit amongst the community who use the metaphor and associated model regularly.

Function: I hypothesize conceptual metaphorical systems represent (to physicists) what diSessa would call primitive encodings of deeply connected knowledge structures [64]. Physicists are able to use these systems to reason productively about a particular situation or problem. These metaphors, which arise from analogical models, may label productive modes of reasoning: Namely, aspects of the positive analogy which may be usefully applied to a well understood group of situations and which may be extended to understand new (unfamiliar) problems.

For example, the ELECTRON IS A WAVE metaphor can be used productively to explain the Heisenberg uncertainty principle:

Prof. A: “I often think of it. . . in terms of Fourier transforms and the reciprocity between the bandwidth of the channel and the length of the signal pulse that can be detected.”

Note the use of words from the base domain of electromagnetic waves: “Fourier transforms”, “bandwith”, and “signal pulse” in particular.

Even defunct analogies (type 2) represent productive modes of thought for physicists. For example, there a class of problems for which it is quite adequate to talk

about HEAT AS A FLUID.

2. Feature: Metaphorical systems are partial in nature. This means that more than one metaphorical system is needed to fully understand a physical concept.

Function: I have observed that physicists switch easily and unconsciously between one system and another depending on the type of question that is asked. 59

For example, in the following extract Prof. D switches back and forth between particle and wave metaphor to describe the process of electrons passing through a Young’s double slit apparatus.

Prof. D: “Of course in any one experiment,. . . you will not observe. . . an interference pattern on the screen [wave metaphor] — if all you do is to scatter one electron [particle metaphor]. Okay? The intensities are just too low [wave metaphor]. And the fact is, you have to have a large number of electrons [particle metaphor], you have to have a beam of electrons [wave metaphor]. Okay? And, uh, each electron will contribute a little piece of the intensity that you see on that screen [particle metaphor]. Uh, so, um, what I envisage is, in fact, a beam of electrons which can be represented by a plane wave [wave metaphor]. . . ”

3. Feature: Metaphors involve the use of the verb “is” rather than “is like”. Metaphors are grammatically relational processes of identiﬁcation, i.e., they are grammatically equivalent to statements of category membership.

Function: I hypothesize that metaphor reflects a particular aspect of an expert physicist’s thought process. The use of metaphor itself rather than simile is significant. Irrespective of deep philosophical discussions about what is “real”, it seems apparent that physicists themselves need to assert something stronger than “like” — they need to assert “is” in their own reasoning process. I suggest that physicists’ need to assert/endow physical systems with an underlying reality in order to think about them, is a fundamental trait of how knowledge is generated in physics, and of human cognition. It says something about how we make sense of the world. It is significant because such assertions may often conceal the vague or partial nature of metaphor itself.

For example, Prof. D provided the following response to the question: What happens to a single electron when it passes through a Young’s double slit apparatus?

Prof D: “. . . [to] understand that experiment, you’ve got to forget about the idea that an electron is a particle. Okay? It is not a particle in that context, it behaves like a wave. So you just think of it as a plane wave 60

[my emphasis] advancing on the, uh, on the two slits, and the interference between the two. . . outgoing beams, just using Huygen’s principle, leads to the famous. . . interference pattern that is observed.”

Note that comparison, “it behaves like a wave,” is followed directly by, “just think of it as a plane wave.”

4. Feature: Using language in physics places great strain on the ability of any language to express ideas. Physicists need to come up with ways of speaking about abstract phenomena and processes. These ways of speaking are grounded in physical experience and are therefore metaphorical by necessity. In summary, the apparatus of language constrains the ways physicists can talk about physical phenomena and therefore constrains the types of models that can be represented in language.

Function: Descriptive analogies (type 3) encoded as metaphors also represent ways of speaking about/describing physical systems. This is very important because there is a limit on what can be represented with language. Such metaphors also give abstract concepts and quantities a grounding in physical reality and physical experience.

Consider for example, a modern physicist’s view of energy in classical physics. A physicist will say that energy is best thought of as a state function deﬁned over the coördinates and momenta of a system of particles. Yet can physicists speak literally about energy as a state function? I suggest that it is simply impossible to come up with grammatical constructions that convey the meaning of energy as a state function. The very best locutions are “energy ﬂows into the system”, or “process X caused the kinetic energy of the system to increase.” In both these cases, metaphorically, energy is begin spoken of as matter and the system as a container of energy. (This is suggested particularly by the use of the adverbs “into” and “of”.) It is no coincidence that these two locutions are identical to 61

examples given by Lakoff and Johnson. The authors describe similar metaphorical patterns in how humans (in English at least) encode physical processes and events as movements of substances into and out of containers.

Physicists are aware of the limitations of their language. In interviews two professors commented about the limitations of language itself. These comments were not prompted in any way by the interviewer. One example was given on page 9. Another comes from Prof. E:

Prof. E: “Well, the only way that we can convince ourselves of the truth of any quantum mechanical prediction is to cast some kind of experiment in classical language, because basically we are not capable of thinking in any other language.”

3.3.2 Ontological Underpinnings

A Lexical Ontology: I hypothesize that the concepts in a physical or analogical model can be arranged into an ontological tree similar to the one proposed by Chi et al. [5]. (See Fig. 2.2 in Chapter 2.) It is necessary to modify Chi et al.’s ontology tree to accommodate one missing category: namely physical states. Thus the mental state category is demoted to stand as a subsidiary member of a broader state category that can also include physical states. (See Fig. 3.2 below.)

Matter Processes States

Constraint- Mental Physical Living Non-living Procedure Event based State state interaction

Figure 3.2: A revised ontology tree

Eugenia Etkina et al. have suggested that physical models can be broken up into a taxonomy of (1) models of objects, (2) models of interactions between objects, (3) 62

models of systems of objects, and (4) models of processes that the objects/system undergoes. I will show (see Table 3.1) that this taxonomy can naturally be mapped to the ontology tree suggested in Fig. 3.2. Etkina et al. suggest the following [96]:

1. Any physical model involves models of localized objects such as point particles, rigid bodies, or unlocalized, unbounded objects such as a ﬁeld.

2. When considering more than one object, physicists describe models of interactions between objects, either qualitatively (“collision”), or quantitatively (e.g., Coulomb’s Law). The authors call laws that describe interactions interaction laws.

3. A set of one or more interacting (or non-interacting) objects make up a system, e.g., an ideal gas, or an ideal ﬂuid.

4. Due to interactions between objects in a system or from external interactions, the system undergoes a process in time. Such processes can have a cause un- speciﬁed or speciﬁed. The authors refer to these as state laws and causal laws respectively. A state law does not include an interaction term, while a causal law does include an interaction term. For example, energy conservation is a state law, while Newton’s second law is a causal law.

In addition to their taxonomy, I am going to suggest that there are two classes of physical variables that describe a system or the objects in it. These are (1) physical properties of objects (such as mass and charge), and (2) state variables that describe a configuration of the system (e.g., position, momentum) or state functions defined over a system configuration (e.g., energy, entropy).

I will now show how Etkina et al.’s model taxonomy can be mapped into the ontology tree shown in Fig. 3.2. This mapping is shown in Table 3.1 below.

Some examples are given in Table 3.2 below: 63

Table 3.1: Table illustrating how Etkina’s model taxonomy ﬁts sucessfully maps into Chi’s (modiﬁed) ontology Ontological Matter Process State category Ontological Non-living Event Procedure and Physical sub- Constraint- State category based Interac- tion Taxonomy objects system interaction causal laws, state vari- element laws state laws ables, state functions

Table 3.2: Examples illustrating how Etkina’s model taxonomy ﬁts into Chi’s (modi- ﬁed) ontology Matter Process State Non-living Event Procedure and Physical Constraint- State based Interac- tion kq q d2x F electrostatics electron, one/many F = 1 2 = net positions, R2 dt2 m proton charged momenta objects Thermo- point ideal gas heat, work PV = nRT Energy, dynamics particles Entropy, Temperature, Pressure, Volume 2 d x Fnet Gm1m2 Dynamics point one/many F = 2 , = , positions, R dt2 m particles particles F = µN energy con- momenta, servation, energy momentum conservation

The only limitation of the ontological classiﬁcation is that it does not include physical properties such as mass and charge.

The categorization of concepts in physics into an ontology tree (as shown in Ta- ble 3.1), will be termed a lexical ontology. Physicists reach agreement about the meaning of terms, although, not without controversy. For example, physicists generally 64

agree that energy is a state function, while heat and work are processes by which energy is transfered into or out of a system. Thus a lexical ontology refers deﬁnitions of physics concepts into matter, processes, and states that physicists would agree with as a community.

Grammar and Ontology: Although physicists can agree on the meaning and ontological classiﬁcation of physics terms, how do they represent the ontology of these terms with language? I hypothesize that the key to answering this question lies in grammar.

Lakoff and Johnson have already suggested that there is a limited set of ontological and causative metaphors that underpin the way humans talk about processes. I want to take this idea further and suggest that every physical model described in language has an ontology and that this ontology is encoded in the grammar of the sentence. This grammatical ontology can be either literal or ﬁgurative (metaphorical). If the lexical ontology matches the grammatical ontology then the sentence is literal. If the lexical ontology does not match the grammatical ontology of the same term in a given sentence, then a grammatical metaphor is present. Ontological and causative metaphors form the building blocks of conceptual metaphorical systems. I suggest that these metaphors may be consistently identiﬁed by using the grammatical/ontological analysis described above. The details of this analysis will be elaborated below.

Consider for example, “John [agent] kicked [process] the ball [medium].” Here “John” and “the ball” are grammatical participants, functioning grammatically as objects or matter. We also recognize that “John” and “the ball” are naturally defined as matter in some sense. Thus the grammatical ontology and lexical ontology match. There is nothing metaphorical in this sentence. Consider now for example, “heat [medium] flows [process] from the environment to the gas.” In this sentence a physicist would recognize heat to define a process of movement of energy into the system (lexical ontology). But grammatically “heat” is functioning as a participant, namely heat is the matter that is flowing. In this case the grammatical function of the term “heat” 65

and the lexical ontology of “heat” contradict each other. The sentence is therefore metaphorical.

I am going to propose the following mapping from the grammar of material processes to the ontology tree shown in Fig. 3.2: Grammatical participants should be mapped into the ontological category of matter. Participants can immediately be separated into living and non-living ontological subcategories: Beneﬁciary, agent, and medium (as it participates in an action process, such as “a force [medium] acts [process]”), can all be thought of as living entities. Range and medium (as it participates passively in an event process such as “heat [medium] ﬂows [process]”) can be thought of as non-living entities.

Certain circumstantial elements can also be mapped to the matter category. The unambiguous one is role. Role describes a role played by one of the participants and can be spotted by the preposition “as”. For example: “Energy [medium] flows [process] into the system [location] as heat [role]”. This locution fits into the “heat as a form of energy” language and so in this case heat should be classified as non-living matter.

The more ambiguous case is the circumstance of location. In the previous example, “Energy [medium] ﬂows [process] into the system [location],” “the system” could be classiﬁed as non-living matter. However location also functions grammatically to make ontological physical states as in “A particle [medium] is located [process] at coördinates (1,1,1) [location].” This will be discussed further below.

The other important type of grammatical process in the discourse of physics is the relational process. Relational processes are processes of being in that they almost always include some form of the verb “to be”. Relational processes have two modes: identifying and attributional. These two modes can appear in a number of contexts. The ﬁrst is in denoting existence. In this situation the attributional mode denotes category membership. For example: “An electron is a lepton” indicates that an electron is a member of the category “lepton”. This is generally recognized by the irreversibility of the sentence. It does not make sense to say “a lepton is an electron”. On the other 66

hand, in the identifying mode the relationship is more akin to a one-to-one congruency, for example, “the neutrino is the lightest known particle.” It makes complete sense to say “the lightest known particle is the neutrino”. The identifying mode can be spotted because the relationship is reﬂexive.

The grammatical apparatus of English used to express physical states is extremely limited. I hypothesize that physical states (as expressed in language) are commonly comprised of identifying relational processes where the second identiﬁer is missing and replaced by a grammatical circumstance of location. Typical examples are: “The electron is in the ground state”, “the particle is located at such and such coördinates.” Such sentences very often involve a grammatical metaphor. Ontologically location is mapped to some sort of physical object or matter, this often conﬂicts with the lexical ontology. These grammatical metaphors correspond directly to the ontological metaphors of Lakoff and Johnson in choice of preposition: “in” implies container, “at” implies point location in either time or space, “on” implies surface. I believe that it is also no coincidence that these statements have a grammatical structure identical to those of mental states. For example, in English we say, “I am in love,” or “I am in trouble”, or “I am in a state of confusion” etc. It seems to me that, unintentionally, physicists have borrowed this metaphor wholesale and blended it with the notion of a physical state, to create a way of speaking about physical states. There simply is no better apparatus in our language!

Ontological processes are realized in speech and writing by grammatical processes. Processes are always denoted by a verb or verb group. In describing the behavior of physical model, I am going to stick to grammatical material processes. Relational processes realize either physical states as shown above, or denote some component of the model in the sense of category membership. In grammar there are two types of material process: action and event. It has already been shown how these two types of process can be used to distinguish between living and non-living matter. If the medium of the sentence is involved in an action process the medium is classiﬁed as living matter. 67

(For example: “A force [medium] of 50 N acts on the box.”) If the the medium of the sentence is involved in an event process the medium is classiﬁed as non-living matter. (For example: “Heat [medium] ﬂows from the hot reservoir into the system.”

Finally one particular grammatical circumstance functions to elaborate something about the grammatical process. This circumstance is called manner and may be recognized by the preposition “by”. For example: “Energy ﬂowed into the system by heat [manner].” Note that here “heat” should not be confused with the grammatical agent. The previous example is NOT the passive tense of “Heat ﬂowed the energy into the system.” This reordering simply does not make sense. In the example here, “heat” functions to describe the process and therefore I will classify such examples of the word heat in the ontological category of processes

The entire mapping from grammar to ontological category is summarized in Ta- ble 3.3 below.

Table 3.3: Summary of the mapping between grammar and ontological category Grammatical function Ontological category (If X functions grammatically as. . . ) −→ (. . . classify X ontologically as. . . ) Agent −→ Matter:living Beneﬁciary −→ Matter:living Medium (action process) −→ Matter:living Medium (event process) −→ Matter:non-living Role −→ Matter:non-living Objects in Location −→ Matter:non-living Process −→ Process Manner −→ Process

Example: In Chapter 5 I will systematically analyze the language physicists use to talk about “heat” in thermodynamic processes. When mapping the grammatical usage of the word “heat” using the ontological mapping shown above, a signiﬁcant portion of the time (greater than 80%), physicists talk about heat grammatically as if it were matter. The contrast between the lexical ontology of physics concepts and their grammatical function in language is shown in Table 3.4 below. I will discuss the implications of this further in Section 3.4.2. 68

Table 3.4: Comparison of lexical and grammatical ontology of thermodynamics Matter Process State Non-living Event Procedure and Physical Constraint- State based Interac- tion Deﬁnitions point many heat, work PV = nRT , Energy, agreed particle point ∆U = Q + W Entropy, to by particles Temperature, physicists Pressure, Volume Gram- point many ﬂows, is matical particle, point transferred, usage heat, particles is dis- based on energy charged common etc. usage of terms in language

3.3.3 Summary

Fig. 3.3 summarizes the theoretical framework developed above. It shows how analogy, metaphor, ontology and grammar ﬁt together.

Analogical encoded in Metaphor Model language as... underlying...

has an

lexicon... If grammar conflicts with

encoded in Ontology Grammar language by...

Figure 3.3: Summary of the role of analogy, metaphor, ontology and grammar

• Many physical models begin as analogical models. 69

• As an analogy becomes established, physicists speak about the analogy using conceptual metaphors.

• Every analogical model has components that can be classiﬁed into a basic ontology tree of matter, processes, and states.

• This ontological classiﬁcation of the analogical model is encoded in the grammar of each sentence that physicists use to speak about a particular physical phenomenon.

• Often the ontology encoded in the grammar may conﬂict with the agreed lexical ontology of a physical model in its modern form. When this happens, a grammatical metaphor is present. These grammatical metaphors underpin the conceptual metaphors that are used to describe a particular analogical model.

3.4 Student Difﬁculties, Student Learning

The central broad question of my thesis is: What is the interplay between the linguistic representations that physicists use and students’ learning and students’ difﬁculties? I will narrow this down to three hypotheses regarding the role of language and learning in physics. These are elaborated in Sections 3.4.1, 3.4.2, and 3.4.3 below.

3.4.1 Student Difﬁculties Interpreting Metaphors

Students struggle to see the applicability and limitations of analogies that they encounter. I suggest the same applies to metaphorical language that they hear and read. These difficulties may manifest themselves as “misconceptions” or student difficulties. Thus some “misconceptions” or difficulties may not be entirely based on physical experience as is commonly suggested in the literature, but also on difficulties stemming from the language that physicists use. I predict that students will overextend and misapply key aspects of metaphorical systems in physics. Instances where metaphors are 70

overextended or taken too literally will be connected with their faulty reasoning.

If this idea is correct I should also be able to predict new “misconceptions” or student difﬁculties that have not been identiﬁed in the literature. I will present an example in Chapter 4.

In order to test these ideas it is ﬁrst necessary to identify if there are really coherent systems of conceptual metaphors in the way physicists speak and write. In Chapter 4 I will show some of the interview data with physics professors that lead me to see that this view of language was really applicable to the discourse of physics. Looking for metaphorical systems is an interesting exercise in itself. It may be possible to identify systems of metaphors of which the speakers themselves are not really conscious. These are generally metaphorical systems encoding descriptive analogies (type 3).

The data in Chapter 4 were obtained in the context of quantum mechanics (QM). QM is relatively free of physical intuition that could come from physical experience of the world. Thus, if “robust misconceptions” exist in QM, they cannot be accounted for as naïve beliefs based in physical experience. QM represents an ideal starting point for testing the hypothesis of metaphorical overextension.

I have also hypothesized that metaphors are based on analogical models. In each case, metaphorical language that works itself into the discourse of physics must come from an analogical model that was made explicit at some point in the history of the development of the field. I will show an example in Chapter 4 where I first identified a metaphorical system in the modern language of QM. This encouraged me then to search the historical papers on QM to find the original analogy upon which the metaphorical system was based.

I will apply the same hypothesis (conceptual metaphors are based on historical analogies) in reverse in Chapter 5. The imagery contained in modern physicists’ language about “force” is so convoluted that it is almost impossible to completely disentangle it without the guidance of history. In this case I will show how, with the help of an analysis of the historical analogies, I was able to disentangle the language and 71

identify metaphorical systems associated with “force”.

3.4.2 Student’s Ontological Confusion

Chi et al. have suggested that many student “misconceptions” can be explained by students’ incorrect ontological classiﬁcation of concepts. For example heat is a process, but students reason with it as if it were matter.

1. In Chapter 5 I will use the ontological coding scheme developed in Section 3.3.2 to analyze physicists’ language about “heat” in thermodynamics and “force” in Newtonian mechanics. I will then show how it is possible to identify the conceptual metaphors in the language using grammatical ontology. Physics textbooks will provide a suitable sample of how physicists talk. I will show that physicists talk about “heat” and “force” in surprisingly consistent ways, both at the level of ontology and at the level of elaborated metaphors.

2. I hypothesize that students are categorizing the concepts of “heat” and “force” into incorrect ontological categories based on the language used to describe them. I will analyze interviews with students solving thermodynamics problems. For example, David Meltzer has observed that many physics students have difficulty solving thermodynamics problems involving heating processes [51]. If language and ontology really matter, then there should be a connection between students’ ability to correctly define “heat” as a process and their ability to solve thermodynamics problems. I predict that students who have difficulties will invoke the idea of “heat” as matter when justifying their reasoning.

In Chapter 5 I will also re-examine some of the classic misconceptions in New- tonian mechanics. I will investigate if there is a pattern of incorrect ontology in students’ reasoning that may be traced directly back to the language used by physicists. If this is the case, I may be able to account for certain classic misconceptions in terms of students’ difﬁculties with language rather than naïve beliefs 72

based on physical experience. The robustness of the “misconceptions” may be explained as well. If the language used to describe certain concepts is ubiquitous, then it is possible that that language serves to simply reinforce the incorrect ontological classiﬁcation.

3. My idea combined with the Sapir-Whorf hypothesis1 suggests that if students with a different native language that has a completely different underlying ontological structure are allowed to develop a language about thermodynamics, their models will be completely different from ours. They may have a completely different set of misconceptions and many of the misconceptions our students display may not appear in their reasoning. Likewise a suitably clever choice of a different language to talk about physics concepts may have a signiﬁcant effect on the types of reasoning that students display. I will analyze two papers, concerning the use of alternative forms of language, through the lens of the linguistic framework I have developed.

3.4.3 Students’ Ontological Groping

In the historical analysis of the development of certain concepts I observed that physicists spend a lot of time trying to figure out how to “talk” about the concepts and what the terms they use really mean physically. In Chapter 5, I will introduce the term ontological groping to describe this observed behavior. I will then extend Nersessian’s continuum hypothesis into this area of language. The continuum hypothesis is the idea that cognition in physics is a natural extension of everyday cognition. This led me to suggest that an analogy could be made between physicists’ and students’ difficulties with language. I will show that some students’ difficulties are in fact an analogous struggle with ontology and terminology (ontological groping) similar to the historical

1The Sapir-Whorf hypothesis suggests that humans interpret their world through the language they use to speak about it. 73

struggles I observed. In Chapter 6 I will extend the idea of ontological groping to understand Bohr’s principle of complementarity.

3.5 Summary

Fig. 3.4 is a diagrammatic summary of the ﬁrst three chapters of my thesis. It is a visual representation of how all the ideas discussed so far ﬁt together. In summary:

Something familiar: Existing physical models, physical experience, sense perceptions ontological frameworks etc... Serve as base for

Physicists good at Physicists engage in Physicist's exploring limitations Characterized by Characterized by protracted ontological analogy and applicability groping Physicists' practice informs what students should do Can be used to understand students' dif Represented by

Equations, Language Graphs, (Metaphors, Diagrams Grammar)

Linguistic representation leads to/is a part of/interacts with... behave like physicists do

Student difficulties and ficulties student learning Students DO NOT

I hypothesize three issues arising from or embedded in language Students are analogically similar to physicsts

H1: Students' difficulties H2: Students' H3: Student's ontological interpreting metaphors: They ontological confusion: groping: This may help us struggle with the applicability Causal and ontological understand certain and limitations of linguistic confusion may arise from difficulties that students representations of a model the language we use have.

Figure 3.4: A summary of my thesis

• Ideas and models in physics are often based on analogical reasoning. 74

• As models are formed, the applicability and limitations of the analogies are explored through experiment. This also involves physicists struggling to understand the meaning of what they observe and struggling with ways of speaking about their ideas.

• Once a physical model becomes an accepted part of the knowledge of physics, it becomes encoded in language. This encoding is captured in grammar and metaphor.

• The nature of linguistic representations lead to a number of difficulties for physics students. Students’ difficulties with linguistic representations are explored in three hypotheses: (1) Students overextend metaphors, (2) students’ difficulty in classifying concepts into the correct ontological category is related to the language they hear and also affects their reasoning processes, and (3) students display ontological groping very similar to difficulties that physicists have trying to find ways express their ideas in language. 75

Chapter 4

Analogies, Metaphors, and Students’ Difﬁculties in Quantum Mechanics

4.1 Introduction

In the previous chapter, I laid out hypotheses about how physicists encode their ideas in language and how students interact with that language. Several examples from interviews with physics professors were provided to illustrate how I came to see physicists’ language as metaphorical.

The importance of metaphor in physicists’ speech, and how students interact with this metaphorical language, will form the basis of this chapter. Rather than try to cover all of the data I collected, I will trace two metaphorical systems in QM, from their origins as analogies through to modern language that physicists use to speak and write about these ideas. These two systems are (1) the POTENTIALWELL metaphor, and (2) the BOHMIAN metaphor. For each of the two systems I will consider a case study of students struggling with the same metaphors.

In this chapter I will show that the two metaphorical systems have explicit analogical origins. The evidence for the original analogies will come from Rudolph Peierls’ autobiography [97], and from a paper by Max Born [71].

It is most important to analyze the modern language of physicists. This analysis will include both an analysis of grammar and a metaphorical analysis. These analyses will together serve to illustrate a number of claims made in Chapter 3. The claims are: 76

1. Coherent and systematic systems of metaphors exist in the language of physicists.

2. The language encodes a representation or representations of a physical model that has an underlying ontology of matter, processes and states.

3. Physicists use these linguistic representations to reason productively about certain phenomena.

4. Identifying the models encoded in the language will give us an idea of the types of possible difﬁculties that students might have in interpreting this language.

The data for the linguistic analysis will come from two sources. The ﬁrst is the interview study with physics professors referred to in Chapter 3. I interviewed each physics professor individually for about one hour to one and a half hours. I asked them to describe and explain various ideas in QM. For example, I asked each professor to explain the Heisenberg uncertainty principle. In another question, I asked them how they would respond to a student who asked, “what is oscillating in a QM wave?” The interview study consisted of ﬁve subjects. The full set of interview questions may be found in Appendix B. The second source of data is a selection of QM textbooks [75, 76, 77, 78, 79, 80].

Next, I want to illustrate how the linguistic framework may be applied to understand students’ difficulties. During initial data gathering I observed two student homework study groups. The first study group was comprised of four junior physics and engineering majors in their first QM course. The second study group consisted of two senior physics majors in their second QM course. Each group consented to be videotaped while working on their QM homework problems. In Chapter 3 I mentioned two half hour episodes of student sense-making that were pivotal in the development of the theoretical framework. In this chapter, I will present an analysis of each episode. I hope to show how the difficulties that the students displayed in these episodes may be 77

interpreted as difﬁculties with language. In particular, difﬁculties with the POTENTIAL

WELL metaphor and with the BOHMIAN metaphor.

Finally, there is very little literature in the field of physics education research related to student’s difficulties in QM. However, I will present an example of a common student difficulty that may be accounted for by the linguistic framework I have developed. The evidence and data for analysis will come from two papers and one Ph.D thesis [93, 94, 95].

4.2 The POTENTIAL WELL Metaphor

4.2.1 Original Descriptive Analogy “Because of the Pauli exclusion principle, the electrons must be spread over the available states; but they settle down to the states of lowest energy, so that as more electrons are added, the energy levels in the band ﬁll up like a bucket ﬁlls with water.” [97]

In this example Peierls makes the analogy explicit. The way he uses it shows that this analogy has a descriptive role.

