Explorations of University Physics in Abstract Contexts

Explorations of University Physics in Abstract Contexts
Digital Comprehensive Summaries of Uppsala Dissertations
from the Faculty of Science and Technology 239
Explorations of University Physics
in Abstract Contexts
From de Sitter Space to Learning Space
ISSN 1651-6214
ISBN 91-554-6713-X
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List of publications
This thesis consists of an introductory text and the following appended
research papers, henceforth referred to as Papers I-IV:
Danielsson, U. H., Domert, D., & Olsson, M. E. (2003).
Puzzles and resolutions of information duplication in de
Sitter space. Physical Review D, 68, 083508
Domert, D., Linder, C., & Ingerman, Å. (2005). Probability
as a conceptual hurdle to understanding one-dimensional
quantum scattering and tunnelling. European Journal of
Physics, 26, 47-59.
Airey, J., Domert, D., & Linder, C. (2006). Representing
disciplinary knowledge? Understanding students’ experience of the equations presented to them in physics lectures.
Extended abstract accepted for presentation at EARLI SIG2 2006 biannual meeting. University of Nottingham. 30
August – 1 September 2006, Nottingham, UK.
Domert, D., Airey, J., Linder, C., & Lippmann Kung, R.
(2006 submitted, revision in progress). An exploration of
university physics students' epistemological mindsets towards the understanding of physics equations. NorDiNa
(Nordic Studies in Science Education).
1. Introduction to the thesis..........................................................................9
1.1 The overarching theme ......................................................................9
1.2 Style of writing..................................................................................11
1.3 The significance of the thesis ...........................................................12
1.4 Research questions ...........................................................................12
1.5 Overview of the thesis ......................................................................14
Part I..............................................................................................................15
2. Exploring aspects of de Sitter space ......................................................17
2.1 Introduction ......................................................................................17
2.2 Basic Cosmology ...............................................................................17
2.3 A (very) brief history of our Universe ............................................20
2.4 Inflation .............................................................................................22
2.5 de Sitter space ...................................................................................24
2.6 A possible information paradox......................................................26
2.7 Resolving the apparent paradox – results of Paper I ....................28
2.8 Experiencing potentially abstract physics......................................29
Bibliography of Part I.................................................................................30
Part II ............................................................................................................35
3. Physics education research – overview and literature review ............37
3.1 Introduction ......................................................................................37
3.2 Overview of physics education research (PER).............................37
3.2.1 What is physics education research? ......................................37
3.2.2 The relationship between PER and physics ...........................38
3.2.3 A selection of main findings in PER........................................39
3.3 Conceptual understanding...............................................................42
3.3.1 Summary, history and theoretical development ....................42
3.3.2 Conceptual understanding of quantum mechanics ...............46
3.4 Physics education research related to physics equations ..............49
3.5 Epistemology, attitudes and beliefs.................................................51
4. Method and methodology.......................................................................55
4.1 Introduction ......................................................................................55
4.2 Methods of data collection ...............................................................55
4.3 Choosing an appropriate methodology ..........................................57
4.4 Phenomenography............................................................................58
4.4.1 The main ideas of phenomenography .....................................59
4.4.2 Data collection...........................................................................61
4.4.3 Data analysis .............................................................................61
4.5 Case studies.......................................................................................62
4.5.1 The main ideas of case studies .................................................62
4.5.2 Conducting a case study ...........................................................63
4.6 Trustworthiness and value in PER .................................................65
5. Exploring university students’ understanding of probability in onedimensional quantum tunnelling ...............................................................67
5.1 Introduction ......................................................................................67
5.2 What is quantum mechanics? .........................................................67
5.3 What is quantum tunnelling?..........................................................70
5.4 Background to Paper II ...................................................................70
5.5 Results of Paper II............................................................................71
5.5.1 Categories of students’ understanding of probability ...........72
5.5.2 Conceptual difficulties..............................................................73
5.6 Discussion of the results of Paper II ...............................................73
6. Exploring the role of mathematics in physics.......................................75
6.1 Introduction ......................................................................................75
6.2 Why does mathematics play an important role in physics? .........75
6.3 What is the role of mathematics in physics? ..................................78
6.4 Exploring students experience of equations...................................80
6.5 Exploring what students focus on when presented with equations
6.5.1 Research questions....................................................................81
6.5.2 What do students focus on? - Results of paper III.................82
6.5.3 Discussion of the results of Paper III ......................................84
6.6 Exploring university students’ epistemological mindsets towards
the understanding of physics equations................................................87
6.6.1 Mindsets.....................................................................................87
6.6.2 Research questions....................................................................88
6.6.3 Results of Paper IV...................................................................88
6.6.4 Discussion of the results of Paper IV ......................................92
7. Conclusions and outlook.........................................................................95
7.1 Concluding remarks.........................................................................95
7.2 Future research ................................................................................95
8. Svensk sammanfattning..........................................................................99
8.1 Introduktion......................................................................................99
8.2 Forskningsfrågor............................................................................100
8.3 Metod och metodologi ....................................................................101
8.4 Resultat............................................................................................102
8.4.1 Att undvika en potentiell informationsparadox...................102
8.4.2 Studenters begreppsmässiga förståelse av kvantmekanisk
8.4.3 Fysikaliska ekvationer och studenters fokus........................104
8.4.4 Fysikaliska ekvationer och epistemologiska attityder .........104
8.5 Några avslutande ord.....................................................................105
Acknowledgments .....................................................................................107
Bibliography of Part II .............................................................................109
1. Introduction to the thesis
1.1 The overarching theme
At one level, my research presented in this thesis is in two very different
fields of physics – theoretical physics and physics education research – so at
first it might seem that these two areas have very little in common. However,
from a learning perspective several common attributes could be brought to
the fore, and I also believe that my research in theoretical physics provided
me with essential insights into physics learning that helped inform my research in physics education.
There is however an explicit common denominator in this thesis that link my
work in theoretical physics and my work in physics education research. This
link is what I have chosen to call potentially abstract physics. What does this
term mean? Let us look at the individual words of this term in order to get an
understanding of the meaning of potentially abstract physics. I assume that
the reader knows or at least has an idea of what physics means, so let us
focus on the other two words: “abstract” and “potentially”.
The word “abstract” can take on many different meanings. To many people
it is simply a synonym of words such as “difficult” and “complex”. This
common usage is, however, not the meaning I would like to attribute to the
word “abstract” in this thesis so let me for the moment focus on pinning
down the meaning of “abstract” which I would like a reader to have in mind.
Let us for a moment turn our attention to a scene where the word “abstract”
is perhaps most well-known: abstract art. What does “abstract” mean in this
context? In the world of art “abstract” normally refers to art that is not a
realistic representation of something concrete. Another way of putting it is to
say that abstract art does not have a recognizable subject that relates to anything familiar. Or to put it even more simply: abstract art is art which depicts
something that we cannot recognize from our everyday life and past experiences. Lifting out the word art from the last sentence brings out the meaning
of “abstract” that I want the reader to bear in mind throughout this thesis:
abstraction is used to characterise something that we cannot recognize from
our everyday life or past experiences.
So the common denominator of this thesis – potentially abstract physics –
involves physics that we cannot relate to our everyday life or past experiences. However, I have not yet explained why I have decided to put “potentially” put in front of abstract physics. The reason for this is that whether
something is perceived as abstract or not is dependent on the individual perceiver. Something perceived as abstract by one person is not necessarily
perceived as abstract by another person. There are, however, parts of physics
that are likely to be perceived as abstract and these parts are what I refer to
as potentially abstract physics in this thesis.
Many branches of physics contain concepts or phenomena that could be
classified as potentially abstract physics – such as electric and magnetic
fields in electromagnetism – but potentially abstract physics is more prominent in some areas of physics than in others.
In classical mechanics, which deals with the motion of objects and the cause
of these motions, we have a lot of earlier experience with these issues that
sometimes help us and sometimes hinder us in dealing with aspects of this
branch of physics. Velocity, acceleration and force are all familiar terms in
our everyday life (even though some terms may take on different meanings
in our everyday life compared to their meaning within physics). It is a completely different story when we get to other fields of physics such as, for
instance, the special and general theory of relativity, quantum mechanics,
cosmology and string theory.
Special theory of relativity contains some highly counter-intuitive phenomena and results, such as the relativity of times and lengths; different people
can measure different times and lengths for an event depending on their relative motion. However, these effects only become noticeable when we are
concerned with speeds close to the speed of light. This is not the case in
most people’s everyday lives – unless you happen to be a particle physicist
working at an accelerator – and we therefore lack previous experience of this
General relativity, which may be characterised as a generalisation of special
relativity, features a geometrical model of gravity that is almost unimaginable from everyday life experience. Within general relativity, space and time
is no longer a passive background but rather constitute a four-dimensional
space-time that is affected by the presence of matter.
Quantum mechanics is concerned with the particles that constitute matter in
nature, and consists of plenty of counter-intuitive and seemingly absurd
ideas that we have no experience with and cannot relate to our everyday life.