4.2.2 Analysis of Modern Language

Grammatical and ontological analysis: When physicists speak of “potential well” and “energy level”, they give energy an existence as water. When physicists speak about quantum particles “leaking through a barrier”, they give the quantum particles an existence as water. When physicists speak of a “potential well”, “potential step”, “potential barrier”, “conﬁnement”, “trap” etc. . . they give the potential energy graph an existence as a physical object.

This ontology is encoded in the grammar. There is however a problem with analyzing physicists’ writing. In many upper level physics textbooks, sentences are comprised of large nominal groups with processes left implicit. Thus to identify the objects 78

and processes in the POTENTIALWELL metaphor, I need to “denominalize” the nominal groups. For example, a complex sentence such as “Potentials like the rectangular barrier or the square well, which are pieced together from constant zero-force sections with sharp discontinuities, do not occur in nature but serve as convenient models.” [78], may be rewritten with fewer nominal groups as: “Potentials like the rectangular barrier or square well are pieced together from constant zero-force sections with sharp discontinuities. These potentials do not exist in nature, but serve as convenient models.” Samples of textbook writing and physicists’ talk from interview data and accompanying grammatical analysis are provided in Table 4.1 below.

Table 4.1: Samples of physicists’ speech and writing for grammatical analysis. Sample of physicist’s speech or writing “Denominalized”/simplified exerpt with analysis “In both cases, a classical particle of to- a classical particle [medium] moves tal energy E. . . moves back and forth be- back and forth between [process] the tween the boundaries.” [77] boundaries [range]. “. . . when you have a confined system, . . . your zero point energy [medium] is . . . [the width of the box is] going to set going to go up and up [process:event]. the scale for what . . . the magnitude of the energy is, so as you confine it [the particle] more and more, your zero point energy is going to go up and up.” - Prof. A, interview study “. . . it has been seen that potential barri- . . . potential barriers [agent] can reflect ers can reflect particles that have suffi- [process:event] particles [medium]. . . cient energy to ensure transmission clas- sically.” [75] “This [wave] packet would move classi- This wave packet [medium] is reflected cally, being reflected at the wall. . . ” [78] [process:event] at the wall [circumstance:location]. “The α-particle then ‘tunnels through’ the The α-particle [medium] then ‘tun- barrier. . . ” [75] nels through’ [process] the barrier. . . [range] “. . . they [α particles] start out with the α particles [medium] leak through energy E inside the nucleus and ‘leak’ [process:event] the potential barrier through the potential barrier.” [80] [range]. “. . . the phenomenon of tunnel- The particle [medium] leaks through ing. . . allows the particle to ‘leak’ through [process] any finite potential barrier any finite potential barrier. . . ” [79] [range]. 79

From the data I have studied, this selection of talk and writing of physicists (Ta- ble 4.1) is representative of the type of language associated with the POTENTIALWELL metaphor. One can see a clear pattern of grammar that can be mapped to the ontological categories of matter and processes. This is shown in Table 4.2 below.

Table 4.2: Ontology Matter Process QM particles, classical particles, parti- moves between, reﬂect, reﬂects at, tun- cles, wave packet, energy, zero point en- nels through, leaks through ergy, energy walls, energy barrier, potential barrier, barrier

The grammatical analysis can tell us more than what the objects and processes are in the metaphorical model. In a particular sentence the particle and its potential energy graph function as separate grammatical participants. The particle most often functions as the grammatical medium and the potential graph as either the range or circumstance of location. (See Table 4.1.) The grammatical processes often describe some sort of interaction between the two participants. Thus the common grammatical structure of the POTENTIALWELL metaphor contradicts the conventional view of the potential energy as a property of the particle or the system.

In summary, the ontological analysis shows us that the potential well metaphor consists of two objects (the particle/wave function, and the potential energy graph) that interact with each other via a number of possible processes such as “tunnel through”, “reﬂect at”, and so on.

Elaborated metaphors: The ontology encoded in the grammar describes the basic objects and processes of the physical model. To understand more subtle properties of these objects, and their interactions, I need to apply a metaphorical analysis. I need to identify the elaborated metaphors, and for this, I need to identify the base of domains of various analogs that go into making up the POTENTIALWELL metaphor.

In this section I will analyze an additional sample of clauses and sentences from a selection of popular introductory quantum mechanics textbooks [75, 76, 77, 78, 79]. I 80

will identify each metaphor that makes up the metaphorical system, and present examples of its use by physicists. Where necessary, I will identify the analogical base from where the words have been “borrowed” to create the metaphor.

• Metaphor: THEPOTENTIALENERGYGRAPHISAPHYSICALOBJECTORPHYS-

ICAL/GEOGRAPHICAL FEATURE.

Examples: “For this reason, the rectangular potential barrier simulates, albeit schematically, the scattering of a free particle from any potential.” [78]

“The perfectly rigid box, represented by a rectangular potential well with in- ﬁnitely high walls, is an ideally simple vehicle for introducing the mathematics of quantum systems.” [77]

“Scattering from a ‘cliff”’. [79]

“. . . even for a total energy of the particle less than the maximum height of the potential hill. . . ” [75]

“What are the classical wave analogs for particle reﬂection at a potential down- step and a potential up-step?” [77]

“Particles of total energy E. . . encounter a potential ‘hole’ of depth V0 and width L.” [77]

Base: The words “barrier”, “box”, “well”, “hole”, “down-step”, “up-step”, “cliff”, and “hill” are borrowed from the category of physical objects or physical/geographical features.

• Metaphor: The previous metaphor THEPOTENTIALENERGYGRAPHISAPHYS-

ICALOBJECTORPHYSICAL/GEOGRAPHICAL FEATURE entails another metaphor: The “walls” of the well or barrier correspond to a physical height above the

ground. In other words, ENERGY IS A VERTICAL SPATIAL DIMENSION in the

Earth’s gravitational ﬁeld. THEPOTENTIALENERGYGRAPHISAPHYSICAL 81

OBJECTORPHYSICAL/GEOGRAPHICAL FEATURE metaphor builds on this spatial metaphor.

Examples: “. . . even for a total energy of the particle less than the maximum height of the potential hill, there is a transmitted wave T .” [75]

“It is instructive to consider the effect on the eigenfunctions of letting the walls of the square well become very high. . . ” [76].

Prof A: “. . . your zero point energy is going to go up and up.”

“. . . ψ1, which carries the lowest energy, is called the ground state. . . ” [79]

Base: The words “height”, “high”, “up”, “lowest”, and “ground” all suggest an analogy between the vertical axis of the potential energy graph and a vertical spatial dimension on the Earth’s surface.

• Metaphor: THEPOTENTIALENERGYGRAPHISACONTAINER. The potential energy graph “contains” or “traps” either the wave function, the particle or the energy of the particle.

Examples: Prof. C: “. . . eventually you’re going to get. . . to an energy range where there is some confinement energy which is the. . . kinetic energy. . . that it costs to keep them . . . confined . . . in whatever confinement, magnetic bottle or some kind of well or something like that, where the confinement energy is comparable. . . to the temperature.”

“The exponential decrease of the wave function outside the square well for the second energy state is less rapid than is the corresponding decrease for the lowest energy state as indicated. . . ” [77]

“Inside the well where V (x) = −V0. . . ” [79]

“. . . bound [energy] states in the. . . well” [77]

“Although E is negative for bound states, it must be greater than −V0. . . ” [79] 82

Base: Words such as “box”, “well”, “conﬁned”, “bottle”, and “bound” all suggest an analogy to some sort of container. Physicists also make a distinction between “in/inside” and “outside” the well: Such adverbial phrases also indicate the presence of the container metaphor.

• Metaphor: THEPOTENTIALENERGYGRAPHISABARRIER.

Examples: This metaphor is suggested by words such as “potential step”, “potential barrier”, and “potential hill”.

• Metaphor: THEPOTENTIALENERGYGRAPHISAHARDCONTAINER/BARRIER

or THEPOTENTIALENERGYGRAPHISASEMI-HARD CONTAINER/BARRIER

Examples: “. . . consider a ‘square’ potential well with inﬁnitely high sides, as indicated in Fig. 3-12(a). This corresponds to a particle bound by impenetrable walls to a region of width 2a.” [75]

“a particle. . . would bounce off, or pass through, the potential” [79].

“We see that, for square wells of ﬁnite depth, the wave function extends slightly beyond the boundaries of the well.” [77]

Base: Words such as “bounce off” or “impenetrable” may be associated with the category of “hardness”. Phrases that suggest that the wave function penetrates into forbidden regions of the graph suggest that the graph is a permeable or semi- hard substance.

• Metaphor: The particle, the wave packet, and the energy are all given an ontological status of matter. More speciﬁcally:

– Metaphor: QM PARTICLES ARE HARD OBJECTS.

Example: “. . . we were able to determine the probability that a particle. . . would bounce off, or pass through, the potential” [79].

– Metaphor: THE WAVE PACKET IS A SOFT OR BREAKABLE OBJECT 83

Example: “For a wave packet incident from the left, the presence of reﬂec- tion means that the wave packet may, when it arrives at the potential step, split into two parts. . . ” [78]

– Metaphor: QM PARTICLES ARE A FLUID

Example: “[α particles] . . . ‘leak’ through the potential barrier.” [80]

– Metaphor: THE ENERGY IS A FLUID

Example: Note the ubiquitous use of the term “energy level”.

See Table 4.3 for a summary of the metaphorical mapping from the domain of physical/geographical features to the domain of quantum systems that involve an interaction between two or more objects.

Table 4.3: Summary of the metaphorical mapping between the base domain of physical/geographical features and the target domain of interacting QM systems Base domain Target domain PHYSICAL/GEOGRAPHICAL INTERACTING QM SYSTEMS FEATURES Physical or geographical fea- −→ Potential energy graph tures Vertical height of physi- −→ Magnitude of energy at a point/region on cal/geographical feature the potential energy graph. Hardness or softness of a wall −→ “Height” of the potential energy graph Container with top face open −→ “Trapping” of QM particles, “bound” states Billiard ball −→ QM particle in some circumstances Soft or breakable objects −→ QM wave function or wave packet Fluid −→ QM particle in some circumstances, or the energy of the particle/system Ball bounding off a wall −→ Reﬂection of QM particle Tunneling/penetration −→ Process by which a QM particle “passes through” a seemingly solid “barrier” Leaking −→ Process by which a QM particle “es- capes” from a QM “container” 84

4.2.3 Productive Modes

How do physicists piece together the grammar/ontology of the POTENTIALWELL metaphor? How do they use the associated imagery to reason productively about quantum systems? From the discourse of professors and textbooks I have identiﬁed the presence of productive modes for the POTENTIALWELL metaphor. These productive modes can be used to reason about physical situations or simply represent ways of talking about physical systems. The productive modes often involve some grammatical process through which the particle/wave function/energy and the potential energy graph interact. I present ﬁve examples below:

1. Squeezing: Squeezing the walls of the well forces the water upwards, thereby raising and spacing out the “energy levels”.

Example: Prof. A: “. . . when you have a conﬁned system, . . . [the width of the well is] going to set the scale for what . . . the magnitude of the energy is, so as you conﬁne it more and more, your zero point energy is going to go up and up.”

2. Stacking: Matter takes up space. Filling up the well/bucket can be used to understand the behavior of fermions.

Example: We already observed Peierls make this analogy explicit [97]. The following is an example of Prof. A. using the mode productively: “if you have fermions then. . . you have to keep stacking the fermions into levels which get more and more elevated in energy. . . ”

3. Tunneling/leaking: The potential energy graph behaves as a physical container or barrier, preventing the escape of the particle. This leads to the ideas of “tunneling” or “leaking”. When reasoning productively, physicists recognize that a higher or wider barrier means less probability of tunneling.

Examples: In describing α-particle decay, Feynman et al. write: “This is called the quantum mechanical ‘penetration of a barrier.’ ” Later when describing 85

what happens to the α-particles, they write: “. . . they start out with the energy E inside the nucleus and ‘leak’ through the potential barrier.” [80]

Grifﬁths shows how this mode can be used productively: “If the barrier is very high and/or very wide (which is to say, if the probability of tunneling is very small), then the coefﬁcient of the exponentially increasing term (C) must be small. . . ” [79].

4. Reﬂecting/scattering: The wave function or particle is reﬂected by or scatters off a hard barrier.

Example: “For this reason, the rectangular potential barrier simulates, albeit schematically, the scattering of a free particle from any potential.” [78]

5. A way of speaking: It is difﬁcult to come up with realistic physical systems of quantum mechanical particles without resorting to lengthly descriptions. (See [77] for examples of such descriptions.) By separating the QM particle from its potential energy graph, physicists are able to talk easily about the particle interacting with an external object (its own potential energy graph). This represents a productive way of speaking about quantum systems, which avoids the difﬁculty of actually having to describe with what the particle is interacting.

Example: During one of the interviews I was asking a professor to describe the process of trapping and cooling atoms to absolute zero.

DTB: “Are the atoms going to jump out, are you not going to be able to trap them?” Prof. E: “No, of course not, you’d just go down to the lowest eigenstate. I mean, I don’t know how they were trapped in the ﬁrst place, but suppose you had them in a square well for example. Okay, we know what the states of a square well are. As long as. . . if the amplitude for all of the atoms is 1 to be in the ground state and 0 to be anywhere else, then you are at absolute zero temperature.”

Notice how the subject spontaneously invoked “a square well” as an example of an atomic trap rather than an actual physical system. 86

4.2.4 Summary

I have tried to illustrate how grammar and metaphor work together to encode the features of a particular descriptive model. Each aspect is necessary and the grammatical and metaphorical analyses together serve to illuminate features that each individual analysis cannot do on its own.

The POTENTIALWELL metaphor is an example of a metaphorical system made up of three ontological metaphors, two of which are encoded in the grammar (THEPO-

TENTIALENERGYGRAPHISAPHYSICALOBJECTORPHYSICAL/GEOGRAPHICAL

FEATURE, and THE PARTICLE/WAVE FUNCTION/ENERGYISAPHYSICALOBJECT

OR MATTER), and one which can only be identiﬁed by looking at the imagery (EN-

ERGY IS A VERTICAL SPATIAL DIMENSION). Other metaphors such as THEPOTEN-

TIALENERGYGRAPHISACONTAINER or THEPOTENTIALENERGYGRAPHISA

HARDBARRIER build on and elaborate this ontology. At the sentence level, we can see how productive modes of reasoning are formed by introducing grammatical processes through which the particle or wave function interacts with its own potential energy graph. Fig. 4.1 presents a visual summary of how the POTENTIALWELL metaphor is structured.

4.2.5 Student Difﬁculties

A study group of four junior students in their first QM course, agreed to be video taped while working on problems. This problem was from French and Taylor [77]. The question was: “What are the classical wave analogs for particle reflection at a potential down-step and a potential up-step?” Notice here the POTENTIALWELL metaphorical system serving a specific function: namely, it describes the shape of the potential energy graph (“potential down-step”).

S1: Well, there wouldn’t be reflection in particle physics on a down-step right? Or even, I don’t think even on an up-step. . . S3: No, there’s reflection on an up-step, total reflection. 87

Grammar encodes two ontological metaphors

QM particle/wave-function/energy is Potential energy graph is a matter/a physical object physical object Features: Hard or soft/breakable Entails another metaphor: Energy is object or liquid. a vertical spatial dimension in a gravitational field

enables

Potential energy graph Potential energy graph is a container is a barrier Features: hard or Features: hard or semi-hard semi-hard

Together, these metaphors lead to productive modes of reasoning and ways of speaking

Ways of speaking: Physicists Productive modes of reasoning: use the potential energy graph as involving different grammatical a short-hand or metonym for the processes actual physical system that the object interacts with.

confining/ stacking/ tunneling/ reflecting/ squeezing filling up leaking scattering

Figure 4.1: Summary of the metaphorical system and its usage by physicists.

S1: Not classical though, right? S2: Not if its less than the energy though. S1: It just slows it down.

In this opening exchange we can observe S1 talking at cross purposes with S2 and S3. S2 and S3 seem to be imagining a classical particle approaching the step and bouncing back (later dialogue show that they do not really shift from this literal view of the situation), while S1 seems to be thinking of a wave approaching with energy greater than the energy of the step. As we see later, S1 is reasoning from picture of a surface water wave passing over a step in a river or sea bed.

S1: Not quite sure what the wave analogs would be. If I had to guess I’d say it would be like sound, like those things that male cheerleaders have, like big cones. S4: Megaphones? 88

S1: Yeah. ’Cause I think, you know,. . . basically a step up or step down in resistance. But I am not quite sure what we are supposed to say about that.

This is the ﬁrst example of an analog from S1. It is interesting that S1 sees the key as a change in resistance (at the end of the ﬁrst exchange S1 says “It just slows it down”), yet he still is the one who proposes a physical form (consistent with the ontology of the graph as a physical object) surrounding the medium rather than a change in the medium itself (which would represent a more obvious change in resistance for the wave).

S2: So they’re saying that there would be reﬂection on a potential up-step like a. . . S1: Yeah, just like a sound, or a water wave or something. S1: Um, well ’cause I know on a potential up-step,. . . like if you just had. . . water and you had, you know, deeper part and a shallower part, and you had a wave, some of it would reﬂect back.

Here S1 applied the metaphor of a physical object again, and proposes a second analog based on the physical form of the graph rather than a change in “density” or “tension” of the medium. Actually, a physical step on a river bed could be a valid example if S1 connected it to a model of how the resistance experienced by a surface wave attenuates with the depth of the water. He does not, and this explains his uncertainty below.

S1: So that’s not too hard to see. But like, I would guess that the same thing would happen if you had a down-step, but that’s not something like I really, I could vouch for. Like I think they’re looking for stuff that like most people know. S2: Is that what its saying? Its coming at it with every energy, like continuous energies, like around the step?

S2’s statement is interesting. The use of “at” and “around” are examples of grammatical location and suggest the metaphor: THESTEPISAPHYSICALOBJECT. S1 shows he is still on the right track when he says:

S1: I think they’re just asking for like, examples from. . . in real life from when a wave. . . goes into a space of less resistance and has reﬂection back. 89

S4: So in classical what would happen at a potential down-step? S1: A potential down-step? S2: It would just keep going. . . S1: . . . It would just speed up. At a potential up-step it would just slow down.

At this point the students decided to give up on the problem and moved on to another without resolving their uncertainty.

4.2.6 Discussion

One alternative hypothesis to explain the difﬁculties presented above could be that the students are unable to interpret the physical meaning of the potential energy graph or are simply not understanding the situation. However, S1’s ability to interpret potential energy graphs correctly and articulate the key to the analogy discounts this hypothesis. The data show that his inability to come up with a productive analog must be based on other factors. My framework explains how S1 is distracted by applying an overly literal interpretation of the POTENTIALWELL metaphor in an inappropriate situation. We see that a way of talking (i.e., describing the potential graph as a “step”) seems to be interacting with students’ reasoning. My analysis (Section 4.2.5 above) shows that the students in this group are searching in the category of “physical objects” for an analogy, in accordance with the underlying ontological metaphor THEPOTENTIAL

ENERGYGRAPHISAPHYSICALOBJECT rather than searching in a more productive category.

As a control I posed the same problem to the professors in the interview study. They all responded that an analogy of an electron beam scattering off of a potential down step is light traveling from a medium with greater index of refraction to a medium with a lesser index of refraction. When asked why changing media was a good analog, most were unable to explain, but continued to elaborate their answer. Only one professor was able to explain why this was a good analogy. Prof. E: “I know because we’ve thought about these things before and its just been classiﬁed in that category.” This 90

statement suggests that physicists are able to automatically search for an analog in a category of analogous processes rather than analogous objects.

I have shown how physics professors can use metaphorical systems to reason productively in certain situations while students take the same representation and apply it too literally and inappropriately in other situations. Strange ideas like the megaphone make sense if we understand the underlying ontology of the graph, spoken of as a physical object. I think that the example of student discourse presented above is a typical example of students’ difﬁculties arising from linguistic representations. If teachers are unaware of the difﬁculties students experience interpreting our metaphorical language, it leaves them less able to understand, interpret, and facilitate student learning.

4.2.7 “Robust Misconceptions” Related to the POTENTIAL WELL Metaphor

Are there “robust misconceptions” in QM? The characteristics of a robust misconception are that it must be (a) common to a signiﬁcant percentage of students in a particular class, (b) reproducible in form and structure across different classes at different insti- tutions in different contexts, and (c) resistant to instruction. One emerging example is presented in Table 4.4 below. Researchers have observed in three studies, that students think that a QM particle loses energy when it tunnels through a barrier.

Although research on students’ understanding of quantum mechanics is in its in- fancy, it appears that students do have speciﬁc difﬁculties that have the characteristics of a robust misconception. McKagan et al. freely use the word “misconception” in their paper. The reader may be able to observe a repeating pattern of reasoning In Ta- ble 4.4: (1) it takes energy for a particle to tunnel through a barrier. (2) Making the barrier wider or higher means that the particle loses more energy/expends more effort when tunneling through. Morgan et al. speculate that students’ idea that energy is lost, may be caused by either (1) intuitive classical ideas about a particle passing though a 91

Table 4.4: Misconception: A particle tunneling through a potential barrier loses energy as it goes through or comes out the other side with less energy than it had previously. Summary from three studies. Authors’ summary and explanation Sample student responses used to justify this explanation. Lei Bao [95] interviewed ten students Bao observed that all three students over two semesters. Three responded gave similar responses. Mike: “. . . less with the incorrect idea that a quantum me- energy so the amplitude will be re- chanical particle loses energy when it tun- duced,. . . Amplitude is reduced be- nels through a potential barrier. cause energy is lost in the passage [my emphasis]. . . ” Jeffrey Morgan et al. [94] found that all Selena: “Uh, because it requires energy six students that they interviewed thought to go through this barrier.” that the particle lost energy when it went Jack: “. . . when the particle of some through a potential barrier. Two of the . . . energy, encounters a potential bar- students had completed a senior level QM rier, there is a possibility. . . that a par- course and four had completed a sopho- ticle will actually just go straight on more level introductory QM course through, losing energy as it does so, and come out on the other side. . . at a lower energy. . . ” McKagan et al. [93] gave a conceptual No interview samples were provided, test to a group of engineering majors (N = but the authors summarize the student 68) and physics majors (N = 64) after they responses as follows: “all students who had completed a modern physics course. selected answers A, B, or E [more One of the questions probed students’ un- than 50% for both engineers and physi- derstanding of tunneling processes. On cists] argued that since energy was lost this question 24% of the engineering main tunneling, making the barrier wider jors were able to answer correctly and and/or higher would lead to greater en- 38% of the physics majors were able to ergy loss.” answer correctly. barrier, or (2) physicists tend to draw the potential energy graph and the wave function superimposed. Thus a decaying wave-function amplitude may be confused with a decrease in energy.

However, McKagan et al. noticed something interesting in their study. In interviews, they discovered that students do not see the potential energy graph as representing the potential energy of the particle in question. They see it rather as some external object with which the particle interacts. The authors describe an example from their interviews: 92

“When pressed, he said that the ‘bump’ was ‘the external energy that the electron interacts with’ and insisted that it was not the potential energy of the electron itself, in spite of the fact that it was explicitly labeled as such in the previous question.”

The authors speculate that statements like “a particle in a potential” may be the cause of this problem.

My analysis supports this idea and provides an explanation for the underlying causes of this student difﬁculty. The problem is much more widespread that just phrases like “a particle in a potential”. As I pointed out in Section 4.2.2, many statements that fall under the category of the POTENTIALWELL metaphor, tend to separate the particle or wave function from its potential energy graph in the grammar of the sentence. Most often the particle/wave function functions grammatically as the medium while the potential energy “barrier” functions as either the range, or circumstance of location. The two grammatical participants then interact with each other by a grammatical process such as “tunnels through” or “is reﬂected”. Factors such as classical intuition and the superposition of the potential energy graph and the wave function may serve to support the idea that energy is lost in the tunneling process. However, the underlying ontological structure of this model is encoded primarily in the language. I hypothesize that the language is the primary source of the students’ incorrect model. Classical intuitions build on and extend this basic model, and graphical representations (such as the superposition of the energy graph and the wave-function) may reinforce it.

Most interestingly, the idea of exhaustion in tunneling seems to appear after instruction. There is no evidence that students enter their QM course with this “ robust exhaustion misconception”. It is important because it is an example of a view that will emerge in this thesis. Namely that many student difﬁculties are possibly representational in nature rather than based on a prior belief system that they bring with them. By this, I mean that, in order to interpret a particular physical phenomenon, students bring particular prior experiences and resources to bear [66] in a complex interaction with the situation as it is presented to them. What experiences and resources are stimulated 93

must depend, in part, on the way in which the physical situation is represented by the instructor. Thus the variability or robustness of students’ understanding may be related to the variability or robustness of the representations they encounter. In summary, the way an instructor speaks about a physical phenomenon may inﬂuence the way in which students “see” that phenomenon.

In the case of the POTENTIALWELL metaphor, I have tried to illustrate how the language used to describe certain QM systems may in fact pose extraordinary difﬁ- culties, especially if students are not aware of how and why metaphorical terms are being used. The metaphorical language, grounded in the classical world, may encourage students to associate extra (classical) properties with the QM system as they try to coördinate these new representations with their prior understanding of the world. These over-extensions of the representation seem to be the source of their difﬁculties.

4.3 The BOHMIAN Metaphor

4.3.1 Introduction

The POTENTIALWELL metaphor, as a linguistic representation, has many of the characteristics of a physical model as described by Etkina et al. [96]. The language describes objects with attributes, and processes by which those objects interact with each other. In contrast, the BOHMIAN metaphor has almost none of those characteristics. It seems to exist in the language of physics solely as a way of speaking. It is an interesting case because it is easy to identify the metaphor, but the original analogy is not. Therefore, in these sections, I will present the linguistic analysis ﬁrst and the study of the analogy on which it is based, second. Although I have called it the “BOHMIAN” metaphor in honor of David Bohm, who advocated the Bohmian interpretation of QM, the entry into the language of physics can be traced back much earlier than this. 94

4.3.2 Modern Language

The BOHMIAN metaphor is identiﬁed in language by words and phrases that suggest that the wave function or quantum state is a container that contains the quan-

tum mechanical particle. There are only two metaphors that makes up the BOHMIAN metaphor:

1. Metaphor: THE WAVE FUNCTION/QUANTUM STATE IS A CONTAINER.

Examples: Noun groups such as “wave packet" or “envelope function” indicate an analogy to a container.

2. Metaphor: THE QM PARTICLE IS A PHYSICAL OBJECT CONTAINED INSIDE

THE WAVE FUNCTION/QUANTUM STATE.

Examples: This is suggested by prepositional phrases such as “in the ground state” in sentences such as “The electron is in the ground state.”

The metaphorical mapping is summarized in Table 4.5 below.

Table 4.5: Summary of the metaphorical mapping between the base domain of containers and the target domain of the QM wave function Base domain Target domain CONTAINERS QM WAVE FUNCTION Bounded region in space −→ “Surface” of the graph representing the wave function Objects inside this bounded −→ QM particles region

Connection to grammar: In the Bohmian metaphor the wave function or quantum state is conceived of as a container containing a particle as a separate entity inside it. The language is based on two sources. The ﬁrst source is an analogy to Einstein’s ghost ﬁeld idea (see Section 4.3.3 below) but the second source is language itself. Cognitive linguists hypothesize that mental states are spoken about in language metaphorically as containers. (See Table 4.6 below.) 95

Table 4.6: The STATES ARE LOCATIONS metaphor [1] Base domain Target domain SPACE STATES Bounded region in space −→ States

For example, if one is depressed one can say, “I am [relational process] in a state of depression [location]”. Such statements seem to all have the same grammatical structure, namely a relational process followed by circumstance of location. It seems as if ontological physical states are expressed by an identical grammatical structure: such as, “the electron is [relational process] in the ground state [location].” It seems as if physicists have unconsciously borrowed this grammar that expresses mental states in every day experience and used it to express physical states in physics. As mentioned in Chapter 3, the metalingual apparatus that we have for realizing physical states in

language appears to be extremely limited. The STATES ARE LOCATIONS metaphor, supported by this unique grammatical structure, is one these limited means of expression. A statement about the physical location of an object within another object would be classified in the ontological category of matter if taken literally. Metaphorically a statement such as “the electron is in the ground state,” is a statement about the energy of a quantum system, and physicists recognize energy as a state function. There is a clear ontological conflict between the literal interpretation of the statement and the meaning that is intended. This leads me to predict that such statements will cause students confusion and may lead to difficulties. In particular, I predict that students will invoke a spatial interpretation of such statements.

4.3.3 The Original Analogy

In the case of the POTENTIALWELL metaphor, the analogy on which it is based, is

relatively well understood. What is interesting about the BOHMIAN metaphor is that the metaphor is easy to identify, but the original analogy is not well known. If my framework is correct and language is built on analogy then an original analogy should 96

exist in the mainstream QM literature. I started searching the original QM papers in the hope that I would ﬁnd some explicit reference to the idea that the wave function could contain the particle inside it. Remarkably, I found such a reference in a paper by Max Born, published in 1926 [71].

“Neither of these two views seem satisfactory to me. [Heisenberg’s interpretation of the wave function and the Schrödinger/deBroglie interpretation of the wave function] I would like to attempt here a third interpretation and test its applicability to collision processes. I thereby pin my hopes on a comment of Einstein’s regarding the relationship between the wave field and light quanta. He says roughly that the waves may only be seen as guiding [showing] the way for corpuscular light quanta, and he spoke in the same sense of a “ghost field”. This determines the probability that one light quantum, which is the carrier of energy and momentum, chooses a particular [definite] path. The field itself, however, does not have energy or momentum.” [71] [Translation by D.T.B.]

There are several remarkable features about this passage from Born:

• Firstly, it lays out the Bohmian interpretation of quantum mechanics twenty-ﬁve years or more before Bohm proposed the same idea, and one year before de- Broglie’s attempt at a “pilot wave” theory. Incidentally it is known that Bohm was inspired to revise his view of quantum mechanics after a period of conversations with Einstein. One could imagine a conversation, similar to the one Born describes, happening between Bohm and Einstein.

• Secondly, when Born says “Neither of these two views seem satisfactory to me,” he is referring to (1) the Heisenberg interpretation of QM which Born describes as “an exact description of the processes in space and time are principally impossible,”, and (2) the Schrödinger/deBroglie interpretation which Born summarizes: “He tries to construct wave groups which have relatively small dimensions in all directions and should, as it seems, directly represent moving corpuscles.” What is remarkable is that in some sense Born is cautioning against overly literal interpretations of (1) an analogy to a classical particle (Heisenberg’s approach), or (2) an analogy to a physical wave (Schrödinger’s approach). Born suggests 97

that both views lead to untenable positions in the physical interpretation of QM and introduces a third model which is essentially a hybrid of the wave and particle analogies. Born makes an analogy to Einstein’s interpretation of light waves and light quanta and applies it to particles with non-zero mass.

Born’s mode of reasoning appears to be metaphorical as well as analogical. He makes an analogy to Einstein’s view of the electromagnetic field as a ghost field, applying this idea to particles that have non-zero mass. But he does not suggest that the wave function is “like a guiding field”, he expresses Einstein’s idea directly as “. . . the waves may only be seen as guiding the way for corpuscular light quanta. . . ” [my emphasis]. For Born to be able to interpret the wave function as a probability distribution, he felt it was necessary to blend together a wave picture and a particle picture with real particles who have definite trajectories determined probabilistically by the wave function. Lakoff and Núñez refer to such a mental construct as a metaphorical blend [1] after the conceptual blend of Fauconnier and Turner [98].

• Thirdly, Born is aware of the limitations of the metaphorical picture he has introduced. In blending a wave and particle picture into a model that looks and feels like a statistical ensemble, Born cautions about taking this “Bohmian” interpretation too literally:

“However, the proposed theory is not in accordance with the conse- quences of the causal determinism of single events.” [71]

4.3.4 Productive modes

One of the difﬁculties with QM is the question of how to speak about quantum processes meaningfully. I suggest that the BOHMIAN metaphor permits a partial solution to this problem. Although Born’s suggestion (intepreting the wave function as a pilot 98

wave) never made it to the mainstream of physics, the associated language is now ubiquitous and used productively by physicists as I will show in the following examples:

• D.T.B.: “Okay, um, I guess, if you wanted to think about how an electron propa- gates would it be. . . It wouldn’t be sensible to talk about it as a wave, you would think more as a particle?”,

Prof C: “Well, I mean, still, you can think of it as a plane wave. Yeah, maybe in an envelope function which makes it into a wave packet.”

• Prof. A: “you can make a packet of photons of different wavelengths to make some sort of localised pulse, so photons were not going to have to be spread out in plane waves or anything like that”

• Prof. C:“. . . think about this thing [electron] being in kind of a wave function which is a superposition of amplitudes to be doing different things. Then its possible for it [the electron] to be going through both slits at once.”

4.3.5 Student Confusion

As part of my study, two senior undergraduate physics majors (in their second QM course) agreed to be videotaped while working on their QM homework together. In this particular session S1 and S2 were working on a problem worked out in class by the lecturer that they did not understand. The question may be expressed as follows: “Given an electron in the ground state of an inﬁnite square well of width L. The walls are suddenly moved apart so that the width of the well becomes 2L. What is the probability that the electron is in the ground state of the new system?”

The two students working on the problem understood the sudden approximation, they calculated the overlap integral and got a numerical answer which was reasonable. Then S1 stopped and pondered that his answer made no sense. He argued that his answer should be zero. A discussion with the observer (D.T.B.) followed. 99

Figure 4.2: Wave function of the electron in the sudden approximation

S1: But I am still confused about what I was. . . saying about if there is a probability that it is in the [sic] ﬁrst ground state — it seems to say that the particle can be where it is not. D.T.B.: Why do you say that? S1: Because we know that the wave function looks like this [points to a sketch similar to Fig. 4.2] — Oh, so its not the probability of it being in the ground state really. . . I think the probability is really. . . I mean, we know that its in this state [points to sketch similar to Fig. 4.2] so it can’t be in the ground state. So it’s zero [the probability].

The discussion circled around this theme for some time. S1 appeared to view the notion of the wave function as container too literally or rather, inappropriately. The wave function limits where the particle can be, but to say the electron is “in the ground state of the new well” does not suddenly permit it to exist outside of the [-L/2,L/2] region. It is simply a statement about measuring the energy of the electron. However, the prepositional phrase,“in the ground state”, is functioning grammatically as a location. One can see the energy measurement spoken about as a location as the key to his confusion. Note how S1 speciﬁcally uses the idea of where the particle is located, and uses the idea of “in” inappropriately. This is not simply a case where he is unaware that the wave function in Fig 4.2 is composed of even eigenfunctions of the new well. I continued to press the student to explain what “in the ground state” really meant until he had a breakthrough:

D.T.B.: If I ﬁnd the particle in the ground state, what does that mean? S1: I don’t see how you can say that you ﬁnd it. . . the energy? You measure the energy. D.T.B.: I measure the energy. . . 100

S1: So. . . what is the probability that the energy of a particle in this state is equal to the prob. . . the. . . D.T.B.: Equal to the energy of the ground state of the new system.

By the end of the discussion, S1 has not reached a full understanding of the language, nor of the fundamental principles of the situation, but we are able to chart his progress. Through questioning him for meaning, he was able to (a) question the overly literal interpretation of the metaphor, and (b) recall that phrases such as “in the ground state” are statements about the energy of the system.

4.3.6 Discussion

What is the source of S1’s difficulties? I believe that the linguistic framework I have developed provides both a reasonable and parsimonious explanation: S1’s argument, that the probability should be zero, draws specifically on the location metaphor. He says, “it [the question, what is the probability of finding the electron in the ground state of the new system?] seems to say that the particle can be where it is not.” This statement suggests that he is viewing the question as a question about the location of the particle. In other words, he is interpreting the phrase “in the ground state” literally rather than figuratively. This is in spite of the fact that he was in his second QM course.

4.3.7 Robust Misconception?

This difficulty with the BOHMIAN metaphor remains undocumented in the physics education research literature. However, in one private conversation with a physics professor who teaches quantum mechanics, he described that he observed the identical difficulty amongst his students with the same sudden approximation problem. Further research is needed, but I predict that if researchers look for this difficulty, they will find it among a certain percentage of students. 101

4.4 Additional Examples

In this section, I present two additional examples of metaphorical systems that I identi-

ﬁed in physicists’ language about QM. The two metaphorical systems are: THEELEC-

TRON IS A PARTICLE and THE ELECTRON IS A WAVE.

4.4.1 ELECTRONISA PARTICLE

Table 4.7 presents some examples from the interview study of how physics professors talk about QM particles as if they were classical particles with deﬁnite trajectories.

Productive modes Linguistic evidence

• Useful and easy way of talk- • Prof. C: “You send in polarised elec- ing about what is happening trons to the left and you send them in in a quantum system to a Stern-Gerlach apparatus and sometimes they go up and sometimes they • Explanation of photoelectric go down.” effect in terms of classical billiard ball type collisions • Prof. B: “Well, you can put light between photons and elec- through a grating and then hit a pho- trons totube where an individual photon has to release an electron.”

Table 4.7: Electron is a particle metaphor. The examples show how physicists use the particle metaphor as a productive way of talking about what is happening in a QM process.

4.4.2 ELECTRONISA WAVE

Table 4.8 presents some examples from the interview study of how physics professors talk about QM particles as if they were classical waves.

4.5 Summary

The following ideas have been presented in this chapter. 102

Productive Linguistic evidence modes Understanding of Prof. B: “. . . you can’t take a photon. . . with a. . . precisely the Heisenberg known frequency — you can’t locate it at a frequency or wave- uncertainty prin- length. . . If you start confining it to a finite amount of space then ciple via analogy the frequency spectrum for that smears out. You must. . . do a to Fourier analy- Fourier transform or a sum of the Fourier series to build up. . . a sis of a classical wave packet.” wave. Prof. A: “. . . I often think of it. . . in terms of Fourier transforms Use of superposi- and the reciprocity between the bandwidth of the channel and tion principle. the length of the signal pulse that can be detected. . . in quantum mechanics, if you think of the wavefunction and its Fourier transform then, again, the width of the pulse is related to, the bandwidth which in this case is the momentum space, the range of momentum states that go into building up that pulse, that packet. So. . . the Heisenberg uncertainty principle is actually another manifestation of something that electrical engineers know — in terms of bandwidth and, and. . . pulse width,. . . that same reciprocity goes.” Prof. C: “. . . you can discuss it . . . in the context of. . . time series. In other words, things that are functions of time. So for example if you. . . have a pulse, tone like a beep! Then. . . if that tone lasts for several seconds it can have a very, very well defined centre frequency, but if it only lasts for a tenth of a second then the definition of the centre frequency is. . . only plus or mi- nus 10% and if you make it so short then basically the spectrum is all over the place. So basically the same kind of notion that when you make a wave packet, now everything is in space instead of in time, if you make your wave packet very long then you can have a very well defined spatial frequency which is to say a very well defined momentum, but now the particle is very spread out.”

Table 4.8: Examples of the ELECTRON IS A WAVE metaphor. All are examples of how the view of the QM particle as a wave is productively used to explain the Heisenberg uncertainty principle.

1. I have given examples where physicists appear to speak and reason using coherent systems of metaphors. The theoretical framework appears to be a valid and a productive way to interpret their discourse.

2. I have shown how these metaphorical systems can be identiﬁed with systematic use of both grammatical and metaphorical analysis. I have shown how the 103

elaborated metaphors build on the underlying ontology encoded in the grammar.

3. In the example of the BOHMIAN metaphor, I have shown how, in some cases, it seems that physicists have appropriated conceptual metaphors from language to

express their ideas. The example with Born and the BOHMIAN metaphor shows how a new idea in physics comes out of a blending of older ideas into a metaphorical blend. Likewise the ﬁnal product of the language is (in this case) a blend between an analogy to Einstein’s ghost ﬁeld and also already existing structures in language that are normally used to describe ontological mental states.

4. Physicists use the metaphorical systems in their language to speak and reason productively about QM systems. I have presented evidence for this from the interview study with physics professors.

5. Physicists understand the applicability and limitations of their metaphorical language. For example, a higher potential barrier means slower rate of tunneling or leaking, but the particles do not lose energy in the tunneling process.

6. Students seem to be struggling with and being confused by overly literal interpretations of the metaphorical language they encounter in QM. The context of QM is particularly convincing because it is difficult to argue that students enter their QM course with preconceptions or misconceptions about QM based on personal experience. Many of the difficulties observed, appear after instruction. It seems more plausible to hypothesize that these difficulties are related to the nature of the instruction itself. The literature, used as data, supports this hypothesis.

7. I have presented one example (the exhaustion misconception) of a documented common difﬁculty that students have with QM and how I can account for their naïve model with the linguistic framework I have developed. Physicists understand that higher barrier means a slower rate of tunneling or leaking. In contrast, students think that the particles get tired. This difﬁculty can be explained by the 104

use and misuse of the metaphorical picture. 105

Chapter 5 Heat and Force

5.1 Introduction

This chapter will present two detailed case studies: The ﬁrst concerns the role of the word “heat” in thermodynamics and its role in students’ reasoning. The second study concerns the word “force” in Newtonian mechanics and its role in students’ reasoning.

The main purpose of this chapter is to show how, using grammar, ontology, and metaphor, I can explain or even predict some of the difficulties that physics students have in learning the concepts of heat and force. Traditionally, students’ difficulties with the ideas of heat and force have been explained as students’ misconceptions, preconceptions or alternative conceptions. The responsibility for the difficulties have been laid squarely at the door of the students and the prior knowledge and experience they bring with them to the classroom. A good portion of the research in physics education has focussed on designing curricula to rid students of their incorrect understanding and facilitate the correct understanding. However, as I mentioned in Chapter 2, these alternative conceptions are often very stubborn and “resistant to instruction”. Many reasons for the stubbornness of students’ alternative conceptions have been proposed. It has been suggested that human belief systems are naturally difficult to change because beliefs are strongly held and only given up with reluctance. Also, it has been suggested that students’ intuitive belief systems are grounded in physical experience, and therefore seem “intuitively obvious” to students, and are a natural place for them to ground their reasoning.

While these ideas may be valid in some cases, I have proposed that this view may 106

not account for all of students’ difficulties. Or at least, these ideas do not present a complete picture of what students are struggling with. In this chapter, I am going to suggest that many difficulties that take on the appearance of misconceptions may be driven by the representations of ideas that students encounter in their physics course, as well as ideas that they bring with them to the physics course. I will show how the language that physicists use to speak about heat and force may have a direct influence on students’ reasoning about these topics.

5.1.1 Issues Concerning Heat in Thermodynamics

What initially piqued my interest in thermodynamics was an extensive study conducted by David Meltzer [51]. In part of this study, Meltzer interviewed 32 calculus-based introductory physics students at Iowa State University. During the interview students constructed a closed thermodynamic cycle through a series of verbal descriptions and diagrams. After each step in the cycle, students were asked to evaluate and compare the work done and energy transfered by heating (qualitatively), and in some cases, comment on the kinetic energy of the molecules of the gas. Once the thermodynamic cycle was completely described, students were asked whether the net work done by the gas, and the total heat transferred to the gas, were positive, negative, or zero. 69% of the interviewees said (incorrectly) that the total heat transferred to the gas for the entire process was zero.

There are many possible reasons why students thought that the total heat for the cycle must be zero. Meltzer explained that students believe that heat is a state functions of the system. I will investigate this idea further and attempt to explain why students might view heat as a state function. It is important to note that Meltzer’s interviews were conducted after the thermodynamics section of the course was complete.

I have already shown that it is almost impossible to describe physical states of a system without invoking some sort of metaphor. The ubiquitous approach in physics seems to be borrowed directly from how mental states are realized in language. In 107

cognitive linguistics two common metaphors are used to describe mental states:

1. THE MENTAL STATE IS A CONTAINER. The mental state functions as a container and a person is either inside it or not. For example, “She is in love.” To express a physical state of a quantum system, physicists often say, “The atom is in the ground state.”

2. THEHUMANBODYISACONTAINEROFEMOTION. For example, “he was ﬁlled with anger,” or “he could not contain his rage.” Physicists tend to speak of state functions metaphorically as substances that are contained within an object. Thus a state function like energy is spoken of as “the amount of energy in the system,” or “the object lost some of its kinetic energy” etc.

I will examine the role of “heat” in the language of thermodynamics, and students’ reasoning as follows:

• First I will examine the historical conceptual development of the idea of heat. Since it seems that much of physicists’ language is based on historical analogies, examining history should reveal the origins of our modern language.

• I will conduct a systematic grammatical and metaphorical analysis of the modern language of physicists. This analysis will reveal two things:

1. The grammatical analysis will allow us to see an underlying pattern of ontology in the way physicists talk about heat, either as an entity on its own, or as a form of energy. In both cases heat is spoken about as non-living matter. Speaking about heat as if it were a substance is at odds with the modern ontology of thermodynamics in which heating is only recognized as a process by which energy can be transferred into or out of a system. (See [15, 17] for example.)

2. A detailed metaphorical analysis of “heat” will show that by far the most common metaphor that physicists use involves heat as a substance and an 108

object or system as a container of heat or heat energy. I will refer to this

as the HEAT IS A SUBSTANCE, SYSTEM IS A CONTAINER metaphorical system. This is the same system of metaphors that physicists use to speak about energy as a state function.

My main hypothesis is that students who display particular alternative conceptions related to heat (namely, they think heat is a state function) have a model of heat that is based primarily in language and that the source of their confusion lies to some extent, in the language which physicists use to talk about heat.

• To test this hypothesis I will repeat Meltzer’s interview study, but also include questions to probe students’ understanding of what “heat” means. If students’ conception of heat as a state-function is driven and supported by the language that physicists use, there are two predictions I can make about this interview study:

1. If the ontological categorization of heat matters at all, I predict that students’ ability to categorize heat correctly should affect their reasoning about thermodynamics problems. If this turns out to be the case, I have a basis on which to question physicists’ failure to make the ontological status of heat clear in their language. (It seems plausible to suggest that if physicists are not being precise about the meaning and function of heat, this must be contributing to students inability to make the categorical distinctions clear in their own minds.) If there is no correlation between ontological categorization and problem-solving, I have to conclude that the ontology coded in physicists’ language is unimportant for students’ learning and problem- solving; or the theoretical framework is not correct.

2. As mentioned before, I will show that physicists speak about heat as a state function through the use of the substance-container metaphor. If the use of this metaphor is inﬂuencing students’ reasoning, then students who 109

think that the total heat transfer for a closed thermodynamic cycle is zero

should back up their reasoning with speciﬁc features of the HEAT IS A SUB-

STANCE, SYSTEM IS A CONTAINER metaphorical system. Namely, the notions of heat as a substance, a system as a container of heat, and the amount of heat contained in a system is indicated by the temperature of the system.

5.1.2 Issues Concerning Force in Newtonian Mechanics

In contrast with the example of heat in thermodynamics, the difﬁculties that students have with the idea of force in mechanics are much more complex, but have been studied far more extensively. The language that physicists use to talk about force is also more convoluted. The complexity of the language represents a challenge to the linguistic framework I have proposed. Initially I was unable to identify a set of underlying patterns to physicists’ language. It was only after I studied the historical development of the concept of force [33] that the patterns emerged. Using the historical analogies related to force, I was able to describe four ontological sub-categories (sub-categories of the matter category) for how physicists speak about force in modern times.

I will examine three well documented “misconceptions” amongst students. A misconception should satisfy the following criteria: They should be shared by a reasonable portion of the class, be common in structure and form across different classes, univer- sities and countries, and should be stubborn and resistant to instruction.

The three misconceptions I will examine are:

1. There is no force in a vacuum.

2. Passive objects don’t exert forces.

3. A constant force is required to maintain a constant speed.

I will show that the ﬁrst two difﬁculties (“there is no force in a vacuum”, and 110

“passive objects don’t exert forces”) can be accounted for as difficulties that stem directly from language. The third difficulty (“a constant force is required to maintain a constant speed”), led me to hypothesize a new aspect to students’ difficulties with language. Namely students are having difficulty defining and refining the meaning and function of everyday terms. As I mentioned before in Chapter 3, I have termed this phenomenon ontological groping. Several famous physicists, including Galileo and Newton, display this ontological groping behavior. For this reason I believe it is a natural human behavior (rather than a prior conception that students bring to the class). If we recognize students’ difficulties as such, we may be able to answer students’ questions and facilitate their thinking better.

5.2 Summary of PER Literature on Language About “Heat”

In Section 2.6.2 of Chapter 2 I reviewed the ideas that physics education researchers have had regarding physicists’ language and how it may affect students’ reasoning. When it comes to language about heat, there is consensus in the literature that physicists’ language can be misleading, but little agreement about why it is misleading or how it can be corrected.

What did I learn from that review?

1. The authors reviewed all agree that heat is not a “thing” and therefore cannot be stored or transferred. This conclusion can be found in all 5 papers I reviewed. The agreement on this issue should be emphasized.

2. The second point of consensus is that all the authors reviewed agree (with greatly varying degrees of explicitness) that heat refers to a method by which energy is transferred into or out of a system. Points 1 and 2 can be considered representative of the “expert” view of the role of “heat” in thermodynamics.

3. All the authors have made suggestions that physicists’ language about heat may 111

be confusing students. Bauman mentions that our every-day language may be a problem too [12].

4. Several authors have made suggestions about how we may correct the language. Some want to use heat as a noun [17], others as a verb [56], and some want to deprecate it [12] in the language or ban it completely from existence [15].

Why, for example, is Leff happy to use the term “heat” as a noun, but considers “heat transfer” to be an oxymoron? Why do some authors want to use heat only as a verb, while others are happy to use it as a noun as long as its process nature is made explicit? Physicists have some ideas that our language about heat is problematic. How- ever their objections are neither coherent, nor complementary. In some cases they seem to be contradicting each other. Yet they are often reaching for linguistic intuitions that are well understood in the linguistic community. I will show how a careful grammatical and metaphorical analysis of physicists’ language related to heat in thermodynamic processes can reveal an underlying coherence and ultimately resolve many of the con- ﬂicting suggestions that have been made about how to improve the language about heat. The basis of this will be the point of consensus, namely that heat is not a thing, but a process. If this is so, the language we use to talk about heat should reﬂect the process nature of heating as closely as possible.

5.3 Historical development

5.3.1 History of caloric

The historical development of the concept of “heat” is a relatively simple, but interesting. It is interesting because it reveals some of the powerful ontological commitments of the physicists of the time.

The caloric theory of heat was developed in the eighteenth century by Joseph Black and Antoine-Laurent Lavoisier. The details of this development are not important. 112

Interestingly, Lavoisier is also credited with conducting a series of experiments that lead to the downfall of the phlogiston theory of combustion. It is remarkable that Lavoisier deposed one theory involving a weightless invisible ﬂuid, only to replace it with another [99].

The ﬁrst serious challenge to the caloric theory of heat came from Benjamin Thomp- son, Count Rumford, in the early 19th century. In observing the boring of cannons he recognized the mechanical equivalent of heat and work. He also realized that the cannon boring experiments were incompatible with the caloric theory since the boring processes seemed to generate an inﬁnite amount of “caloric”. As long as you kept boring, heat kept on being generated.

Remarkably, these experiments did not depose the caloric theory at all. For example, Fourier, in his book on heat conduction, overtly acknowledged that there was some controversy over what “heat” really was, but said, for his purposes, he would treat heat as a material substance [99]. He went on to write down a diffusion equation for heat and mathematically analyze its motion. It is clear from this, that thinking of heat as a material substance, had become a productive mode of reasoning for physicists, irrespective of what the “correct” theory was.

5.3.2 Reiﬁcation of Heat from Substance to Process and Substance

Again

Through the nineteenth century, as the idea of energy slowly emerged, physicists began to talk about heat as a form of energy [25]. What is important to notice from this is that although the analogy to a ﬂuid has disappeared, the ontological status of heat in physicists’ language has remained unchanged. In other words, they continued to talk about heat as a substance, but now as a metaphor, namely as a form of energy.

Only in the late nineteenth century, once kinetic theory began to take shape, physicists began to see heat as a process by which energy could be transferred. Although 113

there was a shift in their understanding, there is very little evidence of a shift in their language. In a lot of the writing of the time, the physicists seem to have simply dropped the word “heat” almost entirely from their language. In cases where “heat” is used, it was still spoken about as if it were a material substance [99].

The term “heat” slowly evolved into a “dual use” term [81]. When Clausius wrote down the ﬁrst law of thermodynamics, he still needed a term to account for the amount of energy ﬂowing into or out of a system by the heating process. So the term “heat” remained in that context. Physicists were able to use “the amount of heat transferred into the system” as a short-hand for “the amount of energy transferred into the system by the heating process.” I will show in Section 5.4 below that this short-hand is representative of the language that modern physicists use to talk about heat.