One of the ideas is, for instance, that a particle can behave both as a particle
and a wave. We clearly have no experience with such an object in our everyday lives. In fact, it is hard to imagine two things more different than a particle and a wave. I will return to the peculiarities of quantum mechanics in
greater detail later in this thesis.
String theory (like quantum mechanics) is concerned with the building
blocks of matter, but at an even unimaginably smaller scale. In string theory,
matter is assumed, at the smallest scale, to consist of, in everyday terms,
“ridiculously small” one-dimensional vibrating strings that interact with each
other. Apart from the difficulties with imagining a one-dimensional object,
string theory also needs nature to have more dimensions than those we can
perceive. For string theory to be consistent, we need dimensions beyond our
three familiar spatial dimensions and the fourth dimension corresponding to
time. This is all clearly beyond our everyday experiences.
While string theory is concerned with nature at its smallest conceivable
scale, cosmology is concerned with nature at its largest conceivable scale. In
cosmology, we try to understand the origin, history, current state and further
development of the entire Universe. The Universe as a whole and the extreme conditions in the earliest states of our Universe are once again things
that we cannot relate to in our everyday life, and thus we have no previous
experiences to rely on and draw from in our sense making.
There are many other areas of physics where we have a similar situation, but
the examples given above should be sufficient to give the reader a sense of
what I mean by potentially abstract physics – physics that potentially can not
be related to our everyday life and earlier experiences. Besides the fact that
these areas probably are the most important cases where abstract physics
prevails, there is another reason for specifically mentioning these areas –
these areas of physics have been fields of interest in my own research in
different ways which I will later spell out in more detail.
1.2 Style of writing
I would like to seize the opportunity in this introduction to discuss how this
thesis is written and why. It is my strong personal belief that this kind of
publication should be accessible for as wide an audience as possible and I
consider making my research accessible outside of the research community
to be an important part of being a researcher. Therefore, this thesis provides
relevant background material in a non-technical way. However, this thesis
should also be of value for peer researchers, by presenting the contributions
of my own research in more technical terms, relating it to previous research
and giving some ideas on future investigations to which my research points.
These two thesis ingredients are not mutually exclusive and I hope that I
have managed to write this thesis in such a way that a wide breadth of readers can find things of interest and value.
1.3 The significance of the thesis
I would argue that this thesis is able to provide a significant contribution to
the development of understanding abstraction’s role in the discourse of physics. The research in theoretical physics corresponds to a description of potentially abstract physics from the perspective of an inside observer, and the
research in physics education corresponds to a description of potentially
abstract physics from the perspective of an outside observer.
More specifically, this thesis can be seen as making research contributions in
three main areas:
Understanding and investigating the nature of some potentially puzzling aspects of de Sitter space in relation to inflationary cosmological models.
Contributing to the exploration of university students’ understanding
of quantum mechanics in the area of quantum tunnelling.
Exploring university students’ experiences of physics equations in
terms of their focus when presented with equations and how they
epistemologically view what it means to understand physics equations.
1.4 Research questions
The first part of this thesis involves theoretical physics research in cosmology. Most contemporary cosmological models involve an era where the expansion of our universe is accelerating. Such an era is called inflation and in
Paper I we investigate some potentially peculiar properties that could occur
in a cosmological toy model where an inflationary phase represented by a de
Sitter space is followed by a particular (and highly realistic) type of phase in
the evolution of our Universe. In this toy model, we pose and investigate an
interesting question:
In a cosmological scenario, where a de Sitter phase is followed by a
phase with a scale factor evolving as tq, where 1/3<q<1, is there a
possibility of information being duplicated, violating the quantum
Xerox principle?
The second part of this thesis consists of research in physics education. Since
I was interested in theoretical physics, a natural step was to investigate from
an educational perspective an area of physics containing some kind of experienced abstractness. Paper II looks at one area of potentially abstract
physics by investigating students’ understanding of quantum tunnelling. The
main research question for this study was:
What is the variation in university students understanding of probability in one-dimensional quantum tunnelling?
From this study, it was clear that it can be difficult to conceptually understand potentially abstract physics and that mathematical representations play
an even more important role in this type of physics than in general. This
initiated an interest in students’ experience of mathematical representations
in terms of physics equations, leading to Paper III and IV. In Paper III, we
are interested in exploring the focus of students in relation to physics equations, and the main research question was:
What do university students focus on when presented with physics
The results of Paper III caused us speculate whether these focuses could be
related to a view of what it means to understand physics equations. So for
Paper IV we are interested in exploring students’ epistemological views of
what it means to understand equations. This is done by investigating three
questions in relation to what it means to understand physics equations:
When students say that they understand an equation, how do they
describe what that means to them?
How can these descriptions be characterised in terms of epistemological mindsets?
Are similar epistemological mindsets prevalent across levels for students at various stages in their academic career?
1.5 Overview of the thesis
In this introduction I have presented the reader with a description of the
overall theme and style of this thesis. I have also described the significance
of this thesis and presented the main research questions that are explored.
Chapter 2 presents the research reported in Paper I as well as a non-technical
background to some of the key aspects of this research. In Chapter 3, a general description of physics education research and its relation to physics as a
discipline is provided, as well as an overview of the literature relevant for
my research presented in this thesis. Chapter 4 presents the methodologies
used in my research, as well as a general discussion of the value and reliability of qualitative research. In Chapter 5, the results of Paper II are presented
and discussed, and a non-technical description of quantum mechanics and
quantum tunnelling is provided. Chapter 6 contains a discussion of why and
how mathematics plays an important role in physics, as well as a presentation and discussion of the results of Paper III and IV. The thesis is concluded
with Chapter 7, which contains some concluding remarks as well as an outlook and topics for future research, and Chapter 8 which provides a Swedish
Part I
2. Exploring aspects of de Sitter space
2.1 Introduction
In this chapter, I will present the reader with a discussion of what my theoretical physics research in Paper I involve. I attempt to do this by providing
the reader with very brief crash courses in my research areas and those areas
significantly related. Section 2.2 presents the basic cosmology necessary to
appreciate the discussion of Section 2.3, where I take the reader on a brief,
and far from complete, historical journey through the history of our Universe. Section 2.4 adds an era to this history called inflation and explains
why this “add-on” is necessary to deal with some problems related to the
history of our Universe discussed in Section 2.3. In Section 2.5 I discuss
some properties of de Sitter space that are relevant for my research in Paper I
and in Section 2.6 and 2.7 I discuss the scenario explored in Paper I and the
main results of that study. I conclude this chapter with some reflective remarks on my research in theoretical physics in Section 2.8.
2.2 Basic Cosmology
For a long time, the theories of physics involved what could be referred to as
absolute time and space. In these theories spacetime is a passive background
against which everything else takes place – a background which is unaffected by whatever takes place against that background. This view changed
dramatically with Einstein’s entrance into the physics arena. In Einstein’s
general theory of relativity, time and space are no longer a “passive” background. Energy (including the mass of objects) affects spacetime in ways
which are dependent on the details of the energy distribution. This is governed by a mathematical relation known as Einstein’s equation. In simple
terms, Einstein’s equation gives us the geometry of spacetime given the distribution of energy.
In cosmology we are interested in the geometry of our Universe as a whole,
which means that the energy distribution we should put into Einstein’s equation is the distribution of all of the energy in our Universe. How could this
possibly be done? This would at first sight imply that we had to know the
exact position and state of every single particle in the Universe. This is of
course an impossible task (personally I find it hard enough to know the exact
positions of important papers in my office!).
Fortunately, observations have indicated that our Universe is highly symmetric on large scales. On these scales our Universe seems to be homogenous (it
has the same matter density everywhere) and isotropic (it looks the same in
all directions). These observations already tell us a lot about the general
geometric properties of our Universe without us having to solve Einstein´s
Another crucial observation, first made by Edwin Hubble in the late 1920s,
is that our Universe is expanding. As time passes, our Universe grows larger
and larger. This is captured by what is known as the scale factor a(t) for our
Universe. As time goes by, the scale factor becomes larger and larger, reflecting the fact that our entire Universe is expanding. The distance between
two comoving points (two points moving with the expansion of the Universe) is proportional to this scale factor. It is not easy to visualize an expanding four-dimensional spacetime, but one way of at least getting a rough
idea (even though the analogy should not be taken literally) is to imagine a
balloon (representing our Universe) with small coins (representing galaxies)
attached to its surface (see Figure 1). As the balloon inflates (analogous to
the expansion of our Universe), the distance between these coins increases.
The increase in distance between two coins could be described by a scale
factor a(t) in the same way as for our entire Universe. One can imagine a
comoving coordinate system drawn on the surface of the balloon. In this
coordinate system, the coordinate distance d between two coins will stay the
same during the expansion since the coordinate system expands as well. The
actual physical distance between the two coins would then at each moment
in time be given by da(t) where a(t) is the scale factor at that given time.