5.4 Modern Language about Heat

5.4.1 Grammar and Ontology

Firstly, do modern physicists speak about “heat” in a systematic way? To answer this question, I surveyed three popular and widely used college level introductory physics textbooks [54, 81, 82]. Williams has argued that such textbooks represent a higher standard of linguistic rigor than the regular talk of physicists [16]. Thus a study of textbooks can at least give us an upper bound on the quality of language used to refer to the concept of heat. From these textbooks, I extracted every sentence that used the word “heat” in it. This amounted to about 1200 clauses in total. To understand the types of possible meaning associated with the word “heat” in these clauses, I mapped the grammatical function of “heat” to the basic ontology scheme suggested in Fig. 3.2 of Chapter 3 using the mapping outlined in Table 3.3. Examples of the mapping and ontological classiﬁcation are given in Table 5.1.

The results of the ontological analysis are presented in Fig. 5.1 [100].

There are two important things to be noted from the analysis summarized in Fig. 5.1. 114

Table 5.1: A representative sample of clauses from three popular introductory college level physics textbooks containing the word “heat”, grammatical analysis of each clause, and ontological classiﬁcation of “heat” according to grammatical function. Clause (grammatical functions indicated) Ontological classiﬁcation of heat

“. . . cylinders [medium] are heated [process]. . . ” Process “. . . energy [medium] needed to heat [process] the water Process [range]. . . ” “. . . heat [medium] ﬂows [process] from. . . hotter cup Matter:non-living [circumstance:location]. . . into. . . cooler hand [circumstance:location].” “. . . heat [medium] is conducted [process] Matter:non-living through. . . material [circumstance:location]. . . ” “. . . stove [medium] is [process] unheated [modiﬁer]. . . ” State:physical “. . . water [agent]. . . loses [process] heat [medium] by ra- Matter:nonliving diation [circumstance:manner]. . . ” “The heat [medium] rejected [process] by this engine Matter:non-living [agent]. . . ” “. . . state whether [heat] [medium] is absorbed by or given Matter:non-living off [process] by the gas [agent].” “. . . hot reservoir [agent]. . . delivers [process] heat Matter:non-living [medium] to each engine [circumstance:location]” “. . . energy [medium] [is] transferred [process] to the ice Matter:non-living [circumstance:location] as heat [circumstance:role]. . . ” “Heat [medium]. . . can be added to or withdrawn [pro- Matter:non-living cess] from the gas [circumstance:location]. . . ” “. . . objects [agent]. . . can exchange [process] energy Matter:non-living [medium] as heat [circumstance:role]. . . ” “. . . energy [medium] cannot escape [process] by Process heat [circumstance:manner] from its surface [circumstance:location]. . . ” “. . . the energy [medium] added [process] to the gas [cir- Process cumstance:location] by heat [circumstance:manner]. . . ” 115

100 89 Cutnell - 726 83 81 clauses 80

Halliday - 245 60 clauses

Serway - 213 40 clauses centage of clauses 20 Per 9 10 5 5 6 8 2 0 2 0 Matter Process State Unclear Ontological Metaphor

Figure 5.1: Classiﬁcation of heat clauses into ontological categories.

1. The grammatical/ontological analysis is able to reveal a clear underlying pattern in physicists’ speech and writing. A naïve metaphorical analysis (following the conceptual metaphors of cognitive linguistics) of the following three phrases, (1) “. . . heat ﬂows from. . . hotter cup . . . into. . . cooler hand,” (2) “. . . hot reservoir. . . delivers heat to each engine,” (3) “. . . objects. . . can exchange energy as

heat. . . ,” would yield the following three metaphors: (1) HEAT IS A LIQUID,

(2) HEAT IS A COMMODITY, (3) ENERGYISACOMMODITY. The grammatical/ontological analysis shows that all three phrases share the same underlying

grammatical/ontological metaphor, namely HEAT IS A SUBSTANCE.

2. I included an analysis of Serway and Beichner to illustrate that is really is possible to speak about heating as a process. A brief analysis of three other textbooks [83, 84, 85] revealed a pattern of usage for the term “heat” similar to that found in Cutnell and Johnson, and Halliday, Resnick and Walker. 116

By far the most dominant sort of sentence, used by Serway and Beichner, took the following form: “. . . the energy added to the gas by heat. . . ”. To remove any possible confusion with passive voice, which might leave the naïve reader to interpret “heat” as an agent, the authors might do even better to write “. . . the energy added to the gas by heating. . . ”.

5.4.2 Subtle Ontological Distinctions in Heat Deﬁnitions

It is interesting to examine six definitions of heat from six popular textbooks. I have classified the first four definitions as “operational” definitions of heat. Namely the textbooks define heat as “energy in transit.” (The physicists’ short-hand for the amount of energy transferred in the heating process.) The last two definitions are “process” definitions since they explicitly define heat as a transfer of energy. (See Table 5.2.)

Table 5.2: Heat definitions from six popular introductory college level physics textbooks. Operational Definitions “Heat is energy that flows from a higher-temperature object to a lower-temperature object because of the difference in temperatures.” [82] “Heat is the energy that is transferred between a system and its environment because of a temperature difference that exists between them.” [54] “Heat is energy that is transferred from one system to another because of a difference in temperature.” [84] “. . . we will define heat as follows: Heat is the energy transferred between objects because of a temperature difference.” [85] Process Definitions “. . . scientists came to interpret heat not as a substance, and not even as a form of energy. Rather, heat refers to a transfer of energy: when heat flows from a hot object to a cooler one, it is energy that is being transferred from the hot to the cold object.” [83] “Heat is the transfer of energy from one object to another object as a result of a difference in temperature between the two.” [81]

Compare the statement, “heat is energy that ﬂows/is transferred. . . ” against “heat is the transfer of energy”. I suggest that statements which suggest heat is synonymous with energy, imply that heat should be classiﬁed ontologically as matter (a form of 117

energy). Statements which suggest that heat refers to a transfer of energy, make the process nature of the term explicit. Giancoli’s deﬁnition of heat [83] is not upheld by subsequent language in the same textbook. A cursory examination of the text reveals a predominantly substance-based language when referring to heat. Serway and Beichner are the only authors of the group who use a process-based language and who suggest the possibility of dual meanings associated with heat when they warn the reader: “We shall also use the term heat to represent the amount of energy transferred by this method” [81].

My examination of more of the language from these textbooks showed a clear pattern of ontology in the language used to talk about “heat”, “work” and “energy”. This pattern of ontology is compared against the accepted lexical ontology in Table 5.3 below.

Table 5.3: Comparison of modern ontology of thermodynamics against the model ontology encoded in the grammar of physicists’ language. Matter Process Physical State Physicists’ model ontology (the way they think) atoms/molecules, or sys- heating, working positions and momenta tem of atoms/molecules of the particles, temperature, energy, entropy etc. . . Physicists’ linguistic model ontology (the way they speak) heat, work, energy, is transferred, ﬂows, is temperature, amount of atoms/molecules rejected, is absorbed, is energy/heat in object. added, is expelled

5.4.3 Prior Research on Students’ Ontological Commitments

As I mentioned in Chapter 2, it has already been suggested that novice physics students enter a physics course with naïve conceptions that “reﬂect an underlying commitment to existing knowledge of material substances” [68]. Miriam Reiner et al. hypothesize that this schema is developed from early childhood since our experience of the world is naturally substance based [68]. We live in a world of physical objects that undergo 118

processes of movement. This ontological commitment has been shown to dominate students’ conceptual understanding. For example, James Slotta, Michelene Chi, and Elana Joram [2] showed that students’ conceptual understanding of “heat” was dom- inated by substance-based conceptions. Chi et al. have hypothesized that conceptual change involves the re-categorization of a concept such as “heat” from the ontological category of matter to the ontological category of processes [5].

At this point I need to re-examine the work of Slotta, Chi, and Joram [2]. The authors suggest a theoretical model for conceptual change as a process of ontological re-categorization. This idea is based on observations that students with robust “misconceptions” often hold substance based notions about physics concepts (“heat” in particular). Expert physicists on the other hand, would regard “heat” as the name for a process of atomic or molecular excitations by which energy is transferred from one place to another. Thus true understanding is achieved when the student reclassiﬁes the concept of “heat” into the process category. The authors predict that “verbal predicates” of novices and experts would reveal a difference in ontological categorization. They ﬁnd that this is indeed so, but at the price of using different units of analysis for the novices and experts. Below I will reanalyze some samples of their data (presented in Table 5.4) using the theoretical framework I have developed. Applied to the experts and novices, a grammatical analysis will reveal little or no difference in how they speak about the concept of “heat”.

The grammatical analysis goes as follows: For novice 1: “. . . heat [medium]. . . is gonna escape [process]. . . ” Novice 9: “. . . air bubbles. . . can absorb [process] the heat [medium]. . . ” Expert 4: “. . . heat energy [medium] is being transferred [process] into. . . ” Expert 3 poses a slight difﬁculty in that he/she is using the phrase “transfer of heat” as a nominal group which functions as the medium. In order to code the function of “heat” by itself the clause needs to be “denominalised”. Thus in this case, as in all cases before, I will treat examples like “. . . the transfer of heat [Nominal group, Medium] is [Process] not as efﬁcient in glass. . . ” as equivalent to “the heat [Medium] 119

Table 5.4: Comparison of sample expert-novice data from Slotta et al. [2] Novices Experts Novice 1: . . . the coffee in the ceramic Expert 3: . . . the transfer of heat is not mug is hotter than that in the styrofoam as efﬁcient in glass, because . . . metal mug, because the heat in the styro- has, ah, more of a crystal structure so, foam cup is gonna escape, because it’s we can have, ah, phonons which just not like, a stryfoam cup is not totally travel along the lattice, moving up the sealed because there’s, like styrofoam lattice quicker. Um, phonons are really has little holes in it, so it, the heat’s just lattice vibrations. gonna go out, escape in the holes so, and the ceramic cup doesn’t have, its just totally sealed tight. . . Novice 9: So if you put something hot Expert 4: . . . heat is a form of energy in there, it would trap the heat better . . . that is being used to agitate these than something ceramic, because ce- atoms even further, meaning that heat ramic doesn’t have like air bubbles in energy is being transferred into kinetic it that can absorb the heat or the cold- or rotational or to even further linear ness. . . kinetic energy or translational kinetic energy. . .

is not transferred [Process] as efﬁciently in glass. . . ” Linguists would regard this as the natural way in which the producer of the sentence would intend the meaning of the nominal group to be interpretted as it refers to some process by which heat is transferred.

If we examine the experts and novices on a level above that of the clause (i.e., on the level of several sentences together) it is quite clear that experts are able to elaborate the details of the heating process, but as we can immediately see, their ontological treatment of the word “heat” seems to be grammatically identical to the novices. It is difﬁcult to judge from the small samples provided, but it appears that experts use the

HEAT IS A SUBSTANCE metaphor less frequently than the novices. I must emphasise that I am not trying to say that experts don’t know what they are talking about. It is clear that they understand heat is a process and they are able to elaborate that process, while novices are not; I am trying to show that their usage of the word “heat” is not reﬂecting that understanding.

In summary, Slotta et al. have observed students’ ontological difﬁculties with heat. 120

I can go further and hypothesize that even if physicists’ language is not the sole cause of this confusion, it is certainly doing little to alleviate it.

5.4.4 Metaphors

As I pointed out in Section 5.1.1, grammar and elaborated imagery work together to produce a metaphorical realization of a physical state in language. The pattern may be seen by referring back to the sample clauses in Table 5.1. For example:

“. . . heat [medium] ﬂows [process] from. . . hotter cup [circumstance:location] . . . into. . . cooler hand [circumstance:location].”

The systems that “contain heat” function grammatically as circumstances of location. These grammatical functions are cued and enhanced with “container” prepositions such as “from”, “to”, “in”, and “out”; as well as verbs (processes) such as “absorbs”, “rejects”, “expels”, etc. . .

Table 5.5 presents a fairly exhaustive list of words that (by the base domain in which they are grounded) cue the presence of the various parts of the metaphorical system.

Reexamining the same data from the three textbooks, I was able to identify six basic metaphors which are used to describe either the movement of heat or the movement of energy by heating in thermodynamic processes. Using Table 5.5, I identiﬁed the metaphor in each clause. The results of the coding are shown in Table 5.6.

By far the dominant metaphor in Cutnell and Johnson, and Halliday, Resnick and

Walker, is HEAT IS A SUBSTANCE THAT MOVES FROM SYSTEM TO SYSTEM AND

THESYSTEM/GAS IS A CONTAINER OF HEAT. This pattern is matched by other textbooks [83,84,85], that were not coded in as much detail. Again, Serway and Beichner were the only exception to the rule. 121

Table 5.5: A list of words that cue certain metaphors Part of Nouns Verbs Prepositions Metaphor Substance amount, quantity Movement ﬂows, moves, between is transferred, pumping, conducted, radiates, is carried, dis- tributes Container reservoir is added, ab- in, into, out, out sorbed, dis- of, to, from charged, expelled, rejected, removed. gain, lose, enters, leaves, extracted

Table 5.6: Metaphorical classiﬁcation of heat clauses Metaphor Cutnell Halliday Serway 1 HEAT IS A SUBSTANCE 13% 13% 6% 2 HEAT IS A SUBSTANCE THAT MOVES 17% 18% 1% 3 HEAT IS A SUBSTANCE THAT MOVES 53% 58% 2% FROMSYSTEMTOSYSTEMANDTHESYS- TEM/GAS IS A CONTAINER OF HEAT 4 HEAT IS A PROCESS 10% 5% 26% 5 HEAT IS A PROCESS WHICH INVOLVES THE 0% 0% 16% MOVEMENT OF A SUBSTANCE (ENERGY) 6 HEATISAPROCESSINWHICHASUB- 0% 0% 39% STANCE(ENERGY) MOVESFROMONESYS- TEMTOANOTHERANDTHESYSTEM/GAS IS A CONTAINER OF THAT SUBSTANCE 7 Unclassiﬁed 7% 6% 10% 122

5.4.5 Summary

The complete metaphorical mapping from the base domain of material substances and containers to the target domain of heat in thermodynamic processes is summarized in Table 5.7.

Table 5.7: Summary of the metaphorical mapping between the base domain of material substances and containers, and the target domain of heat in thermodynamic systems Base domain Target domain MATERIAL SUBSTANCES AND HEATINTHERMODYNAMICSYSTEMS CONTAINERS Material substances −→ Heat, or heat energy Physical containers −→ Thermodynamic systems. E.g., a gas, a metal rod, a “heat reservoir”, etc. . . Movement processes, e.g., ﬂow, −→ Process by which the system internal en- transfer, etc. . . of substances ergy (a state-function) changes due to into/out of containers heat contributions.

The analysis presented in the sections above shows that this single metaphorical system (Table 5.7) dominates physicists’ language about heat and is used very systematically. It is a metaphorical system grounded both in a base domain of material substances and containers, as well as a metaphorical blend with a metaphor that humans use to speak about mental states. The metaphor, in which the person is spoken of as a container of a feeling or an emotion, is common to all western European languages and may well extend beyond that.

5.5 Students’ Difﬁculties: Results and Analysis of the Interview

Study

5.5.1 Description of the Study

I conducted an interview study with 10 physics honors students. All students had completed a minimum of 2 semesters of physics (some were more advanced), had had an exposure to thermodynamics, and had a B+ or better grade in their physics courses. 123

I used the same interview questions as David Meltzer used in his 2004 study. The interview questions may be found in [51]. I also added questions to probe students’ understanding of the meaning and role of heat in thermodynamic processes. The interview used a “think aloud” protocol. Students were asked to respond to the questions and in each case try and express their reasoning verbally as best they could. If students fell silent, I encouraged them to say what they were thinking. Students were also invited to write on paper if they felt it would help them. I allowed students to revisit each question once they had gotten to the end of the interview and invited them to reconsider any contradictions or confusion that may have arisen.

The interviews generated about 10 hours of audio tape and 100 pages of transcript. I narrowed my analysis to students’ responses to question 6 part (ii), and to their deﬁ- nition of heat. Q.6 (ii) went as follows:

Consider the entire process from time A to time D. (ii) Is the total heat transfer to the gas during that process (a) greater than zero, (b) equal to zero, or (c) less than zero [51]?

One student’s transcript (S8) was lost. Certain aspects of his responses were reconstructed from interview notes. Thus S8 is included in some of the analysis, but excluded from parts that involved a detailed grammatical analysis of the transcripts.

5.5.2 Students’ Deﬁnitions of Heat

I classified students’ definitions of heat into three categories. These were called caloric, operational and process. (Cal., Op., and Proc.) These three definitions presented themselves quite obviously from the data. The caloric definition involved a description of heat as a separate entity or heat as a form of energy without restricting the definition to energy movement. The operational definition is the one that often appears in introductory physics textbooks (see Table 5.2). Namely, heat refers to the amount of energy that is transferred to or from a system during the heating process. The process definition only qualified as such if a student could clearly articulate that heat was a process 124

of energy transfer or heat referred to a transfer of energy rather than energy that is transferred. Examples are shown in Table 5.8.

Table 5.8: Students’ definitions of heat Caloric Definition S1: Well, the temperature, which is a measure of the heat. . . of the system . . . I guess I’m trying to think of what heating means. Does that mean like bringing extra heat to the system, or like allowing the system to create heat, because I’m thinking like when you press something in together, like when we press it in, that was like creating more heat, but it wasn’t like we were adding heat. It was being created because. . . S2: temperature is a good indicator of heat. . . Heat is actual energy that gives the molecules the kinetic energy. . . Temperature’s the same, pressure’s the same, so I’m saying it started and ended with the same amount of heat. S5: They [heat and temperature] are directly proportional. If you add heat, you increase temperature, and here heat is added in like when the water is heated up and at the end when its left to cool back down, it releases heat back to the environment. S6: Heat is the energy of. . . from the gas particles moving around. It’s the average kinetic energy so it would be the total kinetic energy of the gas. DTB: Are you saying that the temperature is an indicator of the amount of heat in the system? S9: Yeah, of course! Operational Definition S3: I guess. . . You think about it in uh. . . [heat is] just the energy that’s transferred that causes the temperature to rise. S4: I’m not exactly sure what it means. . . for heat, all I know there is a specific quantity in the equation, I don’t really understand what it is. . . Process Definition S7: It [heat] is just a type of energy. Not even a type of energy. A way of transferring energy from one system to another by thermal contact. S8: Process definition. (Transcript lost.) S10: Well, its the term heat. . . I think I am confusing the term heat somehow. I think I’m just making it equivalent to temperature. . . because its a transfer, heat is a process, transferring energy.

In my analysis, I compared the students’ deﬁnitions against their ability to answer one particular question (Q6 (ii)) correctly. This comparison is shown in Section 5.5.4, Table 5.10.

5.5.3 Responses to Question 6 (ii)

Question 6 (ii) asked students, 125

Consider the entire process from time A to time D. (ii) Is the total heat transfer to the gas during that process (a) greater than zero, (b) equal to zero, or (c) less than zero [51]?

In addition I asked students to explain carefully why they were reasoning the way they were. Two examples of students’ responses to Q.6 (ii) are given below.

The first example is of a student (S9) who reasoned that the total heat transfer for the cycle is zero. His response and reasoning is typical of those who gave the same answer: [13:00] S9: So first the water was heated and heat was transferred to the gas. Then the gas was compressed, so heat was transferred from the gas back to the water. So heat transfer, total heat transfer to the gas is also equal to zero. Why I say this is: In the first case the net work done by the gas on the environment. . . so I say it is equal to zero because, the first step. . . so we said that we heated the water and heat was transferred from the water to the gas and then again during step 4 we allowed it to cool down. We said that we are compress- ing the gas so the heat was transferred back from the gas to the water. So one way and the other way. So they cancel out. So the net work is. . . is zero. And then the next question is total heat transferred to the gas is again the same thing as net work done. The total heat transferred, again it is zero because first there was transfer of heat from water to gas and there was transfer of heat from gas to water. Since we are assuming an idea system without any losses so total heat transferred. . . [26:40] S9: So when there was transfer of heat from the water to the gas and then back from the gas to the water, so I guess they should cancel out each other. Because the magnitudes should be the same, just the direction was different. DTB: So what you are saying basically is that it ended up at the same temperature as it started out with so whatever heat came in must have gone out. S9: . . . gone back. [Revised answer: 45:42] S9 So initially gas gained heat from the water. Then. . . [mumbles to himself] So first heat was transferred from the water to the gas. Then the piston moved out, okay, compressed, then heat generated was transferred back to the water. Okay. So let us say some. . . Let’s take a numerical example. Let’s say 20 J or whatever, was transferred from water to the gas. Then this was spent in moving the piston. Then we did some work to push the piston in. So the question is whether the work [unintelligible] is the same as this 20 J. DTB: There is also a third step where you lose heat remember? The piston is locked and more heat flows out. There’s heat in and 2 heat outs. S9: [Reads to himself] Okay, so if 20 J was transferred to this, then the gas is losing heat. When you compress it, losing heat to the water. During the entire 126

process. . . So we allow it, to lock it in this place and allow it to cool right? So temperature returns back to the same as time A. [Reviews the steps again, noting temperatures.] S9: The total heat transfer to the gas. . . The gas gained heat, of course. Then it spent the heat. I guess again, whatever heat, the gas gained, it should have spent. . . like is there any other source by which the temperature can, like it can spend more temperature than it gained? So I guess it must be the same. DTB: Again, you are basing that argument because the temperature is an indicator of the amount of heat in the system? S9: [Repeats statement] Yeah.

The second example is an example of a student (S7) who was able to give the correct answer with correct reasoning.

[11:00] S7: So now I am thinking since the temperature is the same as in time A I know that the change in internal energy is zero for the whole process. And I also know that the volume is the same as the initial volume. But by the first law of thermodynamics, if the change in internal energy is zero then the magnitude of the heat has to be equal to the magnitude of the work, with opposite signs. I’m just trying to figure out if they are both zero or both non-zero. [13:57] S7: I’m going to say that the heat is negative. First we heat the system. Then the system has to release heat in order to. . . for the system to compress. And later it has to release heat in the end to come back to the initial temperature. So intuitively I would say that the heat transfer is less than zero, which means that the net work done by the gas on the environment has to be greater than zero. [Second look at Q6 (ii): 21:31] [Here Q1,Q2 and Q3 refer to the heat transfers in steps 1, 2 and 3; likewise for work.] DTB: Why should W1 be greater than Q1? S7: Because ∆U is positive and so ∆U = Q + W that is if the system doesn’t have a very important change in potential or kinetic energy. Since ∆U is positive and W is negative, Q is positive so W = ∆U − Q. So let me say it like this: if W = Q in magnitude then ∆U would have to be zero. So, actually this is wrong, this is completely the other way because since ∆U is positive, then Q must dominate over W , so Q must be greater than W . So it was the other way around. And this means that [unintelligible] DTB: Why do you say that? S7: Because Q1 is less than Q2 + Q3. DTB: How do you know that? S7: If the first heat is less than the sum of the other two then I am trying to prove that the heat is negative. I know that the W of the first process is less than the Q of the first process. So if this were true. If Q1 were equal to the sum of the other Q’s, then W1 would be less in magnitude than the sum of the other Q’s. Now we 127

know that Q2 in magnitude is equal to W2 because the change in internal energy of the second process is zero because the temperature doesn’t change. [Third look at Q6 (ii)] S7: So the work done by the gas because of equilibrium, the work 1 has to be FA∆x. Now in process 2 the weight of the piston has also the weight of the leads on top of it. So its FB which is greater than FA. So W2 = FB∆x. And the ∆x is the same so W2 should be greater than W1. Which makes the net W greater than zero and Q less than zero.

My analysis of Q.6 (ii) went as follows. First, I ranked how well the students were able to answer the question. Four groups of solution emerged from my ranking. They are presented below from best (rank = 1) to worst (rank = 4)

1. (Rank = 1.) Two students were able to arrive at a correct solution with some difﬁculty.

2. (Rank = 2.) One student seemed to have the right idea, but was unable to get to a correct solution.

3. (Rank = 3.) Two students had some idea of how to answer the question but were unable to make much headway towards a solution.

4. (Rank = 4.) Five students were unable to answer the question correctly arguing that the total heat transferred for a closed cycle must be zero.

In order to check the reliability of my ranking, I gave the student responses to Q.6 (ii), printed on index cards, to a second rater who knew nothing about the linguistic component of my analysis. I asked the rater to rate the quality of the solutions. The rater grouped the students’ responses into ﬁve groups listed from best to worst:

1. (Rank = 1.) One student solved the problem correctly (remember that S8’s transcript was lost and therefore could not be ranked by the second rater.)

2. (Rank = 2.) One student almost got the right solution, but was somewhat confused. 128

3. (Rank = 3.) Two students had good ideas, but were unable to get a correct solution.

4. (Rank = 4.) One student has some idea but did not get very far.

5. (Rank = 5.) Four students had no idea how to solve the problem.

A comparison of the two ranking is shown in Table 5.9.

Table 5.9: Rankings of students’ ability to solve Q.6 (ii). Student Q.6 (ii) Rater DTB Q.6 (ii) Rater 2 S7 Yes Yes S8 Yes Unranked S3 Almost Almost S4 Some idea Good ideas S10 Some idea Good ideas S6 No (Q=0) Some ideas S1 No (Q=0) No S2 No (Q=0) No S5 No (Q=0) No S9 No (Q=0) No

Next, I examined the transcript of each student and counted the number of times, in their explanation of Q.6 (ii), they invoked the metaphor of heat as a substance contained within a system. Such instances had to satisfy both the ontology of heat as matter and involve the use of a container metaphor associated with the system in question. I then compared these counts against their ability to solve the problem. The analysis is shown in Section 5.5.4, Table 5.11.

5.5.4 Analysis

Analyzing the data poses certain difﬁculties. I want to see whether students’ ability to deﬁne the term “heat” correctly, and the number of times they invoke the HEAT IS

A SUBSTANCE, SYSTEM IS A CONTAINER metaphor correlates with their ability to solve Q.6 (ii). However, it is impossible to objectively attach a numerical value to 129

somebody’s deﬁnition. Likewise, for the other measures; solution quality and number

of invocations of the HEAT IS A SUBSTANCE, SYSTEM IS A CONTAINER metaphor. I decided therefore to analyze that data using the Spearman rank order correlation coefficient. In both comparisons, I cross-checked Spearman rank order correlation coeffi- cients with the more standard Pearson product-moment correlation coefficient. There is some question about the statistical reliability of the Spearman correlation coefficient when there are several items in the same rank.

To compare students’ ability to define the term “heat” against their ability to solve the problem, I assigned a rank of 1 to a process definition of heat, rank 2 to an operational definition of heat and rank 3 to a caloric definition of heat. Quality of solution (Q.6 (ii)) was ranked in numerical order according to the rater’s grouping. This comparison, with ranks, is shown in Table 5.10.

Table 5.10: Rankings of students’ ability to solve the problem (Q.6 part (ii)) compared with their ability to define the meaning of heat. (Process definition (Proc.), operational definition (Op.), or caloric definition (Cal.).) Spearman rank order correlation coefficient: DTB: rs = 0.9364, p = 0.00007 (2-tailed t test), Rater 2: rs = 0.9053, p = 0.0003 (2-tailed t test). Pearson product-moment correlation coefficient: DTB: r = 0.936, p = 0.0001 (2-tailed t test), Rater 2: r = 0.905, p = 0.0003 (2-tailed t test). Heat Defi- Q.6 (ii) Q.6 (ii) Rank Rank Rank nition Rater DTB Rater 2 S8 Proc. 2 Yes 1.5 Unranked 1.5 S7 Proc. 2 Yes 1.5 Yes 1.5 S10 Proc. 2 Some idea 4.5 Good ideas 4.5 S3 Op. 4.5 Almost 3 Almost 3 S4 Op. 4.5 Some idea 4.5 Good ideas 4.5 S1 Cal. 8 No (Q=0) 8 No 8.5 S2 Cal. 8 No (Q=0) 8 No 8.5 S5 Cal. 8 No (Q=0) 8 No 8.5 S6 Cal. 8 No (Q=0) 8 Some ideas 6 S9 Cal. 8 No (Q=0) 8 No 8.5

To compare students’ solutions to Q.6 (ii) against their use of the HEAT IS A SUB-

STANCE, SYSTEM IS A CONTAINER metaphor, I assigned numerical ranks to the solutions in the same way as shown in Table 5.10. I then counted the number of times that each student used the HEAT IS A SUBSTANCE, SYSTEM IS A CONTAINER metaphor to 130

justify their reasoning and assigned a rank based on the relative number of instances of the metaphor. In other words, the least number of instances of the metaphor got rank 1 and so on. The analysis is shown in Table 5.11.

Table 5.11: Comparison of students usage of the HEAT IS A SUBSTANCE, SYS- TEM IS A CONTAINER metaphor in their responses to Q.6 (ii), against their ability to solve Q.6 (ii). Spearman rank order correlation coefﬁcient: DTB: rs = 0.8165, p < 0.05 (non-directional test), Rater 2: rs = 0.8656, p < 0.05 (non-directional test). Pearson product-moment correlation coefﬁcient: DTB: r = 0.817, p = 0.0072 (2-tailed t test), Rater 2: r = 0.866, p = 0.0026 (2-tailed t test). Instances of Q.6 (ii) Q.6 (ii) substance-container Rank Rank Rank Rater DTB Rater 2 metaphor S7 2 3 Yes 1 Yes 1 S3 1 2 Almost 2 Almost 2 S4 3 4 Some idea 3.5 Good ideas 3.5 S10 0 1 Some idea 3.5 Good ideas 3.5 S6 5 5 No (Q=0) 7 Some idea 5 S1 6 6 No (Q=0) 7 No 7.5 S2 7 7 No (Q=0) 7 No 7.5 S9 19 8 No (Q=0) 7 No 7.5 S5 21 9 No (Q=0) 7 No 7.5

5.5.5 Discussion

What are these data telling me?

1. It is important to note that the data do not present a cause effect relationship between the language that students hear and their ability to understand what “heat” means and their ability to solve thermodynamics problems. Such a result is beyond the scope of this study. However, I believe that the data are sufﬁciently compelling to warrant further study.

2. The data show a strong correlation between students’ ability to make surprisingly subtle ontological distinctions and their ability to solve certain thermodynamics problems. This supports Chi’s hypothesis that the ability to classify concepts 131

ontologically is at the heart of students’ ability to reason. It also supports my hypothesis that the linguistic cues distinguishing heat as a process from heat as a substance lie in the ordering of the words in the sentence. In other words, grammar can be seen as the key to ontology. Note how S7 and S10 are able to construct an explanation with the phrase “heat is a transfer of energy”, while other students say that “heat is energy” or “heat is energy that is transferred. . . ”. It is now plausible to suggest that the ontology encoded in physicists’ language may have an effect on students’ reasoning.

3. Students’ use of the HEAT IS A SUBSTANCE, SYSTEM IS A CONTAINER metaphor is strongly correlated with their idea that heat is a state function of the thermodynamic system. The four students who unambiguously argued that Q = 0 for

a closed thermodynamic cycle invoked the HEAT IS A SUBSTANCE, SYSTEMIS

A CONTAINER metaphor the most number of times. This supports my hypothesis that students are imagining heat as a state-function based on a metaphor in which physicists speak about heat as a substance contained within an object (the thermodynamic system.) This is the same metaphor that physicists use to speak about energy as a state-function.

It is also important to note that this result is somewhat surprising. Experts speak about “heat” in thermodynamic processes as if it were a substance moving into and out of systems, while still being able to understand what heat really means and solve thermodynamics problems without difﬁculty. Naïvely, one might predict that students who showed mastery of the subject matter would be indistin- guishable from the novices in terms of their use of language. Rather surprisingly, these data show that students who solved Question 6 part (ii) successfully were much more careful with their use of language. In retrospect, this result is not as surprising as it seemed to me at ﬁrst. Several physics education researchers and cognitive scientists have postulated an intermediate state of knowledge on the road to expertise. (See [101, 102] for example.) 132

5.6 Introduction to Force

In this section and the sections that follow, I will apply the linguistic framework to consider the use of the word “force” in physics texts. In a manner similar to the previous sections, I will code the usage of the word “force” from three introductory physics textbooks. Then I will examine the rest of the sentence structure as it is used to describe how objects interact with each other. I will then review some common student “misconceptions” documented in the PER literature. The strategy will be the same as before. First I will identify the models encoded in the language that physicists use to talk about “force”. Then I will look for evidence (by considering the students’ responses and explanations) that they are often justifying their answers based on models which are directly supported by or derived from physicists’ language. The question again remains: does the language used by physicists when talking about “force” cause the students to adopt the incorrect models? Or does the language simply reﬂect underlying conceptions about force and motion which stem from physical experience and social interaction? What is clear though is that, although physicists may associate precise meanings with their language, their language about force is functionally the same as colloquial usage. What do I mean by “functionally”? I mean that force functions in an identical manner grammatically — either as the medium of the sentence or as an agent. Thus I will argue that at the very least, carelessly used, the language of physicists supports students’ naive models rather than disabusing them of these notions.

The example of how physicists talk about heat and thermodynamic processes is somewhat different from what will be explored in these sections. The example of heat is one in which physicists simply talk inadequately about heat, endowing it with the ontological status of a substance rather than a process. In a sense, the linguistic representation contradicts the expert model on a fairly fundamental level. The way physicists speak and write about force and motion is, however, much more complex and difﬁcult to unravel. The origins of our language about force cannot be traced back 133

to one unique analogy, nor is there an identifiable “correct” and “incorrect” way to talk about force. Some ways are definitely better than others. But the fundamental issue is more that language simply does not support an accurate representation (from a physicist’s perspective) of force and motion. The result seems to be a number of different representations which are seemingly haphazardly cobbled together, sometimes with one sentence containing two or more distinct metaphors in it. There are, however, some clear themes that will emerge. In contrast with the previous sections, I will begin the analysis of force with a much more detailed analysis of the historical record. I need to examine first the history of the development of the concept of force. The purpose of this is to identify the origins of the many themes that appear in the modern language. At first glance it appears that the modern language is incoherent, but viewed from the historical perspective, an underlying coherence will emerge.

5.7 Historical Analogies

The way in which modern physicists speak and write about “force” presented me with a two-year puzzle. The way in which physicists use “force” grammatically, and the many associated conceptual metaphors, seemed to contradict each other and appeared incoherent and inconsistent. This posed a serious challenge to the theoretical framework I developed because I claimed that physicists use systems of conceptual metaphors coherently and consistently.

The solution to this challenge was to turn to the historical record. Since I hypothesized that each conceptual metaphor is based on part of a historical analogy; then if I can identify the original analogies associated with “force”, this might provide the basis to identify if there really are coherent systems of conceptual metaphors that physicists use to speak about “force”.

I approached this problem with the same grounded theory approach described in 134

Chapter 3. My data source was Max Jammer’s book, “Concepts of Force” [33]. Jam- mer’s book describes in detail the historical development of the idea of “force” and how it is bound up in notions of causality. I read and made notes, seeing if one or more analogical categories would emerge.

Examining the history of “force” not only turns out to be a productive exercise in trying to understand how modern physicists view the force concept, but also gives us insight into the development of our every-day language about forces and more generally about causation. The full analysis of Jammer’s text can be found in Appendix C.

5.7.1 The Four Force Analogies

Four fundamental analogies emerged from my analysis of Jammer’s text. The four analogies were:

1. Force is like an animate external entity or agent. This is an analogy to a god or an omnipotent being, or any living external agent. For example, the ancient Greeks wrote about a “world forming spirit”, Aristotle wrote about a “prime mover”.

2. Force is like an passive external medium of interaction. This may be further sub- divided into (a) an analogy to the exchange of a substance between two objects, and (b) an ethereal medium that transmits the interaction between two objects. An analogy is often made to the transmission of sound waves through the air. For example, Crates of Mallos hypothesized that the gravitation effect of the moon on the Earth was transmitted by the air. Ancient Arab philosopher, Al-Kindi hypothesized that force was transmitted by rays of light or heat.

3. Force is like an internal animating spirit. The behavior of an object is explained by an analogy to some human desire. The object is conceived of as possessing a desire to do something. The original analogy is often made to some “internal animating spirit”, or “soul”. For example, Plato’s ideas of attraction and repulsion, grounded in the idea of human desires. 135

4. Force is like a passive internal tendency. The idea of cause is interpreted analogically as an object simply doing what it naturally does. I will call this the “natural tendency” analogy. For example, the ancient Greeks believed that rocks fell towards the ground because they naturally tended to reunite with the Earth. Air bubbles rose in order to reunite with the air.

5.7.2 Historical Development

In Appendix C I present a detailed account of how physicists and philosophers, from the middle ages to the 18th century, tried to develop models of how objects interact and move, based on one of these four force analogies. One can also observe how the writing of these researchers used metaphors that were grounded in one or more of the four analogies. From this analysis, it is clear that each of the four analogies (coded as metaphors) came to represent a useful way of speaking about forces, interaction and motion.

The four basic force metaphors (based on the four analogies) are:

1. FORCEISALIVINGENTITY. This metaphor is grounded in our perception of the world in which living agents can act autonomously and cause change in the environment around them. For example, physicists say “a force acts on an object”.

2. FORCEISAPASSIVEMEDIUMOFINTERACTION. This metaphor is grounded in human perception of how change can be effected through the exchange of an object or a commodity. For example, physicists say “object A exerts a force on object B.”

3. FORCEISANINTERNALDESIREORDRIVE. This metaphor is grounded in our perception of how human emotion and desire can cause action or change. For example, physicists say “the moon is attracted to the Earth,” or “the positive rod attracts the negative rod.” 136

4. FORCEISANINTERNALTENDENCYORPROPERTYOFOBJECT. This metaphor is a blend with common conceptual metaphors in which humans describe “properties/states of objects” as substances contained with in them. For example, physicists say “the tension in the rope. . . ”, or “the force of an object.”

5.7.3 There is No Force

The grounded approach actually revealed a ﬁfth theme that emerges only in the later period of modern physics. I will characterize this as the “positivist” approach to force. F~ In the “weak form” the causal relation ~a = net is rewritten as a deﬁnition of a physical m quantity called “force”: F~ = m~a. In the “strong form”, some physicists have argued that the concept of force should be banished completely from all physics.

In the end, Jammer concludes that modern physicists consider force to be an ex- traneous concept, but at the same time it is a necessary and unavoidable conceptual stepping stone. This is a view that the author of this thesis shares.

5.7.4 Summary

The four force analogies are summarized, with examples of modern metaphorical language, in Fig. 5.2.

5.8 Modern Language: Grammar and Metaphors

For the force study, I used the same three introductory physics textbooks as the heat study [54, 81, 82]. Here there were too many sentences to code every one. I took a representative sample of 200 sentences from each textbook.

In previous cases I have been able to implement grammar and metaphor as two separate, but complementary analyses. The language that physicists use to talk about force does not permit this separation. Grammar and metaphor have to be analyzed 137

Figure 5.2: Summary of the four analogies and examples of modern metaphorical language that uses each analogy as a base Living or Animating Original Analogies: Original Analogies: Empedocles' principle of love and Anaxagoras: "World forming spirit" strife, binding and separating powers. Aristotle: "Prime mover" Plato: "attraction/repulsion" Religious scholars: "animating soul"

Modern Language: Modern Language: A force causes an object to accelerate The moon is attracted to the Earth A force acts on B

External Force Internal Original Analogies: Crates of Mallos: Tides caused by moon - Original Analogies: acts on intermediate air - acts on water. Aristotle: Separation of cause into Al-Kindi: Force propagated by "forced" motion and "natural" tendency. rays of light/heat. Modern Language: Modern Language: The tension in the rope A exerts/applies a force on/to B The force of an object

Passive concurrently to make sense of what is written. For example: “The weight of the Hubble Space Telescope is the gravitational force exerted on it by the earth.” [82], would be analyzed as follows:

1. “The weight of the Hubble Space Telescope [nominal group] is [process:relational] the gravitational force exerted on it by the earth [nominal group].”

2. “The weight of the Hubble Space Telescope. . . ” −→ “of” implies property of

object −→ metaphor: FORCEISANINTERNALTENDENCYORPROPERTYOF

OBJECT.

3. “. . . the gravitational force exerted on it by the earth.” −→ (denominalized sentence) −→ “The earth [agent] exerts [process] a force [medium] on the Hubble

Space Telescope [circumstance:location].” −→ Metaphor: FORCE IS A PASSIVE

MEDIUMOFINTERACTION. 138

From this example we can see that analyzing physicists’ language about force is complex and the ideas of grammar and metaphor need to be used together. Many sentences (such as the one above) contain two distinct metaphors in the same sentence. Such a sentence was coded twice, once for each metaphor.

A few examples of how phrases and nominal groups were coded will give the reader a clear picture of how the language was coded. These examples are shown in Table 5.12.

Table 5.12: Examples of phrases and clauses and how they were coded. Metaphor Phrases, clauses coded as instances of metaphor. FORCEISALIVING “. . . there is a force acting on the object.” [82], “. . . an ob- ENTITY. ject is pulled by several forces. . . ” [54], “. . . only a force can cause a change in velocity. . . ” [81] FORCE IS A PASSIVE “A greater net force is required to change the velocity of MEDIUMOFINTERAC- some objects” [82], “Forces are applied to the block as TION. shown.” [54], “Find the force Pat exerts on the chair.” [81] FORCEISANIN- “Every particle attracts any other particle with a gravita- TERNALDESIREOR tional force” [54], “We are all aware that all objects are DRIVE. attracted to the Earth.” [81] FORCEISANINTER- “The weight of an object. . . ” [82], “. . . the tension in the NALTENDENCYOR vine is 760 N.” [54], “. . . that is greater in magnitude than PROPERTY OF OBJECT. your weight.” [81]

5.8.1 Results of Coding

The results of the coding are shown in Fig. 5.3.

5.8.2 Summary

What is remarkable about this coding is the degree of consistency between the different textbooks. This consistency suggests to me that the coding scheme I devised may be valid. This coding presents the overall structure of physicists’ language about force, but does not delve into details about how the metaphors are put together. Further coding of the language could be attempted in future work. I believe that the historical 139

Figure 5.3: Results of coding.

60 52 Cutnell - 233 clauses 50 45 47 41 Halliday - 230 40 clauses 34 35 Serway - 261 30 clauses

20 18 14

centage of clauses 13 10 Per 0 0 0 0 Force is a living entity Force is an internal desire Force is a passive medium Force is an internal or drive of interaction tendency or property of object Metaphor/Model

development of the idea of force can reveal much more about how the human mind works.

5.9 Students’ Difﬁculties With Force and Motion: Speciﬁc Exam-

ples

If students have alternative ideas that are based primarily on the way physicists speak about force, then I should be able to ﬁnd classic “misconceptions” that have features grounded in one of the linguistic models of force. Mechanics is probably the area of physics that is best studied by physics education researchers. There are many papers published which document students’ difﬁculties and present examples of “typical” student responses in interviews. In this section I followed the following methodology.

1. I gathered as many research papers as I could ﬁnd on students’ difﬁculties with 140

force and motion. I used two methods of search. The ﬁrst was a research database search in ERIC and google scholar. The second was to use PER resource letters such as [103].

2. I then selected only those papers that provided samples of students’ reasoning from interviews with a few exceptions. These exceptions occurred when a particular difﬁculty was only documented in one paper and the authors of that paper did not use interviews as part of their research methodology.

3. Of the remaining group of papers, I broke the data down by student difficulty. (Some papers only covered one difficulty, others covered several difficulties.) I then coded students’ reasoning about those difficulties by placing their sample responses into one of the four metaphorical categories that are used to talk about force.

4. If a particular student difﬁculty is associated with one unique metaphorical category (as seen by students’ explanations of their reasoning), I suggest that the difﬁculty may be language related. Such examples were set aside for closer analysis.

Using this methodology, I found a pattern of three student difﬁculties related to language. These are presented in Sections 5.9.1, 5.9.2, and 5.9.3 below.

5.9.1 There is No Force in the Vacuum

Students were asked to compare the falling time of three different objects (different masses) in a vacuum. Some students reasoned that the time of fall for all three objects should be the same because there is no force in the vacuum. The authors explain further: “It seems that students ‘see’ force as something material which needs a physical support to travel through and exert its action.” Students use this idea to argue that the falling time for the three objects should be the same [92]. 141

The authors provided the following student responses:

• “There is no gravity in a vacuum and therefore the weight no longer interferes with the falling velocity and this explains the same falling time for the three objects.”,

• “if there is no matter inside the tube there cannot be either attractive or repulsive forces nor even gravity (10th grader).”,

• “[Objects in a vacuum are] not acted on by any external force (fourth year university).”

I would explain this strange reasoning as an over extension of the language based model in which students are taking too literally the category of force as matter. (The underlying ontological category that physicists use to speak about force.) Since force is classiﬁed as matter students are arguing that there is no force in the vacuum since a vacuum means the absence of matter.

5.9.2 Passive Objects Don’t Exert Forces

In the metaphor of force as the medium of interaction, the grammatical structure of two interacting objects generally takes the form: “Object A [agent] exerts [process] a force [medium] on object B [range]”. Thus force functions as the medium of the interaction, object A functions as an agent. An agent is classiﬁed ontologically as matter, but more speciﬁcally as living matter; the initiator or doer of the process.

In a cognitive linguistic study of how cause is encoded in the English language, Talmy has suggested that the notion of blocking is one of the fundamental steady state patterns in the semantic category of “force dynamics” [104]. Statements about blocking take on the general form: “The book [medium] stayed where it was [process] because the table was in the way [circumstance]”, rather than “The table [agent] kept the book where it was.” In grammatical terms, statements which ﬁt into the pattern of 142

“blocking” are generally agentless, with the thing that blocks functioning as a grammatical circumstance. Although living entities can block, such behavior is more often associated with non-living entities. In terms of lexical meaning, it is well understood that tables and walls are not alive. Thus, “the table”, functioning as a grammatical agent, is a grammatical metaphor. I can therefore predict that students will have little difficulty with the idea of “my hand” exerting a force, but more difficulty with passive (non-living) objects such as tables and walls. If students use these linguistic models in their models about force, then students will draw on the blocking pattern specifically for inanimate objects because they are not prototypical members of the living matter ontological category. This should be apparent in their explanations of their reasoning. I can predict not only what circumstances are suitable for inducing the blocking primitive, but I can explain why it is happening. Students should have no difficulty understanding that a human can exert a force since lexical meaning and grammatical function are concomitant. Students should have more difficultly accepting the idea that a table can exert a force because it involves forming an ad-hoc ontological category that incorporates living and non-living entities. Samples of students’ difficulties are provided in Table 5.13 below.

The samples of students’ reasoning given in Table 5.13 confirm both predictions. Firstly, the cited studies show that students have little difficulty in understanding that a human can exert a force, but have great difficulty accepting that a table or a wall can exert a force. Secondly, students are clearly making very subtle ontological distinctions in their reasoning:

• Students are making an overt ontological distinction between living and non- living entities. For example: “. . . Those things, the chair, can’t do anything. They are not alive.”

• Students clearly invoke the notion of blocking when talking about non-living objects. For example, “Because the table’s sitting there keeping it from being 143

Table 5.13: A list of studies and quoted student responses Study/Question Typical student response Multiple choice questions given to “57 No quotations given. Students’ re- college-bound [high-school] seniors”. sponses to question 1 are summarised In question 1 students were shown a pic- as follows: “Only 5% of the students ture of a person holding up a barbell and who answered this question did so incor- were asked “What forces are acting on rectly.” Responses to question 2: “. . . 30 the barbell?” In question 2 students were percent of those responding chose an in- shown a diagram of a block on a table correct answer. . . The vast majority of and were asked: “What forces are acting incorrect selections indicated the pres- on the block?” ence of gravity, but no counterbalancing upward force. . . 21% of students who got the ‘table’ question wrong answered the ‘barbell’ question right.” [91] Multiple choice question given to “57 No quotations. The authors summarise college-bound [high-school] seniors”. the results as follows: “Of the students Students were shown a diagram of a who answered, 51 percent failed to iden- woman pushing against a wall. They tify the reaction force of the wall as the were asked “What causes the woman to cause of her backwards motion.” [91] move (slip) backwards (away from the wall) as she pushes against the wall?” Junior-high-school students don’t be- “. . . these things [tables, chairs] are of- lieve that the chair, on which the teacher fering resistance. . . They are not pushing is standing, exerts an upward force on up. . . Those things, the chair, can’t do the teacher. In-class discussion. anything. They are not alive.” [57] High-school students have to draw all “The table is merely in the way.” “Be- forces on a book on a table. Out of cause the table’s sitting there keeping it 25 students who show the gravitational from being sucked, but that’s not a force. force, half leave out the normal force. Because the table is a solid object.” [90] Same high-school students as above “Maybe he is giving it a force. . . But the [90]. Now a book is placed on the out- table, that’s a normal position for the ta- stretched hand of a student. Students ble to be in. It doesn’t exert any force.” have to draw the forces acting on the “If you take the book off, his hand’s go- book. ing to go somewhere, but the table, you take the book, the weight off, and it’s not going anywhere. . . I think that the hand is force and on the table, it’s not force. . . ” [90] 144

sucked. . . ” It appears that the student permits the “misuse” of the table as an agent but then qualiﬁes: “but that’s not a force.” And in another example: “The table is merely in the way.”

Viewed as a problem with language, the table or any other passive (non-living) object should function as a circumstance of the system, not a cause of events. Its almost as if the students are trying to correct our language!

The quotations from the last question are interesting because they exhibit a mixture of force as substance/object as entity, and force as living entity metaphors. This seems to be consistent with the way physicists talk which is approximately evenly divided between these two ontological subcategories. The ﬁrst student is clearly talking about force as a substance possessed by an object and hence argues that the table cannot exert a force. The second student seems to be using similar reasoning, but is also mixing in some ideas of forces as entities by themselves. That is, he/she is talking about the object and the force itself as entities at the same time. This seems to reenforce the notion that passive objects cannot exert forces or in his/her language, “are not force”.

5.9.3 Force Causes Motion

One of the most commonly studied misconceptions or preconceptions in the PER community is the idea that force causes motion, or that a constant force is required to maintain a constant speed. Ibrahim Halloun and David Hestenes have described it as “the prescientiﬁc belief that ‘every motion has a cause’.” [60]

Different researchers observed this difﬁculty in many different situations. For example, in a multiple choice question given to “57 college-bound [high-school] seniors”, students were shown a diagram of a ball with an arrow attached to it, pointing upwards and three vertically ascending points, A, B and C, with A the lowest point. Students had to answer the following question: “A ball is thrown vertically upwards and travels from point A to point B to point C. It eventually reaches a point higher than C. What 145

forces act on the ball? (Ignore air resistance).” The students’ responses to the question were summarised by the authors of the study as follows: “Of all the students who chose an incorrect answer, 82 percent chose D [The downward weight of the ball and a decreasing upward force].” [91]

I am more interested in how students have justiﬁed their reasoning. There are a number of studies that quote typical student responses. These responses are summarized in Tables 5.14, 5.15 and 5.16 below.

Table 5.14: Variety of different studies with students who talk about force as a property of the object. Study/Question Typical student response(s) Students had to draw a FBD for the bob “Fm is the force that makes the pendu- of a swinging pendulum at some point lum swing upward. If Fm weren’t there, in its swing. Many students drew a force the pendulum could never move up to tangential to the arc of the pendulum’s the top of its swing. [86] motion. Students were shown a situation where Students drew an upward Fh, referring a coin was tossed straight up. Students to it as: “force from your hand”, “force had to draw a FBD for the coin when it of the throw”, “upward original force”, is half way up. “applied force”, “the force I am giving it”, “the force up from velocity”, “the force of throwing the coin up”, and “a momentum force. . . acting up.” [86] Mechanics diagnostic test given to “478 “a force of inertia”, “a potential force”, students in “University Physics”. 22 stu- “the force of velocity”, “the speed cre- dents were interviewed one month later. ates a force”, “energy or force you The authors describe the misconception shot it at”, “it’s still got some force as “the prescientiﬁc belief that ‘every inside”, “the force behind it. . . coming motion has a cause’.” from the throw”, “the power also has a force.” [60]

More evidence comes from a study by McCloskey in Genter and Stevens [3]. (See the examples give in Table 5.15.)