Figure 1. An illustration intended to give an insightful idea of the expansion of our
universe in terms of coins attached to the surface of an inflating balloon. It can be
seen from this picture that the coordinate distance between the coins stays the same
since the coordinate system is expanding as well, while the actual physical distance
between the coins increases.
Given the fact that our Universe expands, and that the expansion can be described by a scale factor a(t), it is of course of great interest to determine
what this factor looks like and how it has evolved and will evolve. The rate
of expansion (how fast our Universe is expanding) is normally expressed in
terms of the Hubble parameter H = a /a, where a = da/dt. If we can find the
Hubble parameter, we know the rate at which our Universe is expanding.
Using Einstein’s equation (and reasonable assumptions regarding the matter
and energy in our Universe), it is possible to end up with an equation that
gives us the scale factor a(t) if we know the matter density ȡ in our Universe.
This equation is known as the Friedmann equation:
8SGU / K
3 a
In this equation ȁ is the famous cosmological constant, G is the gravitational
constant and K is a parameter related to the geometry of spacetime that can
be deduced from the values of ȡ and H today. If K=0 our Universe is flat, if
K > 0 our Universe is closed and if K < 0 our Universe is infinite or open. (It
is also infinite for K=0, but the term open is normally reserved for the case
of K < 0). Observations indicates that our universe is spatially flat, corresponding to K=0.
Using the Friedmann equation one can calculate the scale factor and thereby
the expansion rate of our Universe. We “only” have to plug in appropriate
matter densities ȡ. I put only in quotation marks since this is far from trivial.
We have to make educated guesses and theorize the kind and amount of
matter that existed in the early Universe based on cosmological observations
and our knowledge of particle physics.
On top of that, to make the story slightly more intriguing, astronomical observations have indicated that there exists a vast amount of dark matter in
our Universe, matter whose origin and nature still remains a puzzle. The
mysteries do not end there. Einstein included the cosmological constant ȁ in
his original equation of general relativity to get a Universe which, according
to his (and many others) belief, was eternal and had a fixed size. When the
expansion of our Universe was discovered he abandoned the cosmological
constant and is said to have called the inclusion of this parameter “his biggest blunder”. However, in modern quantum field theories a cosmological
constant arises naturally, corresponding to the energy of vacuum, which is
far from an empty, boring place. In these theories vacuum contains a “boiling soup” of oscillating particles being constantly created and annihilated,
giving vacuum an energy, even though the exact origin of and size of this
constant continues to puzzle theoretical physicists. Recent observations [1, 2,
3, 4] indicate that our Universe is accelerating, indicating that there is a
small but non-zero cosmological constant. This has turned the understanding
of the origin and magnitude of the cosmological constant into one of the
most extensively researched areas of theoretical physics. Many researchers
view it as one of the most important issues to address in the field.
2.3 A (very) brief history of our Universe
After providing some basic cosmology I would like to give a brief, schematic description of selected moments of the history of our Universe to give
the reader a sufficient background for Paper I.
The earliest time in the history of our Universe is an area which we know
very little about. General relativity ceases to be useful at the enormous energy scale corresponding to this era corresponds. One could hope that string
theory will provide us with valuable insights into this era in the history of
our Universe. In fact, a lot of cosmological scenarios related to this era have
been suggested by string theory, and string cosmology is a subfield of theoretical physics that has been extensively studied in the last couple of years.
However, these cosmological scenarios are so far little but ideas and conjectures without firm theoretical and experimental bases and only time will tell
whether string cosmology can take us closer to an understanding of this elusive era in the history of our Universe.
This is neither the time nor place for a discussion of the multitude of different ideas that string cosmology has provided for this earliest time of our
Universe. Instead, I will move on to the parts of the history of our Universe
where we believe that we understand the physics at least reasonably well.
The common paradigm for this part of the history of our Universe is called
the Big Bang theory (or Hot Big Bang theory to be precise) or sometimes the
standard model of cosmology.
According to this common paradigm, the expansion of our Universe started
billions of years ago, from a state with extreme physical conditions. At this
stage, our Universe was an enormously hot, dense and small Universe that,
as time passed, evolved to the Universe we live in today. This state occurred
somewhere around 10-12 s to 10-24 s after the creation of our Universe. At this
point in time, our Universe was a “boiling soup” of particles and radiation.
The energy density of radiation was so much larger than the energy density
of ordinary matter that we can neglect the contribution to the energy density
from ordinary matter. Our Universe was in the so called radiation-dominated
For the next 10 000 years or so there was a rather complicated history involving our fundamental forces and particle species that I do not intend to
describe here. One of the important events in this era was the formation of
atomic nuclei from protons and neutrons, a major step towards the formation
of the ordinary matter that fills our Universe today.
When the Universe was about 10 000 years old, the energy density of radiation became equal to the energy density of ordinary matter, and the radiation-dominated era ended, giving way to the matter-dominated era. During
this stage, the expansion rate, and hence the cooling, of our Universe increased. At first, our Universe was still a soup of radiation and matter, but at
the age of around 300 000 years, the shared life of matter and radiation
started to end. At this point in time, the temperature of our Universe had
reached a value where it is possible for the electrons to bind with nuclei to
form atoms. Before this, the energy of the photons in the radiation was big
enough to break up the newly forms atoms, but this ceased to be the case for
the majority of the photons at this time. This process is called recombination
(even though it is a strange name, since the electrons and the nuclei were
never previously combined) and occurred to a larger and larger extent as the
Universe expanded and the photons cooled off. Shortly a state where we had
matter and freely propagating photons was reached. These photons are still
observable today, and constitute what is called the cosmic microwave background radiation (CMBR). This radiation can be observed as a relic from this
time as microwaves corresponding to a blackbody radiation spectrum with a
temperature of around 2.7 K. Why microwaves? As our universe expands, so
does the wavelengths of radiation. Waves with longer wavelengths correspond to lower energy and thus a lower temperature. So what we today observe as microwaves were once waves with much higher energy, which have
since been drained of energy and become microwaves with a temperature of
2.7 K. This cosmic microwave background radiation is a very important
source for hints about the physics involved in the early Universe, and I will
come back to this radiation in next section.
Between this era where radiation and matter decouple from each other and
our present time, structures must have started to form in our Universe. These
structures were the origin of what we today can observe as galaxy clusters,
galaxies and other large-scale astronomical structures. The origin of these
structures is a very complicated and intriguing story and this is not the right
time or place to further discuss these matters. For the interested reader there
is an excellent discussion of the formation of structures in our Universe in
2.4 Inflation
The history of our Universe described in Section 2.3 leaves some important
questions unanswered that need to be addressed. I will not go into detail for
all of these questions, but will, as an example, rather focus on one of these
questions that motivate adding a new era to the history of our Universe, an
era that was not included in the brief historical overview in last section. The
question on which I will focus is often referred to as the horizon problem.
For a discussion of some of the other problems with the standard cosmological model, the interested reader is referred to [5].
The cosmic background radiation mentioned in the last section, which
reaches us from a time when our Universe was around 300 000 years old,
has properties that present a serious problem for the standard cosmological
model described earlier. If we look at this cosmic radiation in various directions, it turns out that this radiation is surprisingly uniform (to one part in
100 000). No matter in which direction we look, the temperature of the radiation agrees to an amazingly high degree. This is puzzling, since if we take
a closer look at the situation in the standard cosmological model, many of
the areas that exhibit this homogeneity could never have been in contact
which each other before the time when the cosmic microwave background
radiation was produced. If we take two regions of our Universe that are so
far apart today that they can not communicate with each other, and track the
history of this origin by using our standard model of cosmology, it turns out
that they could never have communicated with each other. This is illustrated
in Figure 2.
Figure 2. In this picture, the solid line at the bottom represents the birth of our Universe in the standard cosmological model. The grey triangles represent the light cone
of a given point in spacetime – the part of spacetime with which the point can receive information. It can be seen in this picture that points observable by us but
separated by a large distance could never have been able to communicate with each
How can regions of our Universe that have never been able to exchange
information with each other, and that therefore have never been able to reach
equilibrium with each other, have such identical properties? An analogous
example would be an examination where 100 students write an examination
in a large lecture hall. Due to an important call the lecturer in charge of the
exam leaves the students unattended for a short period of time. When she, a
few days later, starts to grade the examination, she finds that all of the students have scored exactly 93% on the examination, all with identical answers. That would be a truly astonishing result if there were no communication between the students while the lecturer was away.
One proposed theory is that our Universe started in such a homogenous state,
but this would require extremely fine-tuned initial conditions and is hardly a
satisfactory explanation. It would be preferable to have an explanation of
why the Universe was in such a homogenous state. Such an explanation –
which also addresses some other problems with the standard cosmological
model – was proposed by Alan Guth in 1980. This involves adding an era to
the description of the history of our Universe, an era where our Universe
undergoes a rapid accelerating expansion. Such a period is called inflation.
In cosmological models with inflation, such an event is proposed to have
occurred very early in the history of our Universe, when our Universe was
around 10-38 to 10-30 seconds old. I will not address inflation theory in any
more detail here (the interested reader is referred to [5] which provides an
excellent summary of inflationary models and ideas), but will move on to
look at how such an era could solve the horizon problem. This concept is
illustrated in Figure 3.