Table 5.16 presents what I consider to be examples of conceptual and metaphorical blending [1, 98]: Students seem to have blended the metaphor of force as a property of motion with ideas about external agents and cause, and extended it to other areas of motion such as circular motion. 146

Table 5.15: Study: McCloskey 1983 in Gentner and Stevens: Student Reponses [3] Question Typical student response Subject describing trajectory “I mean the weight of the ball times the speed of of ball, interviewer asks what the ball. . . Momentum is. . . a force that has been ex- he means by momentum. erted and put into the ball so this ball now that it’s Subject had completed 1 year travelling has a certain amount of force. . . The mov- of college physics. ing object has the force of momentum and since there’s no force to oppose that force it will continue on until it is opposed by something.” [3], (p. 307) Student was asked what is “. . . a combination of the velocity and the mass of meant by momentum. Stu- an object. It’s something that carries it along after dent had completed 1 year of a force on it has stopped. . . Let’s call it the force college physics. of motion. . . It’s something that keeps a body moving.” [3], (p. 307) Subject was asked to ex- “I understand that [friction and air resistance] ad- plain why friction and air re- versely affect the speed of the ball, but now [sic] sistance slow a rolling ball how. Whether they sort of absorb some of the force down. Subject had never that’s in the ball. . . I’m not sure. In other words, taken a physics course for the ball to plow through the air resistance or the friction if it has to sort of expend force and therefore lose it, I’m not sure. . . That seems to be a logical explanation.” [3], (p. 307) Subject was asked to explain “The ball when it was first thrown was provided the parabolic trajectory he with a certain amount of force. . . What’s happening drew for a ball thrown at 45 is that the force is basically being counterbalanced degrees. Subject had com- by gravity and at this point the upward force is still pleted high school and col- stronger than gravity, while here they’re both equal lege physics. and here gravity has become stronger.” [3], (p. 308) Subject was explaining the “Because as it [the cannonball] comes up the force trajectory of a cannon ball from the cannon is dissipating and the force of fired at some angle. Level of gravity is taking over. So it slows down. . . As it education unspecified. makes the arc and begins to come down, gravity is overcoming the force from the cannon.” [3], (p. 308) Subject describing the trajec- “When it leaves the cliff the inertia force—-the hor- tory of a ball rolling horizon- izontal force—is greater than the downward motion tally off the edge of a cliff force. When the horizontal force becomes less the ball would start falling. . . eventually the horizontal force would no longer have an effect, and it would be a straight down motion.” [3], (p. 309) 147

Table 5.16: Study: McCloskey 1983 in Gentner and Stevens: Student Reponses [3] Question Typical Response Subject describing the trajec- The momentum from the curve [of the tube] gives tory of a ball exiting a curved it [the ball] the arc. . . The force that the ball picks tube. up from the curve eventually dissipates and it will follow a normal straight line.” [3], (p. 309) Subject explaining the path of “. . . because of the directional momentum. You’ve a ball twirled in a circle after got a force going around and [after the string the string breaks breaks, the ball] will follow the curve that you’ve set it in until the ball runs out of the force within it that you’ve created by swinging.” [3], (p. 310) Subject explaining the motion “. . . the gravity that pulls it [the ball] to the center of a pendulum gives it enough force to continue the swing to the other side.” [3], (p. 310) Subject explaining the motion the ball stops at the end of the pendulum’s arc be- of a pendulum cause “the force has been expended.” [3], (p. 310) Subject explaining the trajec- “. . . the momentum that is [sic] has achieved from tory of the pendulum bob af- swinging through this arc and should continue in a ter the string is cut when circular path for a little while. . . then it no longer the pendulum is at the lowest has the force holding it in the circular path, and it point of its swing. has the force of gravity downward upon it so it’s going to start falling in that sort of arc motion because otherwise it would be going straight.” [3], (p. 310)

The examples of student’s reasoning given above provide some evidence that students are speaking about force as a property contained inside the moving object. The data in Tables 5.15 and 5.16 show repeated instances of students invoking container metaphors when talking about the force of an object. Table 5.14 shows little evidence of this. However the samples provided in Table 5.14 are only phrases, rather than complete sentences, so it is difﬁcult to draw any conclusions. The idea that force causes motion will be discussed further in Section 5.9.4 below. 148

5.9.4 Ontological Groping: A Reinterpretation of the “Force Causes

Motion” Misconception

The “force causes motion” misconception poses another serious challenge to the theoretical framework. The data show that physicists do talk about force as if it were a property of the object, but only about 15% of the time. It is a present linguistic mode, but not a dominant one. Can it be enough to support and encourage this student difﬁ- culty? In order to understand the difﬁculties that students have with force and motion, I am going to take a second look at the historical development of the concept of force.

In Table 5.17 I have placed, side by side, the words of students and the words of some of the famous players in the development of mechanics.

Table 5.17: Comparison between inventors of mechanics and modern students Student Physicist

“a force of inertia”, “the power also has “The vis insita, or innate force of mat- a force” [60] ter, is a power of resisting. . . ” (New- ton) [33] (p.119) “. . . this vis insita may, by a most signiﬁcant name, be called vis inertiae, or force of inactivity [inertia].” (Newton) [105] (p. 32) “I mean the weight of the ball times the “Bodies of equal weights and moved speed of the ball. . . Momentum is. . . a with equal velocities have equal force that has been exerted and put into forces. . . ” (Galileo) [33] (p. 97) the ball so this ball now that it’s travelling has a certain amount of force. . . ” [3] “. . . a combination of the velocity and “. . . mathematicians. . . estimate the mo- the mass of an object. It’s something tive force by the quantity of motion or by that carries it along after a force on it the product of the mass of the body into has stopped. . . Let’s call it the force of its velocity.” (Leibnitz) [105] (p. 52) motion. . . ” [3] “I understand that [friction and air resis- [talking about a pendulum] “. . . a body tance] adversely affect the speed of the falling from a certain height acquires a ball. . . they sort of absorb some of the force sufﬁcient to return it to the same force that’s in the ball. . . ” [3] height. . . unless the resistance of the air and other slight obstacles absorb some of its strength. . . ” (Leibnitz) [105] (p. 52) 149

I do not believe that the remarkable similarity (between students and the historical figures) of the statements compared in Table 5.17 is accidental, nor an artifact of translation. I am going to propose a qualitative explanation for this similarity. The explanation will be based on Nersessian’s continuum hypothesis [29]. This says that scientific thought is a natural extension of everyday cognition. There is no reason why this hypothesis cannot be applied reflexively (Nersessian in private conversation, 2005). In other words, if we understand why the authors of mechanics made the statements they made about force and the difficulties with which they were grappling, it may give us insight into the cognitive processes of students who are learning the same subject. We must therefore return to a more careful examination of the historical record in order to understand what students are saying.

1. In the 1300’s John Buridan wrote “. . . God, when creating the world,. . . has given to each of them [the celestial spheres] an impetus which kept them moving since then. . . ” [33] (p.70). In this sentence is sounds very much like Buridan is suggesting that the impetus in the body is causing its motion.

2. By around 1600, Galileo spoke of the “impressed force” as an impetus that was gradually consumed by the gravitation force as the stone rose up into the air [33] (p. 100). Here it seems that impetus, or impressed force has shifted subtly from being viewed as a cause of motion to being viewed as a property of motion that changes due to other external factors. It is also clear from Galileo’s writings that he sees this impetus or impressed force as proportional to the mass of the object and its velocity.

3. In 1687, Newton published Principia. It is curious that Newton offers two different definitions of force. The first definition (Definition III) is; “The vis insita, or innate force [or force of inertia] of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.” [105] 150

(p. 32). The second definition (Definition IV) reads, “An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.” [105] (p. 32). Traditionally Definition III is often interpreted as a definition of “inertness” or inertial mass. However, the evidence shows that this view is incomplete. Consider firstly that two definitions before, Newton had already defined “amount of stuff” when he wrote, “The quantity of matter is the measure of the same, arising from its density and bulk conjunctly” [105] (p. 31). Secondly, Newton himself is aware that the force of inertia he has just defined depends on more than the mass of the object. He wrote also as part of Definition III: “the exercise of this force [vis insita/vis inertiae] may be considered both as resistance and impulse; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other” [105] (p. 32). (In simpler words, the force of inertia determines how hard a moving object hits something.) He then goes on to point out that this distinction (between resistance and impulse) is artificial since, “. . . motion and rest, as commonly conceived, are only relatively distinguished. . . ” [105] (p. 32).

What are we to make of this? It is clear from this writing that Newton’s force of inertia depends (in some way that is unspeciﬁed) on motion, as well as the mass of the object. Max Jammer summarizes it best:

“Clearly, in this deﬁnition, force is not conceived as a cause of motion or acceleration. How, then, was it possible for Newton to call the quality of inertia a force? The answer to this question is evident if we regard Def- inition III as a concession to pre-Galilean mechanics.” [33] (p. 120) [my emphasis]

The main points are that (a) this force (vis inertiae) is not a causal agent, and (b) Newton is basically talking about Buridanian impetus, but as an effect, not a cause. 151

4. One year before Newton, Leibnitz published a paper introducing the idea of vis viva or “living force”. This this paper he began by disputing the idea that “the motive force [may be estimated] by the quantity of motion or by the product of the mass of the body into its velocity.” [105] (p. 52). He then went on to show (by reasoning about the speed acquired by freely falling objects of different mass from different heights) that this “motive force” must be proportional to both the mass of the object and the square of its velocity. Leibnitz described what happens to this force in a collision:

“. . . two soft or unelastick bodies meeting together, lose some of their force. . . ’tis true, their wholes lose it with respect to their total motion; but their parts receive it, being shaken by the force of the concourse. And therefore that loss of force is only apparent. The forces are not destroyed, but scattered among the small parts. The bodies do not lose their forces; but the case here is the same, as when men change great money into small.” [33], (p.168) [my emphasis]

The emphasized phrases suggest a commodity metaphor for Leibnitz’s vis viva. It is a metaphor seldom applied to force in modern language, but used almost ubiquitously to talk about energy.

5. After Leibnitz, a debate ensued between two opposing camps. On the one side were those who argued that the “force of a body in motion” should be proportional to mv. On the other side were those who felt that the “force of a body in motion” should be proportional to mv2. This debate lasted for almost 60 years before D’Alembert put an end to it in 1743. D’Alembert argued that the debate between the two camps was nothing more than a semantic one and “a futile metaphysical discussion. . . about words still unworthy of the consideration of philosophers” [105] (p. 57). To do D’Alembert’s argument justice, I shall quote it almost in its entirety:

“The greater the obstacle that a body can overcome, or that it can resist, the greater may we say is its force, provided that, without meaning to express by this word a hypothetical entity which resides in the body, we 152

use the word only as an abbreviated was of expressing a fact, just as we say that one body has twice as much velocity as another instead of saying that in equal times it traverses twice the distance, without intending to mean by this that the word velocity represents an entity inherent in the body.” [105] (p. 56)

In other words, D’Alembert said that, to speak of “the force of an object” is just a metaphor, just as we speak of the velocity of an object. He continued his argument by pointing out that “force” can only be inferred by the interaction between objects, either in equilibrium (say hanging an object from a spring) or in the effect of one object on another in a collision. He rounded off his argument by pointing out that it does not matter if you want to adhere to F ∝ mv, or F ∝ mv2 because at the end of the day, what you measure is Newton’s “impressed force”, and “If we set the same problem in mechanics before two mathematicians, one of whom is opposed to and the other a partisan of vis viva, their solutions of the problem, if they are correct, will always agree. . . ” [105] (p. 58)

6. Soon after D’Alembert, Thomas Young introduced the term “energy” for Leib- nitz’s vis viva very gradually the term “force” became used exclusively to denote Newton’s “impressed force” rather than “force of inertia” or “living force”. (To give the reader an idea of the slowness of this evolution: Joule still used the term vis viva in his writings in 1840, and Helmoholtz’s seminal energy conservation paper of 1847 was entitled “On the Conservation of Force.” [99].)

What can we understand about physicsts’ thought processes from the historical record presented above? Traditionally researchers have taken students’ reasoning about force and rewritten it as a causal relationship, either as Fs = αv, where Fs denotes the ~ “supply of force” and α is a constant of proportionality [106], or as Fnet = m~v [107]. They then proceed to show that this relationship leads to farcical predictions about the motion of objects [107]. 153

At this point, I must make the distinction between the deﬁnition of a physical quan- d~v tity such as ~a = and a causal relationship that relates physical quantities to each dt other [96]. An example would be Newton’s second law that relates the physical quanti- F~ ties of acceleration, mass and force into a causal relationship: ~a = net [33]. In the his- m 1 torical record that I have presented, there is no indication that F~ = m~v or F = mv2 2 were written as a causal relationships. All the evidence (apart from Buridan himself) presented above, shows that physicists were trying to write down the deﬁnition of a physical quantity. Marshall Clagett agrees with my interpretation [108]. In an extensive historical analysis of mechanics in the middle ages, he concluded that impetus underwent an ontological shift from a cause of motion to an effect of motion some time between the late sixteenth century and early seventeenth century (around the time of Galileo).

The conclusion is unequivocal: From Newton onwards, the debate never seems to F~ have questioned the causal relationship ~a = net . This relationship was not in dispute. m The question was simply what physical quantity should we call “force”? Should we talk about a “force” of motion (what we would call now either momentum or kinetic energy)? Or should force be used to quantify the interaction between two bodies? Newton was clearly uncertain on the matter and hence deﬁned the term twice, once as force of inertia, once as impressed force.

D’Alembert clearly states that the debate was a semantic one. I would add that it is also an ontological one. Namely, should we classify force as both as a property of matter and a process, or only as a process of interaction between two objects? Shall we talk about the force of an object, or only about the force that one object exerts on another? Notice also how the two different camps in the “force of motion” debate fall naturally into the two different analogies: Leibnitz’s vis viva falls into the category of “Force is like an animating spirit”, while the “force of inertia” of Buridan, Galileo, Descartes, Newton and others falls into the category of “force is like a passive internal tendency.” They were trying to quantify the “passivity” of an object. Today we call 154

these quantities of motion kinetic energy and momentum respectively.

I am going to refer to this difﬁculty that Newton and and others had with the term “force” as an example of “ontological groping”. I will use “ontological groping” to mean any struggle to deﬁne and categorize ontologically the meanings of physical terms, and I wish to hypothesize the following:

1. Anecdotally, students are well aware that motion continues undiminished when all external resistance is removed. Since I have had this idea, I have used every opportunity to ask my students to do the following thought experiment. In this thought experiment I ask them to imagine a roller-blader on a smooth, straight, level track that goes for a long way. I ask them to imagine that we can remove all sources of friction and if so, how will the roller-blader move after someone pushes them? Every student so far has responded that the roller-blader will move at constant speed until they run out of track. I suggest that most students do not believe that a force is needed to cause or sustain constant speed motion. I am suggesting that many students have a notion of impetus as an effect of motion rather than a cause.

2. The historical debate presented above is a typical example of ontological groping, and should serve as a model for the same debate and struggle that our students are having. Students are not struggling with a cause effect relationship, but are struggling to decide what to call a “force”.

3. The stubbornness of students’ misconceptions about force and motion is based on (a) the clear difﬁculty of the ontological debate (it took the brightest minds in physics almost 60 years to straighten it out), and (b) our failure as instructors to recognize the essence of the difﬁculty that our students are having.

I hypothesize that when a student puts in the “force that my hand gave it” into the free body diagram for a projectile, he/she is asking the instructor, “what should I call 155

‘force’? Is it a property of the object’s motion or an interaction? Does it quantify something about the interaction between two objects or does it quantify a property of the object?” And then under the category of properties, “does it quantify some ‘activity’ of the object (what we would call, with hindsight, kinetic energy), or does it quantify a property of the motion? (What we now term momentum.)” I suggest that physicists’ discourse (the confusing use of four different metaphorical systems) does not support students’ struggle to refine the meaning of the terms. Viewed in this way, students’ confusion is hardly a misconception, but an analogous struggle to refine and define terms, to build on their experience, and refine their every day language. Viewed historically, the difficulty of this struggle cannot be underestimated and may go a long way to explaining the stubbornness of these “misconceptions”. After all, if we as teachers are not answering the real question students are asking, how can we ever expect our students to figure it out?

5.10 Summary and Implications

In this chapter, I have shown that the combination of grammar, ontology, and metaphor can reveal underlying patterns of consistency in physicists’ language. These patterns of consistency may be considered in each case to represent a legitimate model of a physical system. Each model that may be used to reason productively about a physical system (if limits and applicability are well understood) and unproductively (if those limits and applicability are not well understood). I have shown how important it is for students to be able to make subtle ontological distinctions about physics concepts. This supports the claims of [5, 2], and [68] that ontology is important, but also represents a challenge and a direction for future research. It is apparent that physicists are able to mix or blend concepts into different ontological categories, and move easily between ontological categories in the way they speak about physics concepts. It is therefore questionable whether certain physics concepts have deﬁnite ontological categories at 156

all. In any case, the categorization of a concept appears to be context dependent, and more research is needed.

I have also shown how students may be taking cues about ontological categorization from the way that physicists speak. In addition, I have shown how students may attach metaphorical properties to physical systems, based on the metaphors that physicists use. They appear to use these ideas to reason about physical systems.

The linguistic framework can explain why students struggle so much with certain ideas and why many so called misconceptions are so “resistant to instruction”. In sum, physicists’ language does not make ontological distinctions clear. In the case of ontological groping, we, the teachers, are simply failing to hear the questions our students are asking. Namely, students need help sorting out the ontological categories, rather than help being disabused of certain misconceptions they might have. This point cannot be overemphasized. James Zull [109] has written a book explaining how learning works on a neurological level. The evidence presented there suggests that learning does not necessarily involve destroying certain ideas (neurological connections) and replacing them with other ideas (new neurological connections). It involves a grad- ual refinement of neurological connections (weakening of some and strengthening of others). Thus an understanding of students’ difficulties with the concept of force as a problem in ontological category refinement, is more aligned with current neurological understanding of how the brain learns as compared with some previous ideas about conceptual change.

I can make no claim that this approach accounts for all misconceptions or even accounts for all instances of a particular misconception. The goal of a grounded theory is not to explain everything, but to delimit a domain of applicability. It is quite possible that some fraction of students who ask “should I include the force my hand gave it?” are asking an ontological question, as I suggest, while another fraction of students have an impetus model in which force is necessary to sustain motion. But I want to suggest that this new approach opens up a whole new dimension of understanding of 157

what our students are thinking. In many cases it can help us recognise and answer students’ questions in a different and possibly more appropriate way. In many cases this linguistic view can help us account for much of both the inherent stability of students’ responses to a given situation and the contextual dependence of their reasoning and apparently fragmentary nature of their ideas. This thesis does not attempt to explain how ideas are put together on a cognitive/neurological level. Such research is a logical next step to follow. Language may be viewed as a means of processing, constructing and amalgamating ideas and meaning and thus seems essential in understanding fundamental neurological processes. If learning physics involves learning to represent physics, then learning physics must involve a reﬁnement of terminology and cases in language. And part of the teacher’s job must be to support that learning process — something that we, as teachers, are often unaware of.

It is also difficult to say how much of students’ difficulties are caused by the language of physicists and how much is caused by the language that students bring with them to the physics classroom. But it is certainly not exclusively one or the other. Like- wise it is difficult to separate language from cognition. The best we can do is to argue that language gives us a window into the mind. If students and physicists are grounding their ideas (metaphorically, analogically or otherwise) in physical or experiential models of the world, that grounding is being expressed in language (and other representations) and may, in turn, be constrained by language (and other representations). 158

Chapter 6 Summaries, Conclusions and Future Research

6.1 Introduction

In this chapter I will ﬁrst present one example of how the theoretical framework I have developed, may be used to understand the ideas of physics better (Section 6.2). I will then summarize the ideas and results presented in this dissertation, present examples of possible testing experiments that test the theoretical framework (Section 6.3), and I will discuss some of the practical implications of the theoretical framework for teaching physics (Section 6.6). With regards to testing the framework and practical implications for teaching, I will discuss some possible directions in which future research may go.

6.2 Niels Bohr, Complementarity, and the Paradigm Shift in Physics

6.2.1 Introduction

The concept of ontological groping, first covered in Section 5.9.4 of Chapter 5, is a useful one. It is useful not only for understanding some of the difficulties that physics students experience in their physics course, but it provides a productive way to understand some of the difficulties that physicists experience themselves. In this section I will return to quantum mechanics. In particular I wish to re-examine the idea of wave- particle duality and Bohr’s principle of complementarity through the lens of metaphorical language and ontology. I will show how the ideas of metaphor and ontological groping can 159

1. explain why physicists have found quantum mechanics so difﬁcult to understand,

2. help us understand the underlying ontology of quantum mechanics, and

3. provide a means of re-interpretting Bohr’s principle of complementarity and re- interpretting wave-particle duality as a conﬂict of incompatible ontologies.

Niels Bohr’s idea of complementarity is remarkable. I use the word “remarkable” because none of the physics professors I interviewed were able to say what it was, yet Bohr considered it to be his greatest contribution to physics [110]. The statement of complementarity has evolved in precision over many years. Below is Bohr’s ﬁrst published statement of complementarity.

“Indeed, in the description of atomic phenomena, the quantum postulate [deBroglie’s wave hypothesis] presents us with the task of developing a ‘complementarity’ theory the consistency of which can be judged only by weighing the possibilities of deﬁnition and observation. . . [For example] the two views of the nature of light [wave and particle] are rather to be considered as different attempts at an interpretation of experimental evidence in which the limitation of the classical concepts is expressed in complementary ways.” [73], (pp. 55 – 56)

In the sections that follow, I will try to re-interpret this statement, as well as other commentaries on complementarity, through the lens of the theoretical framework I have developed. In doing so, I believe that we can come to a deeper understanding of complementarity and of quantum mechanics itself.

6.2.2 Two Metaphors From Two Different Ontological Categories

In late 1926 and early 1927, Werner Heisenberg and Bohr spend several months trying to come up with a physical interpretation of quantum mechanics. At the heart of their discussion was the issue of wave-particle duality and the contradictions that arose from this. It was these discussions that lead to Heisenberg’s paper on the uncertainty principle and Bohr’s paper on complementarity later in 1927 [110]. Heisenberg later recounted that at the heart of their debate was the question of whether they should think 160

of the electron as a wave or as a particle. Both men exhausted each other after months of discussion and went separate ways to recuperate, without resolving this issue [110].

Why did this issue of wave-particle duality cause so much trouble? I believe that their discussion is another example of ontological groping. A wave is classiﬁed ontologically as a process of movement of some medium, while a particle is categorized ontologically as matter. If we use these two metaphors to describe an electron we run into a conundrum. How is it possible that a single physical concept (an electron) can belong to two distinct ontological categories? In our physical experience there are simply no prototypes of objects that can belong to such a “dual” category. Our human brain seems incapable of forming an ad hoc category that includes both the ideas of matter and process simultaneously. This is a possible explanation of why Heisenberg and Bohr were unable to come to any conclusion.

6.2.3 Revisiting the Partial Nature of Metaphor

In previous historical examples of ontological groping, it appears that the meaning of terms became reﬁned and better delineated through a long process of debate. What is striking about Heisenberg’s and Bohr’s debate was that they got no closer to resolution. The result was a paradigm shift in physics. Heisenberg’s response was to derive the uncertainty relations (using the Heisenberg microscope analogy) and argue that “Quantum mechanics deﬁnitively establishes the non-validity of the law of causality. . . ” (Heisenberg) in [110]. In other words, you can think of the electron as a particle, but you have to surrender the idea of causality. This is exactly the same sort of shift that took place with special relativity, which required physicists to surrender the idea of absolute simultaneity of events.

Bohr’s response was completely different. I believe that he decided that it was necessary to reconsider the actual language of quantum mechanics. Abraham Pais, Bohr’s biographer, documented that Bohr was concerned about the use of language to express ideas in quantum mechanics around the time of his meetings with Heisenberg. 161

In a letter to Kramers, Bohr wrote, “. . . how little the words we all use are suitable in accounting for empirical facts. . . ” [110] More evidence from drafts of Bohr’s complementarity paper show similar thoughts: “All information about atoms [is] expressed in classical concepts. . . ” [110].

6.2.4 Understanding Bohr’s Complementarity

The evidence presented suggests to me that we may interpret Bohr’s complementarity principle through the linguistic framework of this dissertation. Remember that metaphors are partial. They highlight certain aspects of a phenomenon and hide others. It often takes more than one metaphor to adequately represent an idea [38]. In Chap- ter 4 we encountered three metaphors that physicists use to describe an electron. These are, THE ELECTRON IS A WAVE, THE ELECTRON IS A PARTICLE, and the BOHMIAN metaphor. (There is also the SCHRÖDINGER metaphor that was not covered in this dissertation.) Each of these metaphors, by itself, is incomplete. It takes a set of two or more complementary metaphors in order to describe any physical phenomenon. Thus Bohr’s idea of complementarity is not only a revolution in physics, but also a revolution in cognition. Bohr realized an aspect of cognition in physics that has always been present. He realized that our understanding of the world is grounded in human experience and this understanding is reﬂected in and constrained by our language. This theme has recurred repeatedly in this dissertation. I have shown that physicists often lack the language to express their ideas precisely. In cases such as the example of “force”, many different (complementary) metaphors are used to express aspects of cause and effect. Each of these different metaphors has its own applicability and limitations, yet it is the best that we can do with language and cognition that are grounded in human experience.

Bohr realized that to speak about the “electronness” of an electron, physicists could not rely on a single metaphor (either exclusively a particle or a wave metaphor). Be- cause the concept of an electron is beyond the scope of our classical language, we 162

require several complementary metaphors to describe the essence of what an electron is. It does not matter if these metaphors conﬂict ontologically. This resolves the debate over the reﬁnement and delineation of meanings of terms in physics. Bohr’s principle of complementarity suggests that any attempt at coming up with a single “precise” definition of a physical term is doomed to failure. A physical term (such as “an electron” or “force”) is endowed with multiple aspects of meaning, grounded in the conceptual metaphors that physicists consciously or unconsciously use to describe these terms.

6.3 Testing the Grounded Theory

6.3.1 Summary of the Grounded Theory

The main points of the formal grounded theory laid out in this thesis are as follows:

1. Physicists’ language represents physical models. These models are based on historical analogies or on common language that has been co-opted.

(a) Physicists’ language encodes a model ontology of matter, processes, and states in the grammar of each sentence.

(b) Other features of the analogical model are encoded in elaborated metaphors that are based on the underlying model ontology encoded in the grammar, and/or on features of the base domain of the original analogy.