Figure 3. In the inflationary scenario the solid horizontal line no longer represents
the birth of our Universe, but simply corresponds to the end of inflation. We can
extend the light cones into the inflationary era, where they eventually overlap if
inflation persists for sufficiently long time, making communication between distant
points possible in this early era.
Before inflation, our Universe was extremely small and all points in our
Universe could easily communicate with each other, exchanging radiation
and thereby achieving a uniform temperature. This resulted in a Universe
with homogenous properties – the same homogeneity that we can observe in
the cosmic microwave background radiation today. When inflation occurred,
our Universe expanded enormously and a many points became separated by
such large distances that they could no longer communicate with each other,
and no longer needed to do so in order to avoid the horizon problem, since
they had already been communicating and creating the homogenous state of
our Universe before the era of inflation.
So, a cosmological model where our Universe undergoes a period of inflation early in its lifetime provides a solution to the horizon problem. There
are other problems with the cosmological standard model, such as the monopole problem, the flatness problem and the origin of our large-scale structures in our Universe which also seem to get satisfactory solutions by assuming an inflationary phase (details can be found in [5]). In summary, it seems
like we need a period of inflation in the history of our Universe to get a satisfactory description of how our Universe has evolved into what we see today.
2.5 de Sitter space
As described in the previous section, inflation involves an accelerating expansion of our Universe. In Paper I we study a particular type of accelerating
spacetime known as the de Sitter space. This spacetime has received a lot of
attention lately for several reasons. First of all, observations [1, 2, 3, 4] indicate that our Universe currently has an accelerating expansion, which might
be due to a positive cosmological constant, making this spacetime very interesting to take a closer look at. Moreover, de Sitter space plays a central
role in many inflationary scenarios and research [6] has shown that due to
inflation the cosmic microwave background radiation could provide us with
valuable insights into the physics of our early universe at extremely high
energy scales. A third reason for the interest in de Sitter space is that it turns
out to be very tricky to implement string theory and quantum gravity in de
Sitter space. In the case of a negative cosmological constant, a spacetime
known as anti-de Sitter space (AdS), a lot of progress has been made in
terms of holographic dualities (see e.g. the references listed in [7]) and the
hope has been that similar ideas could be applied to de Sitter space as well,
but so far de Sitter space continues to be elusive. A final reason for the interest in de Sitter space is that de Sitter space could have important parallels
to the physics of black holes, due to the presence of cosmological horizons.
The spacetime known as de Sitter space is the maximally symmetric vacuum
solution to the Einstein equation, in the case of a positive cosmological constant. Geometrically, de Sitter space can be represented by a hyperboloid
embedded in flat Minkowski space and in the case of a four-dimensional de
Sitter space, which is our interest in Paper I, this hyperboloid is described
X 02 X 12 X 22 X 32 X 24
The quantity R is known as the de Sitter radius and we will have more to say
about the significance of this radius later on. From this basic expression, we
can equip de Sitter space with many different coordinate systems. The
choice of coordinate system depends on what we are interested in exploring
as far as de Sitter space is concerned. For cosmological purposes and global
questions, a suitable set of coordinates is the Friedmann-Robertson-Walkercoordinates, where the metric of the de Sitter space becomes:
ds 2
dt 2 a ( t ) 2 (dr 2 r 2 d: 2 ),
where a(t)=Ret/R is the scale factor for the de Sitter space. Another useful set
of coordinates is the static coordinates, which are useful when we want to
adapt a local perspective such as when an observer is put in de Sitter space
and we are interested in exploring the experiences of this observer. For these
coordinates we have the metric:
ds 2
r2 ·
r2 ·
¨¨1 2 ¸¸dt 2 ¨¨1 2 ¸¸ dr 2 r 2 d: 2 .
R ¹
R ¹
Perhaps the most interesting thing that becomes visible in these coordinates
is that when r=R (i.e., the radial coordinate equals the de Sitter radius) we
find that there is a cosmological horizon – the de Sitter horizon. What is the
significance of this horizon? If we were to put an observer in a de Sitter
space, this observer would all the time be surrounded by a cosmological
horizon beyond which the observer can not get any information of what is
going on. Even though de Sitter space itself is infinite, the observer is
shielded from all but a finite portion of de Sitter space.
Another well known situation where we have a horizon, is when we consider
black holes. For these there exists an event horizon which prevents anything
that happens to end up inside of this horizon from ever making their way out
again from the black hole. In de Sitter space, we seem to have the opposite
situation: anything sent out through the horizon is doomed to be lost forever.
If this spacetime remained a de Sitter spacetime this would not be much of a
problem, unless you happened to drop one of your favourite CDs and it vanished at the horizon. In Paper I, however, we present a scenario where a de
Sitter space is replaced with another spacetime and now the existence of this
horizon gives rise to a potentially puzzling situation, which is discussed in
detail in the following section.
2.6 A possible information paradox
In our scenario in Paper I, a de Sitter spacetime is momentarily replaced by
an era where the scale factor grows like tq where 1/3<q<1, referred to as the
post-de Sitter phase. This can be viewed as a “toy model” for an inflation era
and the replacement of this era with an era of matter or radiation domination.
This kind of transition is an integral part of all inflationary theories and is
known as reheating. During this transition the vacuum energy is transformed
into ordinary matter and radiation through mechanisms which are thoroughly
discussed in, for example, [5].
In this scenario a situation that threatens one of the fundamental principles of
quantum mechanics can occur. As described in the previous section, anything that happens to end up beyond the de Sitter horizon is no longer in the
causal patch of an observer situated in de Sitter space. An object which is
released by an observer in de Sitter space that crosses the de Sitter horizon is
lost forever, and so is any information contained in the object. However, if
the de Sitter space is suddenly changed into a spacetime which expands at a
slower rate than the lightcone for an observer in this spacetime, the object
and any associated information are no longer gone forever. The object can
actually return to the part of spacetime that an observer can access. This is
still not a problem. The object and the information becomes accessible again
– so what? The problem lies in the fact that horizons emit radiation. Stephen
Hawking showed in a seminal paper that black holes in fact are not completely black. Radiation is emitted from the event horizon of the black hole.
If we throw an object into a black hole, microscopic description of black
holes makes it reasonable to believe that this radiation is able to carry the
information contained in the object.
If black holes emit radiation from a horizon and that radiation carries information about what was thrown into the black hole, the same should be true
for the horizon that is surrounding each observer in de Sitter space. In fact,
analogous to the case for black holes it is possible to attribute a temperature
and an entropy to the de Sitter horizon, so these horizons seem to have many
things in common. The essence of this is that the information contained in
an object that leaves the causal patch bounded by the horizon can be sent
back towards the observer as radiation from the horizon. It is however im26
portant to stress that even though the radiation can carry information, it is by
no means obvious that an observer can extract that information.
Assuming that the information about the object is sent back towards the observer as radiation, and assuming that the observer is capable of extracting
this information from the radiation, we do have a problem. In our scenario
there is the possibility for the object to become accessible to the observer
again when the de Sitter phase is turned off. So the radiation from the de
Sitter horizon carries information about the object and the object itself eventually becomes accessible to us again. This means that we apparently have
managed to duplicate information, something that is strongly prohibited by
the basic laws of quantum mechanics and often referred to as the quantum
Xerox principle. Figure 4 illustrates this apparent threat to the quantum
Xerox principle.
Figure 4. The thin line at the bottom represents the observer and the thick line describes the horizon from beyond which we can not get any information. During the
de Sitter phase the horizon stays at a constant distance from the observer, and during
the phase following the de Sitter phase the horizon grows linearly with time (corresponding to the fact that as time passes, information from regions further and further
away has had the time to reach us). The dashed line describes the motion of an object that is sent out through the de Sitter horizon only to later on, when we no longer
are in the de Sitter phase, return to within the area from which the observer can
obtain information. At the moment when the object passes through the de Sitter
horizon we put t=0, then to and tin then correspond to the time at which the de Sitter
phase is turned off and the time at which the object become accessible to the observer again, respectively.
2.7 Resolving the apparent paradox – results of
Paper I
As discussed in Paper I, there are fortunately several ways of avoiding the
possible duplication of information outlined in Section 2.6. One possibility is
that the radiation does not carry any information about the object, which
would eliminate our paradox, but which also would have severe implication
for information in other situations, such as black holes, where horizons are
present. Another related possibility is that even though the radiation carries
information, it is not possible to extract this information from the radiation.