2. When students learn physics, they have to decode physicists’ language. In this decoding process:

(a) Students may have difﬁculty seeing the applicability and/or limitations of the models encoded in the metaphorical language. The evidence that I have presented in Chapter 4 shows that students over-extend and misapply the linguistic models that they hear or read. 163

(b) If the ontology encoded in physicists’ language conﬂicts with or leaves ambiguous the accepted model ontology, then this may confuse students and lead to them categorizing physics concepts into incorrect ontological categories. I have shown that incorrect ontological categorization can affect students’ reasoning about physics problems.

(c) Both physicists and students appear to be engaged in ontological groping, trying to define and refine the meaning and function of terms. These difficulties should be recognized as such rather than labeling them as “misconceptions”.

There are many possible tests of the formal grounded theory presented in this thesis. I will present two.

1. The first possible test is to compare student learning in a course where good and consistent language was used against another course in which traditional language was used and see if there is a notable difference in the incidence of certain “misconceptions” amongst the students. This is however, not a simple experiment to do. Firstly, it is difficult to control the language used by the lecturer and teaching assistants, secondly, there is no textbook (that I know of) that completely supports the use of “good” language about force and heat. Thirdly and most importantly, language, as a representation, does not function in isolation. Lemke has suggested that students need to coördinate many different representations to create understanding [10]. If different representations of the same concept conflict with each other, then I doubt whether any effect on the students will be observed. They are more likely to remain confused. To test the grounded theory this way, linguistic and non-linguistic representations must be coördinated.

2. The second test is more of a direct test of the Sapir-Whorf hypothesis. If language matters, then a paradigm shift in language should lead to a paradigm shift in the way that students think. Such a testing experiment would involve either 164

a group of students being obliged to engage in the use of different language to describe a physical phenomenon, or an ethnographic study of students learning physics in a different language (probably their native language). This language would have to be sufﬁciently different in grammatical and metaphorical structure from English that the learners might classify physics ideas differently.

In Sections 6.4, and 6.5 below, I will present one example of the ﬁrst test and two examples of the second test described above. Although the examples I will present do not present any statistically valid results and thus do not allow us to draw any real conclusions, they represent examples of directions in which my future research will go.

6.4 Coördinating All Representations of Physics: The Ultimate Test-

ing Experiment

It is my belief that in order to test the ideas of this thesis, one must do much more than merely speak better. In order to test whether students “robust misconceptions” are par- tially driven by how we speak about and represent physical ideas, all representations used to describe this idea should match, as closely as possible, the underlying expert model ontology. Such a testing experiment is not easy. It would require the undivided attention and coöperation of the instructor in charge, all of his/her teaching assistants, and a textbook and/or workbook that matched all the representational requirements. This treatment group would have to be compared against two control groups: both students who received traditional instruction with traditional representations and students who received reformed instruction with traditional representations, to see if representation has any effect at all on students’ understanding and the persistence of these misconceptions.

The way in which we represent ideas must necessarily go beyond the scope of this thesis. I have to introduce a new hypothesis: students often reason by forming categories and prototypes of those categories based on ﬁrst exposure. (This hypothesis 165

is based on observational data that are too extensive to include in this thesis.) Thus the representations employed must not only match the expert model ontology, but must work in such a way as to prevent naïve categories from being formed.

For example: To test the hypothesis that students’ naïve impetus model of force and motion is driven, in part, by the representations physicists use and their failure to address students’ questions (namely, “what is your deﬁnition of force?”), the following steps need to be taken:

1. Everyone involved in teaching the course needs to only speak about “the force exerted by object A on object B.” Such a phrase must be used exclusively to talk about force. This will promote the interaction view of force, namely as a process of interaction between two or more objects. When drawing a free-body diagram, the teacher needs to ask, “what other objects is the object of interest interacting with?” rather than “draw the forces acting on the object of interest” [111].

2. The involved instructors must never used the terms “normal force”, “tension”, “centripetal force” etc. . . These terms will promote categorization of different types of force as separate entities, rather than one process of interaction that we call “force” [81]. Such distinctions may be introduced later after students have developed an initial understanding. Naming, grouping, and categorizing must happen later in the learning process [109]

3. Diagrammatic representations and equations about force should always have the force written with two subscripts to denote what is exerting a force on what. For example, for a block resting on a table, traditionally the forces involved in the free body diagram are denoted by N~ for “normal force” and W~ for “weight”. ~ ~ These should rather be written as Ftable on block, and FEarth on block, respectively [111]. Such a reform of representation both promotes the interaction nature of force and discourages students from categorizing the force exerted by the table and the force exerted by the Earth as different categories. 166

4. At least one explicit discussion needs to take place in the classroom or lecture hall about what the meaning of “force” is. In this discussion the instructor should mention that physicists struggled for many years over what to call a “force”. Either a property of the object’s motion or a means to quantify the strength of an interaction between two objects. Physicists eventually settled on using the term force to describe the interaction between two objects and “the force of an object’s motion” is now called “momentum” and “kinetic energy”. In this view forces or interactions between objects cause an object to lose or gain momentum or kinetic energy.

Since no textbook exists at this time that follows all of the requirements listed above, a study is almost impossible to conduct. Fortunately, however, I was able to teach an introductory algebra-based physics course over which I had full control. (Physics 193 & 194.) In this course students did not use a textbook, but used, instead, Alan Van Heuvelen’s and Eugenia Etkina’s “Active Learning Guide” [112]. This workbook incorporated all the representational reforms mentioned above. Time was also devoted to training teaching assistants about the required representational reforms. The course followed the ISLE format [4]. Students were expected to learn physics through a process of observing, proposing multiple explanations, testing different explanations with testing experiments, and applying physical models in different situations. At a suitable point in the course, I held an in-class discussion about the meaning of the term “force”, emphasizing that it refers to an interaction between two objects rather than a property of motion of an object. Physicists have identiﬁed two “useful” (conserved) properties of motion and call them “momentum” and “kinetic energy”. Unfortunately I was unable to assemble a control group for this study. Thus it is impossible to draw any statistically valid conclusions from the results I am about to present. However, the results are sufﬁciently interesting to warrant further investigation.

Fig. 6.1 presents a question I used in the second-midterm of the Physics 193 course. At this point the students had completed sections on Newtonian mechanics, momentum 167

and energy. In this question I added one distractor that showed a force of inertia Finertia pointing in the direction of motion.

Figure 6.1: Question from second mid-term, physics 193

I am going to compare the results of students’ responses to the question in Fig. 6.1 against a similar question used at New Mexico State University to elicit the same dif- ﬁculty amongst students [7]. The question is shown in Fig. 6.2. This study involved a group of 152 engineering students who had completed one semester of physics. 114 students had completed a traditional course (their responses are shown in column 1 of Fig. 6.2) and 38 had received reformed instruction (Overview, Case Study (OCS) physics [113]) 8 months earlier (their responses are shown in column 2 of Fig. 6.2).

Van Heuvelen’s study provides me with the two required control groups. The ﬁrst are the responses of traditionally instructed students using traditional representations, the second group are those who received reformed instruction, but with traditional representations. The responses of these two groups of students to the question in Fig. 6.2 are compared against the responses of Physics 193 students to the question in Fig. 6.1 in Table 6.1.

I believe that these results are sufﬁciently different to warrant further investigation. 168

Figure 6.2: Pendulum question from Alan Van Heuvelen’s paper [7]

Table 6.1: A comparison of students taught with traditional force representations and reformed force representations. Course description % of students who chose FBD showing force in direction of motion Traditional course for engineering stu- 60% dents, traditional representations (n = 114) Reformed course for engineering stu- 16% dents (OCS physics), traditional representations (n = 38) Physics 193, reformed course (ISLE [4]), 3.5% reformed representations (n = 170) 169

Although the courses, circumstances, and populations are different, the engineering courses were calculus-based and represent a higher level physics course as compared to the 193/194 course, which was algebra-based. In addition, the question is slightly different. The pendulum question (Fig. 6.2) presents more opportunity for students to chose the inertia option by random guessing. But even with the double chance factored in, the 193 students chose that option less. 6 students out of 170 represents a number far below random guessing. In addition, the loop the loop question (Fig. 6.1), given in the 193 course, is more sophisticated and has more chance for confusion. The pendulum question allows some chance of elimination guessing which should improve the performance of a small percentage of students. My future research should include a properly controlled study to examine the effect of representation and its interaction effect with instructional methods.

6.5 Different Language, Different Concepts, Different Understand-

ing

In relation to students’ difﬁculties with the concept of heat and work in thermodynamics, as far as I know, no-one has tested whether speaking about heat as a process makes a discernible difference to students’ understanding of the subject. Again, such a testing experiment would require a wholesale shift in representations from language, to textbook diagrams that seem to indicate objects containing heat.

However I found two cases, reported in the literature, of different language being used to talk about thermodynamics. In each case, the difﬁculties that students had in developing a conceptual understanding were remarkably different from the difﬁculties with which we normally see students struggling [114, 115]. 170

6.5.1 Kaper’s Study about Thermodynamics

In a pair of experiments, Wolter Kaper and Martin Goedhart set out to test a hypotheses related to the development of a scientiﬁc language and conceptual change about thermodynamics. I must emphasize that their work was not intended to test the theoretical framework of this dissertation, but I will show how their results can be interpreted as supporting evidence for my ideas [114].

The authors’ hypothesis was that, by introducing an “intermediary language” about thermodynamics (a language of limited validity, grounded in everyday language) and presenting students with examples of the limitations of that language, students would develop a language more aligned with current scientiﬁc language and understanding about thermodynamic processes. In summary, they predicted that the intermediary language, combined with judicious choice of examples, would facilitate a transition from everyday experience to scientiﬁc understanding.

The first experiment: The first experiment started with a group of 65 first year chemistry students in a thermodynamics course. The students were instructed using the standard “forms of energy” language. In this language, kinetic energy, potential energy, chemical energy, heat are all considered forms of energy. Processes involve the transfer or transformation from one form to another. After this a group of 5 students were selected to participate in a set of tutorial sessions given by Kaper. In these sessions students were exposed to the limitations of the forms of energy language. In particular, students had to solve a problem that revealed that the amount of heat transferred in a thermodynamic process starting at some initial state and ending at some other final state was path dependent. (Two different paths a and b were shown.) Kaper and Goedhart expected that, presented with these challenges to their understanding, students would revise their language to begin to talk about heating as a process and realize that heat carries a different status as compared to kinetic and potential energy. Instead of changing their language, students protested that if different amounts of heat were transferred for the two different paths (a and b), then the system could not have 171

arrived at the same ﬁnal state and therefore the author of the problem had made a mistake. Students clung remorsefully to the idea that heat was a state function solely determined by the temperature of the system.

The second experiment: In the second experiment another group of 5 students received special tutorial instruction from Kaper. In these sessions, Kaper introduced an entirely new intermediary language, namely the “exchange value” language. In this language state variables of a system are spoken of has having an exchange value. For example the exchange value of height h is mgh, or gravitational potential energy, the 1 exchange value of velocity squared (v2) is mv2 and so on. Any exchange value that 2 “behaved like a store” (of value) was recognized as a “state quantity”, while those that did not “behave like a store” were recognized as “process quantities”. In tutorial discussions, students were able to realize that there are not different forms of energy, but “always only one and the same energy” [114]. Students were ﬁnally confronted with the same problem as the students in the previous experiment. Kaper and Goedhart describe the resulting ﬁnal transition to a thermodynamic language as a non-event. Confronted with the path dependence of heat, students were immediately able to see that the term “heat” did not behave like a store, and was not a legitimate exchange value of temperature. Therefore heat should be seen as a means of exchange rather than an exchangeable quantity itself. The authors feel that the language transition took place earlier when students realized that they did not need forms of energy language, but could view energy as one intrinsic value associated with certain state variables of a system.

My interpretation: First of all, these experiments did not contain a large enough sample of students to draw any statistically valid conclusions. With that in mind I believe that these two experiments represent an excellent test of the theoretical framework of this thesis. I have contended that language is a essential key to understanding in two respects. Firstly language cues and reenforces a model ontology of matter, processes, and states through the grammatical construction of the sentences. Secondly, from a 172

metaphorical perspective, the metaphors used, ground the physical ideas in an analogical base domain which can be used to reason either productively or unproductively about the phenomena under consideration.

The “forms of energy” language grounds energy ideas in a base domain that forms the basis of a number of powerful conceptual metaphors of everyday language. Namely the movement of substances into and out of containers. This is grounded in the intuitive notion of conservation of matter and this matter ﬁlls up the available space. In the “forms of energy” language, physicists have borrowed this conceptual metaphorical system and adapted it to realize physical states of a system. In this language heat is spoken of as a form of energy and therefore should naturally function as a state function, like any other form of energy. I would therefore predict that students will generally have difﬁculty in transitioning to a more “correct” language of thermodynamics since the idea of heat as a process quantity directly contradicts the forms of energy language.

In the “exchange value” language, a completely different metaphorical system is activated. One that is grounded in the notions of money, intrinsic value, and exchange of commodities according to that intrinsic value. It should be relatively easy to extend this idea to non-linear exchanges (for example, the exchange value of v2) and to talk about different means of exchange. Thus when students found that “heat” was not a legitimate “exchange value” of temperature, it is not surprising that the ontological shift was much easier. While different commodities (state variables) still function grammatically as ontological matter, in the exchange value language, processes are described in terms of different means of exchange rather than the movement of substances associated with the “forms of energy” language. The notion of a value attached to a commodity (energy) is should still be classed ontologically as matter, but in terms of elaborated conceptual metaphors, the idea of value is far more sophisticated, abstract, and well developed amongst the students. Kaper and Goedhart’s study shows three things:

1. The study represents a legitimate test of my theoretical framework and the results 173

that emerged are pretty much as I would have predicted. Thus I am able to explain why a different choice of “intermediary language” could have such a profound effect on students’ understanding in terms of the different ontological and metaphorical associations that the language activated.

2. The study shows how profoundly language matters for students in how they interpret the data that they are presented with. In the first experiment, the students simply refused to accept what they were seeing. Rather than change their conceptions they simply filtered out the conflicting data. This process is well understood neurologically [109]. Therefore the study serves as a more direct test of the Sapir-Whorf hypothesis that humans interpret the world through the language systems they use. We do not only describe what we see, but we reinterpret what we see through the language chosen to describe it!

3. This study gives a hint as to how to use language better in teaching. Sometimes the language we have is adequate. Sometimes a new language, that can activate a different metaphorical system, can serve our pedagogical purposes better than the language we currently use.

6.5.2 Hewson’s Study of Sotho Speakers

The second study I wish to consider is a study of the scientiﬁc understanding of heat in thermodynamic processes amongst rural Sotho speakers in South Africa conducted by Marianna Hewon [115]. The Northern Sotho live in a hot, semi-arid part of South Africa where adequate water resources are a source of constant concern. In this environment, they have developed the following conceptual metaphor about “heat”. Cool- ness is associated with calm and health, while hotness is associated with anger, sick- ness, and exhaustion. The underlying mechanism is that “heat” is said to cause “agitated blood”. For example, an impatient or restless person is described as “hot” and it is explained that their blood is in an “agitated” state. Hewson suggests that this metaphor 174

is grounded in their experience of the harsh environment.

Hewson interviewed 20 subjects who had had no formal schooling in thermodynamics. Subjects ranged from high school children to factory workers. The majority had completed 8 years of formal schooling. Interviews were conducted in English. First, Hewson checked to see if the conceptual metaphor of heat (described above) was present. She found it present in 16 out of the 20 subjects. Then Hewson asked the interviewees about their understanding of heat in thermodynamic processes. Subjects were asked to explain the phenomena of water, placed on a burner, coming to a boil, the conduction of heat along equal length rods made of different materials, and the expansion of solids, liquids, and gases when heated [115]. From the responses of the subjects to the interview questions Hewson judged a remarkable number (12 out of 20) of the subjects to have kinetic or pre-kinetic views about heat. For example, one in- terviewee said, “I think heat makes the molecules that are in [the material]. . . to move faster. . . and when they are moving fast they are causing more pressure because they bump against each other.” Only 2 subjects had a consistent caloric view of heat.

This result was compared against similar studies given to Western school children. In one study, for example, 80% of school children in the 12 – 13 year-old age group were shown to have a consistent caloric conception of heat [116].

This study represents another test of the ideas of this thesis and also a test of the Sapir-Whorf hypothesis. Namely, humans interpret what they see through the language that they speak. Reality is mediated by the cultural metaphors. In this case it is well documented that the Northern Sotho have a powerful conceptual metaphor associated with the concept of heat, that heat causes a state of agitated blood. The presence of this metaphor would suggest (if the Sapir-Whorf hypothesis is correct) that Sotho speakers would associate increased agitation with an input of heat in a thermodynamic process. This appears to be the case from Hewson’s study. It is a view that is much closer to the “correct” expert view of heat in thermodynamic processes as compared to the caloric view that is so predominant in western thought. 175

6.6 Implications For Teaching

It may seem as though I am embarking on a crusade to change the language of physics. Far from it. From the metaphorical analysis of the language of physics it seems clear that part of the success of physics stems from the way in which abstract concepts are metaphorically elaborated as familiar substances. I would argue that in many cases, attempts to change how physical theories are articulated may be doomed to failure. The evidence reviewed in this dissertation suggests that making sense of the unfamiliar in terms of the familiar through the medium of metaphor is one of the most fundamental traits of human cognition and without it, science as we know it could not take place. Both the applicability and limitations of any analogical or metaphorical picture are important. Humans make sense of their world, not only through exemplars, but by considering what a concept is not [117].

I wish to suggest however, that physics educators take more holistic view of what it means to learn physics. They should pay more attention to the linguistic aspect of learning physics. Learning physics is more than using equations to solve back of chapter problems. It is also more than gaining an understanding of concepts or a deep understanding of the connections between concepts. Part of the act of learning physics may be elaborated by an analogy to mastering a second language: To master another language you need to do more than learn the words, you need to master the subtle nuances of the native speakers’ speech. I have tried to demonstrate through grammatical and metaphorical analyses, that the frames of physicists’ speech, although powerful and useful, can also be misleading and confusing. Physicists necessarily articulate their ideas in ﬁgurative language and as a result they do not always say what they literally mean.

I have given examples where physicists’ metaphors may be overextended by being taken too literally. If students do not realize that metaphors only highlight aspects of 176

meaning, it could explain many aspects of students’ confusion. Many student “misconceptions” may be better accounted for by considering the possibility that students are taking the language they hear and read too literally.

The linguistic aspect of learning physics should be considered and explored. Can we teach physics students to stop and ask: “What does that phrase actually mean?” And approach answering that question with language comprehension skills as well as knowledge of physics. Can such a metalingual awareness help students to become deeper and more understanding participants in the knowledge structures of physics? More research is obviously needed.

6.6.1 Can We Speak Better?

On the subject of using better language to talk about physical ideas and processes, I suggest that in some cases we can do better. The framework I have developed provides clear and general guidelines for better language use. These are laid out below.

1. Identify the matter, processes and states of a particular physical model that you want to talk about. This is what I have termed the “lexical ontology” of a model.

2. Try to use language that is consistent as possible with the lexical ontology of the model. It is, of course, impossible to achieve perfect alignment between grammatical usage and lexical ontology, but the closer the alignment, the less the cognitive load placed on the students. Some practical examples are shown in Table 6.2

3. Where ever there are examples of inescapably ambiguous language, it cannot hurt to have an explicit discussion of what that language really means. For example, when a physicist says that object A exerts a force on object B, it does not mean that object A has to be alive. It also does not literally mean that a material “force” is exchanged in any way between the two objects, it means that two objects undergo a process of interaction. In a case like that of the potential well 177

Table 6.2: Examples of common sentences that do not reflect the underlying lexical ontology of a particular model and suggestions for their improvement. Typical bad sentence Improved version of sentence Heat flows from A to B. Energy flows from A to B by heating. Energy flows from A to B as heat. Energy flows from A to B by heating. Object A did work on object B. Object A transferred energy to object B by working The tension in the string is. . . The force that the string exerts on the pendulum bob is. . . The normal force of the table is. . . The force that the table exerts on the book is. . . Consider all the forces acting on A. Consider all the other objects that A interacts with.

metaphor, I believe it might help to have a discussion about the imagery associated with the metaphor, what it means, and what it does not mean. Students need to be given the opportunity to explore the applicability and limitations of such a metaphorical system.

6.6.2 Language and the Road to Understanding

Evidence has been presented that there seems to be a natural order to learning [109]. When a physicist or physics teacher stands up in front of a class they speak in the language of the expert. They use metaphors whose meanings are not always obvious. They speak with large nominal groups and use nominalizations of processes that are seldom made explicit. And if the meaning is made explicit, probably not more than once. Studies of the brain reveal, that we do not learn this way. Understanding and reflection comes before terms can be named and categorized [109]. Naming is the last step of the process. This is one of the most significant effects of our language. By naming and defining first, we are probably making the task of learning and understanding more difficult [11]. Learning takes place through a process of exploration of the applicability and limits of the models that students learn. The same exploration needs to take place in the context of language. I suggest that we can alleviate many student 178

difﬁculties by actively facilitating students’ exploration of the meaning of the language they read and hear.

Part of the usefulness of the theoretical framework I have developed is the following: A linguistic representation of a model is a representation like any other. A grammatical and metaphorical analysis will allow us to understand the sorts of model(s) that may be constructed from the words we use. We can then judge what sort of representation we have, what its limitations are, how it might be misunderstood by students. What do we , the physics experts, leave implicit in our words and students may miss etc. . . ?

6.6.3 The contextual dependence of students’ knowledge

It has been noted that students’ knowledge and reasoning processes (in contrast to expert knowledge and reasoning) tend to be context dependent [64]. Although priming is well understood in terms of spreading activation models of the brain [118], the actual cause of the priming in testing physics students is not well understood. Here a linguistic view may be useful. It may be possible to analyze the language with which a particular question is phrased, pick out the underlying metaphors and ontology in that language and look for correlations between the models implied by the ontology/metaphors and the ontology/metaphors in students’ responses to the question. If students’ responses are indeed inﬂuenced by the ontology and metaphors implicit in the questions we ask, such a view would help in the design of summative assessment tools and maybe even open up a new linguistic component of assessment. I suggest that we examine the linguistic component of context more closely.

6.7 Conclusion

In this dissertation I have used the techniques of cognitive linguistics and systemic functional grammar to develop a new perspective on how knowledge is structured in 179

language used by physicists. I have established a linguistic view of communication in physics in terms of metaphors, metaphorical systems, ontology, and grammar. I have shown that the unconscious metaphors underlying expert physicists’ talk and writing may have an influence on student learning and understanding of physics. The multi- layered metaphorical systems implicit in the discourse of physics also reveal interesting aspects of how physicists encode their knowledge and make models of physical systems. This understanding may be used to help make the knowledge structures of physics as transparent and comprehensible as possible to physics students. The model of metaphorical language and thought in physics established in this thesis may also be used to make predictions about sources of students’ difficulties, possible teaching strategies and the contextual dependence of novice knowledge in physics. These ideas obviously need to be tested further. As the applicability of this theoretical framework is established, it could help to increase our understanding of communication between physics instructors and physics students and provide new insight into some of the difficulties students have in learning physics. 180

Appendix A

A Glossary of Terms

Base: I will use this word ubiquitously for any sort of comparison (analogical, metaphorical or otherwise) between two distinct ideas or conceptual domains. The base is the familiar, well understood (often concrete) domain that is begin use to interpret or classify an unfamiliar (often abstract) domain (the target).

Cognitive apprenticeship is a model of learning in which students become apprenticed in the tools and techniques that the experts use in order to learn. In analogy with the way in which a carpenter might become apprenticed in using the tools of the carpentry trade. The key to apprenticeship in physics is not to tell students about ideas of physics, but to have them learn the ideas of physics by doing it themselves (as if they were physicists doing research).

Conceptual metaphor, also known as a conventional figurative is a metaphor that has been used so much in language and is so ubiquitous to the group of language users that it is no longer perceived as a figure of speech, but is viewed as a statement of fact. Some examples of conceptual metaphors identified by cognitive linguists are

TIMEISMONEY and ARGUMENT IS WAR [38].

Grammatical metaphor is a grammatical device where one part of speech replaces another in a different grammatical class. The most common example of grammatical metaphor is nominalization. Nominalization refers to a process whereby a noun or noun group is substituted for a verb or verb group. For example, “here (energy) [noun] (is transferred) [verb group] (internally) [adverb] . . . via the external force,” is reused later as “We (want to relate) [verb group] (the external force) [noun group] . . . to (the 181

internal energy transfer) [noun group].” [54] vol 1. p.184. Notice how . . . energy is transferred internally. . . ” has been replaced in the second sentence by a single noun group, “the internal energy transfer”.

The ideational function of language is a meta-function of language whereby language serves to describe “patterns of experience” [52] in the world.

The interpersonal function of language is a meta-function of language whereby language functions as an exchange between speaker and listener through which the speaker can issues commands, demands, requests, offers of exchange etc. . .

Legitimate peripheral participation is a model of learning in which the learner is viewed as starting out as peripheral to a knowledge system (made up of practitioners and their knowledge). The learner moves from peripheral to central participation in the system if their place in the system is legitimate and their “centripetal” movement (towards becoming deeper participants) is properly supported by the expert practitioners.

Lexical difficulties are difficulties students have with the meaning of a word in physics. Underlying this is an assumption that physicists can agree on the meaning of a term and therefore could, in principle, enter the term into an imaginary dictionary of physics terms. Hence the term lexical difficulties. Difficulties can arise from either (1) if physicists cannot agree on the definition of a term, or (2) if the every day meaning of a term and its technical definition in physics are different or contradictory [16].

A nominal group is a phrase that performs the grammatical function of a single grammatical participant. For example: “{The boy who kicked the ball} [medium] is [process] very tall.” (The nominal group is indicated in curly brackets.)

Nominalization refers to the process by which a complete sentence is turned into a single nominal group functioning as one grammatical participant in a sentence. For example, the progression from “The boy [agent kicked process] the ball [medium]. He is very tall.” to “{The boy who kicked the ball} [medium] is [process] very tall.” is an example of nominalization.

Ontology is the study of existence. How things exist or how we see the existence 182

of the world around us. Throughout this thesis I will adhere to a very simple division of the world into three ontological categories (categories of existence): substances, processes and states. Chi et al. [5] have argued that there are subcategories within each broad category. For example, one could divide the substances category into animate and inanimate substances.

Ontological groping is a term I have introduced in this thesis. It describes a process I have observed in the historical record of physics discoveries. Essentially ontological groping is what physicists do when they try to ﬁgure out how to talk about terms and the meaning of terms that they introduce in physics.