In Paper I we show that even if we try really hard to get a duplication of
information, by assuming a worst case scenario: that the radiation does carry
information, that we can extract this information from the radiation by using
some kind of detector and that the object does become accessible again, we
can avoid the apparent paradox. If we start a clock at the time the object
crosses the horizon in the de Sitter phase, then the time it takes for the object
to become accessible to the observer again, tin, is shown to be:
t in
§ (1 q ) x Re t 0 / R
(qR )1q ¸¸
(qR )
1 /(1 q )
t 0 qR
where x is the comoving coordinate of the object, R is the de Sitter radius
and t0 is the time at which the de Sitter phase is replaced by the post-de Sitter
phase with the scale factor a(t) ~ tq. We then provide an estimate of the
minimum time, IJ, needed to measure the information in the radiation from
the de Sitter horizon (once again assuming that this is possible) which turns
out to be
IJ ~ R3
where R once again is the de Sitter radius.
To be able to extract information from the radiation, we thus need the time t0
to be at least IJ.
Replacing t0 in the expression for tin with IJ and keeping the dominant term
we get the time of return to be
tin ~ e R ~ eS,
up to factors of the order of one in the exponential (where we have used that
the entropy S in the de Sitter space is given by ʌR2). This time is, however,
nothing but the Poincaré recurrence time for our de Sitter space. This is the
time it takes for a trajectory in phase space (a space whose dimensions corre-
sponds to the positions and momentum for all the components of the system
and where a trajectory represents a possible evolution of the system) for an
isolated finite system to return arbitrarily close to its initial value. This
means that discussing experiments lasting longer than the recurrence time is
meaningless, since the system effectively has lost its memory. Since our
detector obviously has less entropy than the entire de Sitter space, we are
considering an experiment lasting far longer than the recurrence time of the
detector, thereby indicating that the experiment does not make any sense! It
makes the retrieval of information from the radiation an impossible task and
prohibits our scenario from violating the quantum Xerox principle.
I would like to conclude my discussion of Paper I with some remarks. First
of all, in our scenario, we have assumed a worst case scenario, where the
radiation does carry information, that we can extract this information from
the radiation by using some kind of detector and that the object does become
accessible again. We showed that we can resolve the paradox in this worst
case scenario, but there might be mechanisms that prevent this worst case
scenario from occurring in the first place, e.g. that it is not possible for the
observer to extract the information during the de Sitter phase (which however would mean that an observer in the de Sitter phase would not experience unitary evolution until the post-de Sitter phase).
Secondly, it would be wonderful if we could rigorously analyse this process
in detail using e.g. string theory. However, as already mentioned, there is a
long way to go before we (if we ever) have a happy marriage between string
theory and de Sitter space. This means that any discussion of the physics in
de Sitter space has to involve semi-classical arguments, general discussions
and a certain amount of qualitative arguments. These explorations are however still valuable, by providing insights into the properties of de Sitter space
that informs the quest for a realisation of a more fundamental theory in de
Sitter space.
2.8 Experiencing potentially abstract physics
My research presented in this chapter and in Paper I has taken me on a journey through the landscape of theoretical physics. On this journey, I learnt a
lot about general relativity, cosmology and string theory, but I also experienced some of the challenges of learning potentially abstract physics. Hence,
I believe that, apart from subject knowledge, my research in theoretical
physics has provided me with valuable insights into the process and dynamics of making the learning of potentially abstract physics possible. These
insights are arguably invaluable to both my physics education research and
to the development of my understanding of the teaching and learning of potentially abstract physics.
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Part II
3. Physics education research – overview
and literature review
3.1 Introduction
In this chapter I attempt to provide the reader with a brief overview of what
physics education research involves as well as a selection of some of the
main findings of physics education research. I also dedicate a section of this
chapter to a discussion of the relationship between physics education research and physics. After these introductory discussions, I proceed to a more
detailed literature review of the research in physics education that is closely
tied to the research presented in this thesis. The main purpose of this review
is to situate my research in physics education and to provide some of the
background to why we chose to conduct the studies included in this thesis.
3.2 Overview of physics education research (PER)
3.2.1 What is physics education research?
Physics education research (PER) is a relatively young but rapidly growing
area of research. The main research focus of this area is to obtain a better
understanding of the teaching and learning of physics and the factors that
affect these processes. The growth of this research area reflects a dissatisfaction with the way physics is being learnt and taught, and represents a realisation that it is not only important to produce new physics knowledge, but also
to make physics knowledge accessible in as fruitful, successful and stimulating a way as possible. In PER, detailed and systematic studies of the learning
and teaching of physics are conducted that can influence the creation of
more effective learning environments.
The first studies in PER originated in physics departments, from a concern
about the knowledge (or rather lack of knowledge) students seemed to have
acquired from their learning of physics. These studies were, due to being
situated in physics departments, typically not grounded in any educational
theoretical frameworks, but consisted mainly of empirical investigations of
students’ level of knowledge and understanding of specific concepts in physics. Today, research into the teaching and learning of physics has developed
into a scholarly inquiry, and a full-fledged, diverse research area has
emerged. The rest of this chapter aims at providing the reader with an overview of PER and the areas that are closely related to my own research presented in this thesis in particular.
3.2.2 The relationship between PER and physics
Every now and then, PER researchers have to answer to the question of
whether PER should be considered a subfield of physics. I think you can get
different answers to such a question depending on whom you ask and in
what context the question is asked. I do not believe it to be the most relevant
question to ask. Let me try to elaborate on why I believe this to be the case.
Educational research differs from research in physics in several important
aspects. In physics, the main way to collect data is to conduct measurements
which provide the researchers with quantitative results. In PER, a multitude
of ways of collecting data exists (e.g. interviews, video-recording, in-class
observations) and the results are most often qualitative. In physics, there is a
strong agreement on the interpretation of concepts (such as an electron)
while this is not necessarily the case in PER (for example, learning). These
differences and others (see for instance Aalst, 2000) should not be alarming
or surprising. First of all, PER is a young research field still in its infancy
which has not had the time to reach the same mature state as physics. More
important, though, is that the research questions of physics and physics education research are vastly different. Naturally enough, different types of research questions need different approaches and provide different types of
results. In PER we study human individuals and human interaction and attempt to answer questions about the learning and teaching of physics for
which quantitative results, in most cases, are neither relevant nor sufficient
to provide satisfying answers.
Due to these differences, I believe that the relevant question might not be
whether PER is a subfield of physics or not, but whether PER research
should be conducted at physics departments. Aalst (2000) provides three
reasons why this should be the case:
(1) PER contributes positively to the teaching of physics at all educational levels;
(2) curriculum development should take into account the context in
which the physics is taught and should involve both practical and
theoretical knowledge of the teaching and learning of physics. This
practical knowledge is available in physics departments;
(3) subject matter knowledge (content and epistemological) is considered to be a very important factor as far as the improvement of physics education is concerned and physics departments are ideally
suited to provide this knowledge.
In summary, I argue that PER research is most fruitfully conducted in physics departments, since both physicists and educational researchers can benefit from each other’s expertise in a dynamic dialogue. It is important to stress
that this does not mean that physicists should become educational researchers or vice versa. However, for this cross-fertilisation to be successful, educational researchers should have a solid knowledge of physics and physicists
should have familiarity with PER and a willingness to reflect on their teaching and to try new instructional approaches. I believe this to be a possible
scenario and therefore I strongly believe that PER research should be situated in physics departments as an integral part of physics.
3.2.3 A selection of main findings in PER
Physics education research is a relatively young research area, but has nonetheless been able to provide important insights into the teaching and learning
of physics. In this section I attempt, in broad terms, to highlight and exemplify some of the most prominent findings. An excellent and extensive summary of many of the main contributions PER has made can be found in Redish (2003) and comprehensive overviews of physics education research,
which provides a flavour of the various areas that have been studied, can be
found in Duit (2004), McDermott and Redish (1999) and Thacker (2003).
One of the most important things that can be concluded from the amassed
body of physics education research is the limited success of the traditional
teacher-centred transmission approach to teaching, as well as the limited
success of the traditional way of assessing students’ knowledge. As far as
assessment is concerned, research in PER has shown that very little insight
into students’ actual understanding is provided by the traditional way of
assessing physics knowledge and that there is very weak correlation between
the ability to solve examination-type problems and an understanding of the
physics involved. Several studies have shown that although successful on the
exam, students’ still have many difficulties in relation to many important
concepts in physics. Furthermore, the traditional role of the lecturer as a
transmitter of knowledge has, in many contexts, been shown to be inferior
and less successful than more interactive ways of learning physics, as far as
successful student learning is concerned. Many studies have also investigated the effectiveness and fruitfulness of laboratory work and tutorials as
well as suggested new ideas for curriculum development. Other findings
have suggested alternative ways of presenting physics that are not as heavily
rooted in mathematics as it traditionally is. In summary, PER suggests that in
many contexts the traditional way of presenting and assessing physics
knowledge needs a transformation towards a student-centred, interactive
environment which promotes successful learning and assesses the things we
want our students to learn.
Hake (1998) provides a good illustrative example of this area of research,
which highlights the fact that traditional methods of presenting physics
might not be the most efficient approach as far as students’ understanding of
physics is concerned. In a survey involving more than 6 500 introductory
physics students, the results of two conceptual tests which probed students’
conceptual understanding of key concepts in mechanics were analysed together with the results of a test of problem solving in mechanics. The most
interesting aspect of this analysis was that Hake compared the results for
students who had been taught in a traditional way with the results for students who had experienced a more interactive learning environment. The
results showed, in Hake’s own words, that, “the use of IE [interactiveengagement, author’s remark] strategies can increase mechanics-course effectiveness well beyond that obtained with traditional methods” (Hake,
1998, p. 71).
Another important main finding in PER is that many students have negative
experiences of the culture of physics. Narrow, abstracted, boring, “male”,
and irrelevant are some of the epithets assigned to physics by students. On
top of that, teachers are experienced as uncaring, disdainful and impersonal
and perceived to teach in ways which are disconnected from real life with a
lot of things taken for granted. Taken together with the fact that in many
universities, fewer and fewer students want to study physics, the significance
of these results cannot be underestimated. We need to make the culture of
physics appealing to the students, so that they feel that learning physics is an
exciting endeavour in a stimulating environment.
A very good illustrative account of students’ experiences of the culture of
physics, which I could recommend to anyone involved in the teaching of
physics, is provided by Thomas (1990). This study is first and foremost a
gender study, but it also excellently explores students’ experiences of being
a science student. Looking at Thomas’ discussions with students about their
experiences of studying science, and physics in particular, it is clear that
much remains to be done in order to create a stimulating learning environment in physics. Something that can be found in many students’ experiences
is a sense of physics as a subject which does not require thought and understanding, and where there rarely or never are any discussions of physics going on. Illustrative examples are provided below with quotes from students
in Thomas’ study.
I went to all the lectures and they’re easy to go to, because you’re spoon-fed,
they don’t sit back and they don’t philosophize, a lot of it, it’s all material on the
board. (Thomas, 1990, p.57)
It’s just proved my ability to learn chunks of knowledge, chunks of pages and
books, and reproducing them the day after, then forgetting it, then you have to
learn it again for next year’s exams. (Thomas, 1990, p.57)
One of the most interesting things found in Thomas’ (1990) study of physics
students’ experiences of studying physics was students’ comparison between
how they experience physics and how they experienced studying physics.
This highlights a need for making the physics culture more stimulating.
Tomas found that most students
…were very enthusiastic about physics. Physics was perceived as exciting, progressive and fundamental in contrast to other disciplines which were perceived
as routine, static and lacking substance. Yet their experience of studying physics
was far from exciting. (Thomas, 1990, p. 56-57)
Apart from the unsuccessfulness of the traditional ways of teaching physics
and the negative experiences of the culture of physics discussed earlier in
this section, a third major finding in PER involves student and teacher epistemology. Epistemology is a concept which refers to how knowledge and the
acquisition of knowledge (i.e., learning) are viewed. Research has shown
that both teachers and students exhibit a wide variety of different views of
what physics is and what it means to learn something, both in general and in
physics in particular. Moreover, these views have been shown to affect how
physics is taught and learned. Thus it seems that apart from the content
knowledge per se, we also have to pay attention to epistemological views, in
order to get as full a picture of the mechanisms and factors that affect student
learning as possible, in order to be able to provide a successful learning environment.
Paper IV in this thesis explores epistemological views, and a detailed overview of the research in PER involving epistemology is given in Section 3.5.
Here, I will only provide the reader with an illustrative example of this area
of research in PER – a study by Lising and Elby (2004). In their study, they
investigated the influence of epistemology on learning by conducting a case
study on an introductory university physics student. By analysing videotaped
class work, written work and interviews they could conclude that there is a
direct, causal relationship between a student’s epistemological views and her
learning of physics. In particular, her reasoning was either formal/technical
or everyday/intuitive, with very few links between these ways of reasoning
that could nurture a cross-fertilisation of ideas. The activation of either of
these seem to be dependent on her beliefs about what type of reasoning were
“appropriate”, i.e., her beliefs about what physics and learning physics involves. Drawing on these results, Lising and Elby (2004) suggest that “curriculum materials and teaching techniques could become more effective by
explicitly attending to students’ epistemologies” (p. 1).
This section has provided the reader with a flavour and broad overview of
some of the main findings of PER. In the rest of this chapter I will go into
much more detail and try to situate my own research in PER, as well as review areas of PER which are closely related to or have been important for
my own research
3.3 Conceptual understanding
3.3.1 Summary, history and theoretical development
One important ingredient in developing a good understanding of the critical
issues in the teaching and learning of physics is to explore how students
perceive and understand central concepts. This area, known as conceptual
understanding, was one of the first areas to be explored within PER and still
continues to be an important area of PER. Here, the nature and prevalence of
students understanding of particular physics concepts as well as difficulties
and “misconceptions” (also referred to as “alternative conceptions” or “alternative frameworks”) associated with these concepts are explored and
compared to what could be called appropriate disciplinary concepts.
Studies dealing with conceptual understanding can be found in many areas
of physics (see McDermott and Redish, 1999, for a comprehensive overview) but the most extensively studied area of physics is classical mechanics, where most of the initial studies took place. Here, studies have explored
students’ conceptions of kinematical quantities (velocity and acceleration)
and their graphical representations (e.g. Trowbridge & McDermott, 1980;
Peters, 1982; Bowden et al., 1992), while others have explored force, energy
and momentum (e.g. Halloun & Hestenes , 1985; Gunstone, 1987; Thornton
1997). This research led to an appreciation of the fact that students do not
enter their learning of physics as a tabula rasa. Based on their experiences,
students bring prior conceptions to their learning of physics. So, if the students’ minds are not a blank sheet, what do they bring in terms of previous
knowledge? As more and more studies were conducted it became apparent
that what students bring are more or less consistent fragments of knowledge
and concepts which, to varying degrees, differ from the conceptions of experts.
The further exploration of these issues led to several important contributions
to the development of PER. One of these contributions involved considerations of how to best address the conceptual difficulties found. One approach
to this, which could be described as a primitive model of learning, is known
as the conceptual change model (e.g. Hewson, 1981; Hewson, 1982; Posner,
Strike, Hewson & Gertzog, 1982). In this model, the conceptual structures
held by the students are restructured in order to allow the acquisition of science concepts. Duit (1999) describes this as learning pathways from preinstructional conceptions to the science concepts to be learned. In this model,
this restructuring occurs when a student becomes dissatisfied with his/her
prior conception and an alternative conception exists that is perceived as
intelligent (non-contradictory and understandable), plausible (believable)
and/or fruitful. These criteria are seen to be mediated by a set of epistemological commitments referred to as the conceptual ecology (e.g. metaphysical beliefs, prior experience, analogies).
The change of conception that occurs if a competitive conception is deemed
to be more intelligent, plausible and fruitful than the conception previously
held, is seen as (adapting ideas from Piaget) involving either accommodation
(also known as strong/radical knowledge restructuring or conceptual exchange) or assimilation (also known as weak knowledge restructuring or
conceptual capture). Assimilation refers to the recognition that an event fits
an existing conception as well as a selective ignorance of discrepancies that
are not prominent. Thus, the existing conception is not changed dramatically,
but rather enlarged by incorporating new aspects. Accommodation is more
drastic and involves a change in beliefs about how the world works. This
enables an event to be assimilated that could not have been assimilated until
accommodation occurred.
The conceptual change model is considered to be a powerful framework for
improving science teaching and learning and has, since its birth, been continuously debated and modifications, alternatives and additions have been
suggested (for example, Linder, 1993). Duit and Treagust (2003) provide a
good summary of this development.
Another example of an approach towards understanding and addressing conceptual difficulties is known as elicit, confront and resolve (Shaffer and
McDermott, 1992). Here, a clash between a student’s conceptual understanding and what could be referred to as an expert’s conceptual understanding
are deliberately introduced, making it necessary for the student to somehow
resolve this mismatch, thereby replacing their inappropriate conceptual understanding with a, from a disciplinary point of view, more accurate understanding.
These two examples illustrate the fact that the interest in a deeper understanding of the conceptions held by students, and how these conceptions
could be changed, initiated a more theoretical orientation in PER. At the
same time an awareness that there were other, previously unexplored (at
least in PER) factors that influenced the learning of physics (such as epistemology – discussed in detail in Section 3.5) arose. Moreover, a lot of ideas
from psychology and other areas of science education started to make their
way into PER. Taken together, these were the seeds of an emerging theoretical interest in PER at this time.
One major contribution to this theoretical orientation resulted from critical
scrutiny of the dominant views of students’ conceptual understanding.
This dominant view involved viewing students’ conceptions as “misconceptions” that had to be confronted and replaced with more appropriate
conceptions. At this time researchers such as Smith et al. (1993) and diSessa (1993) started to approach student conceptions from another viewpoint. Students’ conceptions were no longer seen as erroneous conceptions
that needed to be replaced by conceptions mimicking those of experts, but
seen as learning seeds – resources for learning which could be developed
through instruction.