Scaffolding is a metaphor from construction sites. Essentially when an instructor supports a student’s thought processes (without telling them what to do) by helping them to think or providing extra resources that the student does not possess, we say the teacher is “scaffolding” the student or the student’s learning process. Its support, but not any sort of support. Telling the student the answer to a problem would not be considered scaffolding, but reminding her/her of possible problem solving techniques he/she has learned that might be useful to tackle the problem would be considered legitimate scaffolding. Essentially scaffolding refers to the act of a teacher providing a support structure to aid students’ reasoning or learning. The term “scaffolding” also implies the removal of the scaffold at a later time when the student has developed a deeper understanding and more familiarity with the ideas.

Target: I will use this word ubiquitously for any sort of comparison (analogical, metaphorical or otherwise) between two distinct ideas or conceptual domains. The base is the familiar, well understood (often concrete) domain that is begin use to interpret or classify an unfamiliar (often abstract) target domain.

The textual function of language is the meta-function of language whereby the language functions to convey a message. 183

Appendix B Interview Questions About QM for Physics Professors

B.1 Questions About Professor’s Own Research

This is a series of questions to look at the epistemological issues behind major conceptual change.

• Can you please describe some aspect of your research

• How do you generate new ideas?

• When you see a new system in your research, so you describe it in terms of something familiar? Eg: “system X is behaving like an harmonic oscillator.”

• How do you know that a particular picture is applicable? What cues you to think about an harmonic oscillator when looking at this system?

• Do you try out different pictures? If so, how do you decide which pictures are good and productive and which should be thrown out?

• Will you use a certain picture even if it has serious limitations and why?

B.2 Questions About the Heisenberg Uncertainty Principle

• How do you understand or justify the Heisenberg uncertainty principle?

– In particular would you consider it a fundamental concept or something incidental or an artefact of the mathematics? 184

– How would you explain it physically? How does it manifest itself in real physical experiments or processes?

• What do you think of the validity of explaining the Heisenberg uncertainty principle using the idea that measurement of the system “disturbs it”? In other words, Heisenberg’s (and others) argument went something like: if you want to measure the position of an electron, you will have to scatter a photon off it. Because of their comparable size, this scattering signiﬁcantly affects the trajectory of the electron, hence introducing an uncertainty in the measurement of the electron’s subsequent position and momentum. Is this description

1. Conceptually correct?

2. A good and/or effective way to explain the uncertainty principle to beginning students?

3. Conceptually incorrect and should not be presented to the students at all.

• If you had to give a lecture on the uncertainty principle to students taking their ﬁrst serious course in quantum mechanics, how would you explain it/elaborate on it?

Here are some questions whose goal is to examine epistemological issues behind various ways of visualizing the Heisenberg Uncertainty Principle:

• How does the Heisenberg Uncertainty Principle apply to the cooling of atoms to absolute zero? Does it represent a fundamental limit beyond which one cannot go?

• Talk about the cooling from the perspective of single atoms interacting with photons. What does localizing the momentum (bringing the atom to rest) imply for its position? 185

• Talk about localizing a single QM particle in a box and shrinking the walls. What happens?

• Talk about the cooling experiment from a statistical standpoint. Even if some atoms disappear out of the system, surely some will stay behind?

B.3 Questions About Visualization or Mental Images

• Can you draw a picture of a photon? If not, can you describe how you visualize it? Can you draw a picture which explains or represents the sort of mental imagery you use to picture a photon?

• How do you explain that a photon and a wave picture of light are compatible? In other words, they give the same predictions for measuring things like intensity.

• How do you visualize/draw/imagine an atomic orbital?

B.4 Questions About the Schrödinger Equation

• How would you derive the Schrödinger equation:

– To yourself if you were stranded on a desert island and had forgotten it?

– To quantum mechanics students you are teaching?

• Do you know how Schrödinger came up with his equation himself?

B.5 Questions About the Probabilistic Interpretation of the Wave

Function

• How do you justify or understand the motivation for the probabilistic interpretation of the wave function yourself? 186

• How would you explain or motivate the probabilistic interpretation of the wave function to your students?

• Probe further as follows: Can you explain what motivated Born to propose the probabilistic interpretation of the wave function in contrast to Schrödinger’s interpretation of the wave function as representing “matter waves”?

B.6 Questions About Wave-Particle Duality

• What does the term “wave-particle” duality mean to you?

• What does "wave-particle" duality mean experimentally? How does it manifest itself in quantum mechanical experiments?

• Consider the following two statements and pick the one which you feel is more satisfactory:

1. An electron behaves either a wave or a particle depending on the type of experiment you subject it to. The electron is sometimes a wave and sometimes a particle.

2. An electron possesses wave-like properties and particle-like properties, but neither of these two descriptions completely adequately describes the essence of what an electron really is.

• Please describe what is happening with a simple Quantum Mechanical double slit interference experiment with electrons. Describe what happens when a single electron interacts with this system.

• Explain what you understand about Bohr’s principle of complimentarity. Why did he introduce it and what is it good for in physics teaching today? 187

B.7 Questions About Measurement

Please explain the Schrödinger’s cat thought experiment:

• What is the point?

• What does it mean?

• Why did Schrödinger invent it?

B.8 Unclassiﬁed Questions

• Can you suggest some classical wave analogs of a beam of electrons scattering off a potential down-step?

• What is the importance of various formulations? In classical mechanics we have Newtonian mechanics, Hamilton’s Principle, the Principle of Least Action, the Hamilton-Jacobi Equation and others. Why learn all these different approaches, what does it do for you? Compare this situation to quantum mechanics. We have many different formulations: many worlds, sum over histories, path integral formulation and so on, yet students only learn the Schrödinger formulation with maybe a bit of matrix mechanics and the Copenhagen interpretation. Why is it so restricted? 188

Appendix C Analysis of Historical Accounts of Force

All quotations are from [33]. From the comparative analysis, four categories of analogy emerged. The analysis is presented below.

C.1 Origins of the Four Analogies

C.1.1 Origins of the “force is like an animate external entity/agent”

analogy

The earliest examples of this idea are:

• Anaxagoras’ theory of mind [33] (p.25). Anaxagoras wrote about external agents, extended in space, and a “world-forming spirit” (p.26). In particular, I would argue that the usage of the word “nous” or “mind” implies an interpretation of the divine spirit in terms in human desires or human will.

• Aristotle evoked a “prime mover” to explain the motion of the heavenly bodies. In Aristotle’s view, the heavens are separated from the earth and not governed by the same physics.

C.1.2 Origins of the “force is like an passive external medium of

interaction” analogy

• Jammer observed that the ancient Greeks saw a connection between tides and movement of sun and moon. Aristotle suggested that there is no action at a 189

distance. This inspired Crates of Mallos to argue that the moon acts on the intermediate air which in turn acts on the water. So, in effect, air is the medium of the interaction. This suggests a possible analogy to the process of sound (p.42)

• Poseidonus wrote about a “doctrine of tonos” — a universal tension between all matter. Another way to see this is in terms of the concept of sympathy in the sense of plucking one string on a harp may set another vibrating. Later Alexan- der of Aphrodisias proposed an “all-pervading ﬂuid”, pneuma, that conveys the symphathy or “sympatheia” (p.43).

• Al-Kindi “conceived force as an entity propagated by rays. Not only light, and perhaps heat, according to Al-Kindi’s view, but also every other type of force.” (p.57).

C.1.3 Origins of the “force is like an animating spirit” analogy

Empedocles’ described a principle of love and strife as binding and separating “powers”. Later, this idea was interpreted by Plato as the agents of attraction and repulsion. These ideas represent an analogy for cause in human emotion. In Jammer’s view, nature is endowed with soul and life that animates and causes things to happen. In these models the animating spirit is intrinsic in matter according to Jammer’s interpretation.

C.1.4 Origins of “force is like an internal passive tendency” anal-

ogy

The basic idea of the ancient Greeks was that rocks fall because of their inherent tendency to unite with the earth. Likewise air bubbles rise in water because their natural place is in the air. Aristotle used the word “dynamis” do describe this (p.34).

Aristotle took this natural tendency analogy and extended it. He suggested that 190

there are two types of motion: “natural”, and “forced”. So, things have a natural tendency inherent in them. But to make an object go against its natural tendency requires an external force. It cannot be detached from the original object, but is somehow outside of it. Thus he seems to have combined two separate analogies into one notion of causation.

C.2 Historical Development of the Analogies

There are a number of models of force and motion that are grounded in three of the four categories that emerged from my analysis. There are, however, no scientific models of force based on the “force is like an animate external entity/agent” analogy. Interest- ingly, the idea of force as an animate external agent will turn out to be one of the most pervasive ways of speaking about “force” in physics. Throughout the historical record quoted in Jammer’s book, a lot of philosophers, probably under religious influence, tend towards the divine external agent model of force as a way of speaking. However, an omnipotent divinity is not really a falsifiable hypothesis. This probably explains why scientific explanations based on this idea did not materialize as serious theories in the historical progress of physics.

In this section I will present, in turn, how each of the historical scientiﬁc models of force falls naturally into the emergent categories described in Section C.1. I will then show how each of the themes evolved into a way of speaking.

C.2.1 Scientiﬁc development of the “force is like an animate exter-

nal entity or agent” analogy

As mentioned above, this analogy was never used as a serious scientiﬁc explanation for anything. Yet physicists seem to have co-opted the analogy and used it as a conceptual metaphor from early on. It appears historically as a useful way of speaking. For example: 191

• Galileo’s teacher, Buonamici viewed force as: “an agency that caused unnatural motion and was, so to say, an intruder in the otherwise harmonious system of natural processes.” (p.95).

• Leonardo da Vinci seemed to combine force as external agent and invigorating spirit. He wrote: “Force I deﬁne as an incorporeal agency, an invisible power, . . . imparting to these an active life of marvellous power. . . ” [33] (p.96).

What I ﬁnd amazing is that by the time we get to Descartes, he is already using force metaphors that directly contradict the model he is trying to explain. For example: “But I desire now that you consider what the gravity of this earth is, that is to say, the force which unites all its parts, and which makes them all tend toward its center. . . ” He then goes on to elaborate a model of vortices which sounds very much like a precursor to the ethereal models of force propagation. In this sentence, however, the word “force” functions grammatically as the agent. Thus suggesting an ontological metaphor of force as animate external agent.

C.2.2 Scientiﬁc development of the “force is like an passive exter-

nal medium of interaction” analogy

• Early models of force as an external material substance come from Bacon. Jam- mer argues that Bacon was inﬂuenced by Al-Kindi. For example, Jammer quotes Bacon’s writing: “For every efﬁcient cause acts by its own force which it produces on the matter subject to it, as the light of the sun produces its own force in the air and this force is light diffused through the whole world from the solar light. This force is called likeness, image, species, and by many other names, and it is produced by substances as well as accident and by spiritual substance as well as corporeal” (p.59). 192

• In a similar time period to Oresme and others, John Buridan seems to have suggested a notion of a field of forces: “It was assumed that the virtus caelestis permeates all space and thus exerts its influence on the bodies, more or less like a stationary field of forces.”

• Girolamo Fracastoro suggested the mechanism of sympathetic activity and combined with with Bacon’s “species” which was transmitted between bodies: “When two parts of the same whole are separated from each other, each sends toward the other an emanation of its substantial form, a species propagated into the in- tervening space; but the contact of this species each of the parts tends toward the other in order to be united in one single whole; this is the way to explain the mutual attraction of like to like, the sympathy of iron for a magnet being a typical example” (p.73). This example is interesting since it combines aspects of force as a medium of interaction, an also as an internal tendency, but at the same time, interpreted in terms of human desire for union. There are various variations on this theme which Jammer classiﬁes as “attraction of the parts by the whole” (p.80). In all of these cases force is external.

• Max Jammer says that: “. . . Kepler clearly envisages the forces of attraction exerted by the earth on a stone as magnetic lines, or chains, as he says, thereby approaching Gilbert’s conception of gravity as a magnetic emanation.” (p.84). In Kepler’s later work in seeing a connection between the distance of the planets from the sun and their speed, he introduced force as an “intermediary concept” (p.88). Thus force becomes bound up in notions of causality. These notions were probably already there, but Jammer does not really focus on this. Kepler wrote: “The planets are magnets and are driven around by the sun by magnetic force.” p.89. Jammer summarises this idea as follows: “Kepler imagined these magnetic forces, emanating from the central body such as the sun, to be like giant arms propelling the planets on their appropriate orbits.” (p.89). 193

C.2.3 Scientiﬁc development of the “force is like an animating spirit”

analogy

• During the time of Buridan’s impetus model of motion, religious scholars took over Buridan’s impetus, viewing the impetus as analogous to the soul that animates the human body. For example, Guido Ubaldo del Monte wrote: “When we say that a heavy body tends by a natural propensity to place itself in the center of the universe, we mean to say that this heavy body’s own center of gravity tends to be united with the center of the universe.” (p.78). This is a view of force as something internal, yet conscious and deliberate.

• Kepler also made a contribution to this branch of reasoning when he wrote: “Gravity is a mutual affection among related bodies which tends to unite and conjoin them. . . no matter whereto the earth is transported, it is always toward it that heavy bodies are carried, thanks to the faculty animating it.” (p.85).

• Gilbert echoed these ideas when he wrote: “. . . it is in bodies themselves that acting force resides. . . ” (p.86) Note the use of the word “acting”, which suggests more than just a passive tendency.

• Gilbert’s view must have also inﬂuenced Nicolas Oresme who describes “the concept of force as an inherent activity, causing the motion of the body.”

C.2.4 Scientiﬁc development of “force is like an internal passive

tendency” analogy

• On the side of viewing force as an inherent property of the motion of an object, Buridan was a major contributor when he suggested an explanation for circular motion of bodies on the celestial plane as “. . . an impetus that kept them moving. . . ” (p.70) 194

• Albert of Saxony elaborated the view a little more eloquently: “. . . When God created the celestial spheres, He put each of them in motion as He please; and they continue in their motion still today by virtue of the impetus which He impressed on them. . . ” (p.70)

• Kepler vacillated a lot according to Jammer. In his early work, Kepler suggested that: “Gravity is not an action but a passivity of the stone which is attracted.” (p.82). Later on however Kepler starts to argue that forces of attraction exerted by the earth are external as was mentioned earlier.

C.3 Development of the Modern Language about Force

C.3.1 Newton

Let us now consider Newton’s writings. First of all Jammer argues that Newton seems to be making some concessions to history because he appears to adhere to the dual force model. Namely a separation into inertial forces and active forces. As Newton puts it: “. . . [the] innate force of matter, is a power of resisting by which every body, as much as in it lies, continues in its present state. . . ” (p.119). Basically, Newton seems to view inertia as a force. Newton then provided a separate deﬁnition for an impressed force: “An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.” (p.121) Note that Newton uses a nominal group (“impressed force”) as the grammatical medium of the sentence. This deﬁnition for “force” is very important because it represents a process conception of force, rather than seeing force as a thing. This is suggested both by the accompanying “impressed” and by the suggestion that “impressed force is an action” (my emphasis).

What is striking is Newton’s use of unconscious metaphors as in the following example: “A body, acted on by two forces simultaneously. . . ” (p.128). Here force is functioning grammatically as the medium in an action process. Thus, in my coding 195

scheme, force should be interpreted ontologically as living matter — the initiator and cause of motion. Newton appears to me to present a prototypical example of how modern physicists talk. Max Jammer describes it best himself:

“While he speaks of ‘bodies attracting each other,’ ‘the attractions of one corpuscle towards the several particles of one sphere,’ of ‘mutual attraction,’ and uses similar expressions that could easily have misled the reader to the assumption that he conceived of the forces involved as innate in matter and acting at a distance, Newton nowhere in the ﬁrst edition of the Principia made a statement to this intent.” (p.137).

In fact, Newton suggested the possibility of an ethereal medium, but in the end expresses his wish to abstain from judgement and also clarify his language when he says:

“I here use the word attraction in general for any endeavor whatever, made by bodies to approach each other, whether that endeavor arise from the action of the bodies themselves, as tending to each other or agitating each other by spirits emitted; or whether it arises from the action of the ether or of the air, or of any medium whatever, whether corporeal or incorporeal, in any manner impelling bodies placed therein towards each other.” (p.137)

C.3.2 Alternative conceptions of force

By far the strongest emergent theme following Newton, was the conception of force as some sort of excitation in an ethereal medium. After Descartes, Robert Hooke was one of the earliest to suggest an ethereal model: He suggested that massive objects “execute small, imperceptible, and rapid vibrations or spherical pulsations. . . ” (p.189). Objects are drawn towards the earth in analogy with the experience that small ﬂoating objects move towards the center of water waves.

Euler also speculated:

“But if you suppose that the intermediate space is ﬁlled with a subtile matter, we can comprehend at once that this matter may act upon the bodies by impelling them. The effect would be the same as if they possessed a power of mutual attraction. Now, as we know that the whole space which separates the heavenly bodies is ﬁlled with a subtile matter called aether, it seems more 196

reasonable to ascribe the mutual attraction of bodies to an action which the aether exercises upon them, though its manner of acting may be unknown to us, than to have recourse to an unintelligible property.” (p.194)

The development of this theme and its extension to electrodynamics and the development of the ﬁeld concept shall be presumed to be sufﬁciently well known and does not bear repeating here.

During Newton’s time and subsequently, there seems to have been a long debate between two groups of scientists. On the one side were those who said F = mv2. Leibniz started this idea. It is my thesis that this equation was never seen historically as a causal relation between physical variables, but as a definition of a physical quantity, F . Viewed through the development of the four themes, this seems to be an attempt to quantify that inherent activity in an object. In other words, a quantification of the “force is like an animating spirit” analogy. The connection of the equation to this analogy is suggested by the following examples: Max Jammer describes Leibniz’s understanding of force as a “. . . concept of force [that] we call today kinetic energy, but conceived as inherent in matter and representing the innermost nature of matter.” (p.158). Belanger described this definition of F as “living power” (p.166). Leibniz’s idea was championed by John Bernoulli, William Jacob s’Gravesande, Christian Wolf, Geog Bernhard Bulfinger and Samuel König.

On the other side of the argument were those who viewed force as the “quantity of motion” F = mv. (I will refer to this group as the “Buridanians”.) As mentioned before, I do not believe this equation to represent a causal relation. (As conceived by its authors.) In my view, this equation is also a deﬁnition of a physical quantity. The Buridanians were simply trying to quantify the analogy of force as a passive property or tendency of the object’s motion. This view was championed by the Abbe de Catelan, P. Jean Simon Mazeire, Colin Maclaurin, James Stirling, Samuel Clarke, and Jean Jacques d’Ortons de Mairan.

When studying the language of the Leibnizians I can identify the emergence of 197

the FORCEISCOMMODITY metaphor. For example, Leibniz wrote: “. . . two soft or unelastick bodies meeting together, lose some of their force. . . ’tis true, their wholes lose it with respect to their total motion; but their parts receive it, being shaken by the force of the concourse. And therefore that loss of force is only apparent. The forces are not destroyed, but scattered among the small parts. The bodies do not lose their forces; but the case here is the same, as when men change great money into small.” p.168. It is a metaphor that appears in very few modern sentences about force. However it is a metaphor that has been transferred wholesale to physicists’ language about energy.

What is most important to note at this point is the following: In the modern view of these equations it is often suggested [107] that they represent a physical law, for example, that a constant force is required to maintain a constant velocity. In my view the statement F = mv, placed in its correct historical context, is not a relation between physical quantities, but an attempt to deﬁne a new physical quantity in the same sense of a modern physicist writing p = mv. The debates between those who want to deﬁne the quantity of force that a body is endowed with as mv and those who want it to be mv2 do not represent any attempt to write down a mathematical relationship. Rather, both are trying to describe some aspect of the property of the object which is both related to how big it is and how fast it moves. (forcefullness, power etc...). Jammer summarises: “The conception of force as the primordial element of physical reality, advanced by Leibniz, Boscovich and Kant and their followers, was not very fruitful and productive for the advancement of theoretical physics.” (p.187).

The ﬁnal piece of the language puzzle comes from Boscovich. By appealing to continuity, Boscovich showed that all interactions involved some sort of action at a distance rather than direct contact. This included collisions. He then introduces the idea of “centers of force”. This idea carried all the way through to the 19th century. For example Tyndall says: “What do we know of the atom from its force?” (p.184). This small piece of history may represent the origin of some of physicists’ most common modern language about force. It is a way of speaking that suggests that force is a 198

substance residing inside the body, but then emerges and travels or is transmitted to another body. It seems to represent a metaphorical blend of the internal property and external medium metaphors.

C.4 The Fifth Theme: There is No Force

Berkeley was one of the ﬁrst scientists to suggest that force is more of a crutch than anything else. Jammer puts it best: “ ‘Force’ in his view has the same status in science as the notion of ‘epicycles’ in astronomy.” (p.204). Berkeley wrote: “Force, gravity, attraction and similar terms are convenient for purposes of reasoning and for compu- tations of motion and of moving bodies, but not for the understanding of the nature of motion itself.” (p.204) And later “Those who assert that active force, action, and the principle of motion are really in the bodies, maintain a doctrine that is based upon no experience, and support it by obscure and general terms, and do not themselves understand what they wish to say.” (p.206). And later still: “The physicist observes a succession of sense data, connected by roles, and interprets that which precedes in their order as the cause and that which follows as the effect. It is in this sense that we say that one body is the cause of the motion of another, or impresses motion on it, pulls it or pushes it.” (p.206).

Jammer summarizes the idea: “To introduce the term ‘force’ as an explanatory element in the theory of physical science means to develop a misleading vocabulary. . . ‘force’ is a construct of the conceptual scheme of physics and should not be confounded with metaphysical causality.” (p.206 & p.208).

Maupertuis’ had a similar view of force as an “analogy to the sensation we have when moving an object from its place or arresting the motion of another that we ascribe a similar state of affairs to the phenomena of physical motion.” Jammer goes on to elaborate: “However, says Maupertius, since we cannot emancipate ourselves completely from the idea that bodies exert mutual inﬂuences upon each other, we may 199

continue to use the term ‘force’. But we should always remember that the concept of force is but an invention to satisfy our desire for explanation.” (p.209)

Kirchoff, Hertz, and Mach took this idea and tried to eliminate the concept of force from physics altogether. From these physicists, force is a derived notion, from the product of mass and acceleration. As Mach eloquently put it: “I hope that the science of the future will discard the idea of cause and effect, as being formally obscure; and in my feeling that these ideas contain a strong tincture of fetishism, I am certainly not alone.” (p.221).

Hertz said that: “It cannot be denied that in very many cases the forces which are used in mechanics for treating physical problems are simply sleeping partners, which keep out of the business altogether when actual facts have to be represented.” (p.226) In Hertz’s scheme “We thus say that the motion of the ﬁrst body determines a force, and that this force then determines the motion of the second body. In this way force can with equal justice be regarded as being always a cause of motion, and at the same time a consequence of motion.” (p.229) In terms of language and functional grammar, Hertz is saying that the term force can function as agent or medium of the sentence with equal validity.

C.5 Summary

I have shown in this section four historical analogies used to understand and describe the concept of force. Three of these analogies seem to have been used by scientists to ground models of motion. The fourth, I hypothesize, must have fallen by the wayside due to its lack of falsiﬁability. In Chapter 5, I use this historical analysis to help classify modern language about force. Later in Chapter 5, I focus on the apparent confusion and debate about the meaning of force and what it may mean for our students’ learning of Newtonian mechanics.

Several philosophers mentioned in Jammer’s book seem to have been grasping 200

towards the modern neuroscientiﬁc and cognitive linguistic view that concepts are grounded analogically or metaphorically in our physical experience and emotional desires. DuBois-Reymond says:

“so far as it [force] is conceived as the cause of motion, is nothing but an abstruse product of the irresistible tendency to personiﬁcation which is impressed upon us; a rhetorical device, as it were, of our brain, which snatches at a ﬁgurative term, because it is destitute of any conception clear enough to be literally expressed. . . ” (p.235)

Pearson says in kinder terms:

“Primitive people attribute all motion to some will behind the moving body; for their ﬁrst conception of the cause of motion lies in their own will. . . Slowly, scientiﬁc description replaces spiritualistic explanation. The idea, however, of enforcement, of some necessity in the order of a sequence, remains deeply root- ing in men’s mind, as a fossil from the spiritualistic explanation which sees in will the cause of motion. The notion of force as that which necessitates certain changes or sequences of motion, is a ghost of the old spiritualism.” (p.235).

The feeling is that “While the modern treatment of classical mechanics still admit- ted, tolerantly, so to say, the concept of force as a methodological intermediate, the theory of fields would have to banish it even from this humble position [if a unified field theory could be suggested]” [33] (p.264) 201

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Vita

David T. Brookes

Education

1999 - 2006 Ph.D. degree at Rutgers University, USA. 1997 - 1998 MA, (Physics), Physics Brandeis University. 1996 - 1997 MSc. With distinction, University of Cape Town, South Africa. Dissertation on relativistic heavy ion collisions. 1995 BSc (honors) (Theoretical Physics) University of Cape Town, South Africa. 1992 - 1994 BSc (Majors: Physics, Applied Mathematics, distinction in Applied Mathematics), University of Cape Town, South Africa.

Publications

E. Etkina, A. Van Heuvelen, D. T. Brookes, and D. Mills, Role of Experiments in Physics Instruction - A Process Approach, The Physics Teacher, 40 351–355 (2002). D. T. Brookes, E. Etkina, and S. Barnhart, Integrating Video Technology Effectively into Instruction, in Society For Information Technology and Teacher Education, 14th International Conference (2003). D. T. Brookes, G. K. Horton, A. Van Heuvelen, and E. Etkina, Concerning Sci- entiﬁc Discourse about Heat, in Proceedings of the 2004 Physics Education Research Conference, edited by J. Marx, P. Heron, and S Franklin (American Institute of Physics, Melville, NY, 2005). D. T. Brookes and E. Etkina, Do Our Words Really Matter? Case Studies From Quantum Mechanics, in Proceedings of the 2005 Physics Education Research Confer- ence, edited by P. Heron, J. Marx, and L. McCullough (American Institute of Physics, Melville, NY, 2006). G. J. Aubrecht, Y. Lin, D. Demaree, D. T. Brookes, and X. Zou, Student Perceptions of Physics by Inquiry at Ohio State, in Proceedings of the 2005 Physics Education Research Conference, edited by P. Heron, J. Marx, and L. McCullough (American Institute of Physics, Melville, NY, 2006).