Apart from this shift in how student conceptions were viewed, another interesting theoretical development started to emerge at this time – a more finegrained view of conceptions taking into account the context and dynamics of
conceptions. In a seminal paper, diSessa (1993) suggested a model where
introductory physics students’ conceptions do not form a coherent, organised
structure, but manifest themselves as loosely related, highly contextdependent pieces of knowledge based on students’ prior experience that are
applied in a context-dependent manner. These loosely connected pieces of
knowledge are called phenomenological primitives, or p-prims for short. In
diSessa’s model these p-prims are refined and developed – not replaced – in
the process of learning. These p-prims are a theoretical construct that is
viewed by many researchers as a more efficient way of approaching students’ conceptions than by looking at the conceptions themselves, due to the
fine-grained nature of the p-prims as well as the possibility of taking context
and dynamics into account.
The introduction of p-prims by diSessa (1993) was the starting point for a
growing interest in the mechanisms of student learning and initiated an increasing theoretical interest in cognitive models for student learning. To give
an example, in Hammer (2000) and Hammer and Elby (2000) a more finegrained structure of epistemological beliefs was introduced known as resources – in many ways analogous to the introduction of p-prims for student
Building on the ideas that emerged during the development of cognitive
mechanism of learning that followed diSessa’s introduction of p-prims, as
well as on ideas from psychology, sociolinguistics and neuroscience, Redish
(2004) proposed a “supertheory” for students’ learning of physics. This theory is built on various cognitive mechanisms with one of the key concepts
being resources – which are now seen as the building blocks for both students’ epistemological views and student’s conceptions. These resources can
be seen as a development of diSessa’s ideas about p-prims, consisting of
loosely connected intuitive ideas that are activated or deactivated depending
on the context. In this model, the key to successful learning of physics is to
activate the appropriate resources in a given context. Hence, successful
teaching should aim to provide a learning environment which activates these
appropriate resources.
In summary, the interest in students’ conceptual understanding has provided
PER with a lot of theory development apart from the obvious contribution to
our understanding of students’ content-knowledge. Today, many areas of
physics have been explored as far as conceptual understanding is concerned
and explorations of students’ conceptions in previously unexplored areas
continue to interest many PER researchers. At the time of this writing, areas
where students’ conceptual understanding has been explored is electromagnetism (e.g. Cohen, Eylon & Ganiel, 1983; Rainson & Viennot, 1997), relativity (e.g. Hewson, 1982), light and optics (e.g. Watts, 1985; Goldberg &
McDermott, 1986; Grayson, 1995), thermodynamics (e.g. Erickson, 1979;
Rozier & Viennot, 1991) and waves and sound (e.g. Linder & Erickson,
1982; Linder, 1993; Wittmann, Steinberg & Redish, 1999). The common
denominator in all of these studies is the fact that they might no longer play
as prominent a role as the early studies of conceptual understanding in mechanics which initiated several important theoretical orientations in PER.
However, they all make important contribution by identifying student diffi-
culties with dealing with various aspects of key concepts in physics, thereby
informing the teaching and learning of physics.
There is one area where conceptual understanding has been explored which
is not mentioned above. This area is quantum mechanics. Paper II of this
thesis belongs to this category and in the following section I attempt to provide the reader with a detailed overview of research into students’ conceptual understanding of quantum mechanics.
3.3.2 Conceptual understanding of quantum mechanics
Despite the unquestionable importance of quantum mechanics in many areas
of modern physics and technology – implying the importance of both the
teaching and the learning of quantum mechanics – physics education research in the area of quantum mechanics has not been given the same attention as it has in classical areas of physics such as mechanics, electromagnetism and thermodynamics. There has, however, been a steady increase in
interest in research in this area over the last few years and one can only hope
that this increasing interest will persist.
The study in Paper II of this thesis deals with university students’ conceptual
understanding of quantum mechanics and most of the early research carried
out in this area has involved pre-university students. Some examples are
Niedderer, Bethge and Cassens (1990) and Mashhadi (1995) where secondary school students’ view of the atom has been investigated. Other examples are Müller and Wiesner (1999) (included in Zollman, 1999) which explores conceptions of quantum physics and the work of Ireson (2000) where
pre-university students understanding of quantum mechanics has been explored in broad terms.
In the realm of university physics, conceptual understanding has looked at a
range of areas and concepts in quantum mechanics. One such area, which
seems to be conceptually challenging to students, is the wave-particle duality
and the wave nature of matter. A study by Johnston, Crawford and Fletcher
(1998) concluded that the university students involved in the study had difficulty describing what characterises a particle or a wave. Another study by
Vokos et al. (2000) investigated university students’ understanding of the
wave nature of matter in the context of interference and diffraction of particles and concluded that students had difficulty interpreting interference and
diffraction in terms of a wave model and furthermore, that students lacked a
proper understanding of the de Broglie wavelength.
A peculiar property of quantum mechanics is the occurrence of indeterminacies (often referred to as uncertainties). These indeterminacies are a fundamental “built-in” part of quantum mechanics and Johnston, Crawford and
Fletcher (1998) have found that students find it difficult to distinguish the
quantum indeterminacies from measurement uncertainties.
Other examples of studies of conceptual understanding in quantum mechanics are Vokos et al. (2000), where university students’ understanding of the
wave nature of matter in the context of interference and diffraction of particles has been investigated, Singh (2001), which deals with how undergraduate students deal with concepts related to quantum measurements and time
development, Steinberg et al. (1999), which investigate the influence of students’ understanding of classical physics when learning quantum mechanics
and Bao and Redish (2002), which is a study of university students understanding of classical probability and the implications of this understanding
for teaching quantum mechanics.
Besides these empirical explorations, listings of “misconceptions” from the
researchers’ own experience can also be found. An example is Styer (1996),
who lists 15 common “misconceptions” related to the measurement process
in quantum mechanics and the nature of quantum states as well as miscellaneous misconceptions.
It is not only researchers in physics education who have explored students’
conceptual understanding of quantum mechanics. There has also been research on students understanding of quantum mechanics in chemistry education, although a lot of the research consists of pedagogical suggestions and
inventions (for an overview, see Fletcher, 2004). One of the areas that have,
however, been explored in a systematic way is orbitals. In Tsaparlis (1997),
successful physics students’ understanding of orbitals were explored and the
results showed that students have a number of difficulties associated with
orbitals such as confusing orbital representations and failing to recognise the
approximate nature of orbitals in many-electron atoms.
Apart from looking at individual concepts or phenomena in quantum mechanics, there has also been research aimed towards constructing a quantum
mechanics concept inventory – similar to the FCI (Halloun & Hestenes,
1985) in mechanics – to probe students understanding of basic quantum mechanical concepts. One such survey is the Quantum Mechanics Concept
Inventory developed by Falk (2004), which is “currently not being refined
nor extended” (Falk, personal communication) and another one is the Quantum Mechanics Conceptual Survey (QMCS) developed and continued to be
developed by McKagan and Wieman (2006).
There is also a large body of PER research in quantum mechanics that suggests various ways of presenting the material that are supposed to enhance
the learning of the students. However, many of these publications make very
little or no reference to supporting empirical research. An overview of this
research as well as a comprehensive general literature review as far as PER
and quantum mechanics is concerned can be found in Fletcher (2004).
Since a lot of the physics education research in the area of quantum mechanics had been carried out at the pre-university level and with some areas of
quantum mechanics being under-explored, we felt that there was a need for
further exploration of university students’ conceptual understanding of quantum mechanics. For Paper II we chose to focus on a phenomenon known as
quantum tunnelling (described in non-technical terms in Section 5.3), an area
of quantum mechanics where, to our surprise, no previously published research could be found at the time we started our study. Quantum tunnelling
is an area very well suited for an exploration of students’ conceptual understanding of quantum mechanics, since it deals with many of the basic concepts of quantum mechanics and involves a number of counter-intuitive results.
In the last few years, quantum tunnelling has started to receive well-deserved
attention from the PER community and a number of interesting results have
been found as far as students conceptual understanding of quantum tunnelling is concerned. One such result is that many students carry a conception
that tunnelling causes particles to loose energy. This conception has been
reported and explored in Paper II of this thesis and in Wittman (2003) as
well as Wittman et al. (2005) and has been confirmed in two quantitative
surveys (Falk, 2004; McKagan & Wieman, 2006). Several studies (Paper II
in this thesis; Wittman, 2003; Wittman et al., 2006) have made it plausible
that one likely source of such a conception is the way diagrams of quantum
tunnelling are drawn.
Other findings related to quantum tunnelling involve conceptions where
students believe that a wave packet is either reflected or transmitted as an
entity (Paper II in this thesis; Falk, 2004), that only particles with “high
enough energy” are transmitted (Paper II in this thesis; Falk, 2004) and that
probabilities for different energies determine whether a particle manage to
tunnel or not (Paper II in this thesis; Falk, 2004; McKagan & Wieman,
2006). McKagan and Wieman (2006) have also shown that students have
difficulty interpreting what the potential energy in potential energy diagrams
(such as the ones used to represent a barrier in quantum tunnelling) means.
In summary, quantum tunnelling seems to be conceptually challenging in a
variety of ways for the students, thereby providing a suitable “laboratory”
for exploring students’ conceptual understanding of quantum mechanics.
The results of the study in Paper II of this thesis, which explores students’
understanding of probability in quantum tunnelling (of which some results
briefly have been mentioned here) is presented and discussed in detail in the
last two sections of Chapter 5.
3.4 Physics education research related to physics
Looking at mathematics, there has been significant progress in understanding students’ use of mathematics in the context of mathematics courses.
These results are both useful and interesting, but situated in pure mathematics. In order to research the role of mathematics in physics, studies are likely
to be most fruitfully conducted in the context of physics. For these reasons,
the rest of this section focuses on research on mathematics in the context of
As described in detail in Section 6.3, physics can be viewed as a discipline
concerned with describing the world by constructing models – the end product of this modelling process often being equations. These equations encode
the relationships inherent in the modelling process in the language of mathematics and therefore play an important role in the representation of knowledge of physics in most situations where physics is taught or learned. Therefore, physics equations are an important aspect to study as far as the role of
mathematics in physics and the learning of physics are concerned.
Despite their importance, physics equations have received surprisingly little
attention in educational literature. Looking at physics equations, there are
several things that come to one’s mind as potentially interesting aspects to
explore. As far as the symbolic structure of equations is concerned, Herscovics and Kieran (1980) and later Kieran (1981) have investigated students’ interpretations of the equal sign, concluding that many students view
the equal sign as a symbol meaning “do something”, although it is not clear
from this research whether this is a harmful interpretation. Another study
which has looked at signs inherent in physics equations is provided by Govender (1999). This study conducted a phenomenographic study of university
physics students’ experiences with sign conventions for quantities such as
displacement, acceleration and force. One of the findings is that students do
not realise the arbitrary nature of sign conventions. The main finding is however that the transition from one-dimensional motion to two- and threedimensional motion poses some difficulties to the students. Govender attributes this to an incomplete understanding of the relationship between a vector
component and a scalar component. Related to this is a conception found
among the students that only vectors are associated with algebraic signs. A
suggested way of dealing with this is to start with two- and threedimensional motion and consider one-dimensional motion as a special case
in order to get a more coherent introduction to vectors in kinematics.
There has also been research which has looked at students understanding of
the variables involved in physics equations. Clement, Lochhead and Monk
(1981) videotaped college students solving word problems and identified
difficulties in translation from a verbal representation to an mathematical
representation in terms of algebraic symbols. Rozier and Viennot (1991)
showed in the context of algebraic relationships in thermodynamics that
some students find it hard to parse the relationships between variables in
problems which involve multiple variables and their relationships. Problems
with multiple variables have also been identified by Steinberg, Wittmann
and Redish (1997) who found – in the context of mechanical waves – that
students have difficulty understanding the meaning and internal structure of
equations involving functions of more than one variable. Based on these
findings, a tailored tutorial aiming at addressing these difficulties was developed.
Other studies in PER have looked students’ use of physics equations. A lot
of studies have indirectly investigated equations while looking at problem
solving (for an overview of research in problem solving, see the review in
Hsu et al., 2004) and Tuminaro (2004) has discussed a cognitive framework
for analyzing and describing introductory physics students’ use of mathematics in physics, which uses cognitive mapping between knowledge elements
and student reasoning as the analytical framework. Another significant study
as far as the use of physics equations is concerned is the work of Sherin
(2001), who has examined students’ ability to construct equations appropriately describing a given physics situation, claiming that much of the reasoning related to equations draws on different kinds of equation templates that
carry specific meanings for the students.
All of these studies have a common denominator: they either focus on the
structural elements of physics equations or on students’ use of physics equations. Looking at this research it occurred to us that there were several important aspects of physics equations that remained unexplored. Three such
aspects that we became interested in exploring were what students focus on
when they are presented with physics equations, how students view what it
means to understand an equation and how well students understand physics
equations. The first two of these questions: “What do students focus on when
presented with physics equations” and “What does it mean for students to
understand physics equations” have been explored in Paper III and Paper IV
3.5 Epistemology, attitudes and beliefs
As mentioned earlier, early PER into student learning at the university level
traditionally had a strong focus on students’ difficulties and “misconceptions” associated with specific physics concepts. As described by Hammer
(2000) this work “has been productive for curriculum development as well
as in motivating the physics teaching community to examine and reconsider
methods and assumptions, but it is limited in what it can tell us about student
knowledge and learning” (2000, p.52).
In order to complement this work, and in line with the growing interest in
constructivism that has emerged, studies on university students’ epistemology (beliefs about knowledge and knowing) and the relationship between
epistemology and learning have started to become an important and increasingly explored area of educational research. For example, Linder (1992)
illustrated how teacher-reflected epistemology could be a further source of
conceptual difficulty for students. Hammer (1994) showed that introductory
physics students can be characterized as having beliefs about knowledge and
learning and that these beliefs affect their work in physics courses, i.e., the
way they learn physics. Similar results have been found by Redish, Saul and
Steinberg (1998) and Roth and Roychoudhury (1994). Lising and Elby
(2004) found a causal relationship between students’ epistemology and
learning behaviour and May and Etkina (2002) found correlations between
students’ conceptual understanding and epistemological beliefs.
Research in epistemology within physics education has generated several
areas of research, apart from exploring and establishing the relationship between epistemological beliefs and learning while highlighting the importance of taking student epistemology into account. Two of these are a closer
and more fine-grained look at what epistemological beliefs involve and the
development and implementation of surveys to probe students’ epistemological beliefs in physics.
The first of these areas involves a closer look at the notion of epistemological beliefs. In Hammer (2000) and Hammer and Elby (2000), a more finegrained description of epistemology than “beliefs” is suggested. It is argued
that by using “beliefs”, epistemological perspectives in certain situations run
into the same difficulties as descriptions of students’ conceptual learning in
terms of “misconceptions”. In both cases there is no account of how “misconceptions” or “beliefs” evolve and how different contexts affect these
“misconceptions” and “beliefs”. It is argued that an exploration of such
situations would need more fine-grained elements, similar to diSessa’s
(1993) introduction of p-prims (phenomenological primitives) as a more
fine-grained account of conceptions. In this spirit, Hammer and Elby argue
for and present examples of “epistemological resources” – a more finegrained account of epistemological beliefs (Hammer & Elby, 2000, 2002,
2003; Elby & Hammer 2001). In this model these resources are more basic
epistemological units that get activated in a context-sensitive manner –
sometimes appropriately, sometimes not. An example of such a resource is
“Knowledge as propagated stuff” where someone who invokes this resource
treats knowledge as something that is transformed from a source to a recipient. The idea of epistemological resources is only one piece of a larger anatomy of epistemological views, and Hammer and Elby (2002) elaborate of
the idea of resources by introducing another concept – “epistemological anchors” – concepts or analogies that cue the activation of appropriate resources. The search for a more fine-grained and detailed view of epistemological beliefs is beyond doubt some very interesting research in progress.
There has also been research which looked at epistemology from different
perspectives and provided useful insights into the taxonomy of epistemological beliefs. An interesting concept as far as student epistemology is concerned is the notion of pedagogical commitments, which was introduced and
further developed by Hewson (1981, 1985). Pedagogical commitments involve what an individual believes counts as a successful approach or explanation in a given field, as well as the individual’s more general view of what
knowledge involves. As described in Hewson and Hewson (1984), such
epistemological commitments may be a very important component as far as
students’ learning is concerned. An example of this importance can be found
in relation to my own research: consider a student whose epistemological
commitment to understanding an equation involves being able to use the
equation to solve problems. This student might only be focusing on that particular aspect of an equation, thereby overlooking other important features of
the equation.
An interesting question in relation to student epistemologies is the robustness of these epistemologies, i.e., can we tailor our teaching of physics so
that students develop more sophisticated and appropriate epistemological
views? May and Etkina (2002) attempted such an approach and concluded
that this indeed seems to be a difficult task. However, Linder and Marshall
(1998) investigated the possibility of affecting students’ epistemological
views of science and of learning through stimulating students to adopt a
meta-cognitive perspective of science and of their own learning in an introductory physics course. They concluded that such an intervention led to a
more appropriate view of science and learning, and that it could “profoundly
influence students’ conceptions of science and conceptions of learning”
(Linder and Marshall, 1998, p. 116). Thus, affecting students’ epistemolo-
gies seems to be a delicate matter, likely to be heavy dependent on the context and the ways of attempting to achieve this.
In the same way as the amassed indications of students’ conceptual difficulties in mechanics led to the construction of large scale surveys to probe this
further, the PER movement towards a larger focus on epistemological issues
has led to the development and implementation of several epistemological
survey instruments whose purpose is to assess and probe students’ attitudes
and beliefs about physics, physics knowledge and learning in physics. These
include the MPEX (Redish, Saul & Steinberg, 1998), EBAPS (Elby, n.d.),
VASS (Halloun, 2004) and CLASS (Adams et al., 2004) surveys, which all
have shown that students carry a range of epistemological beliefs as far as
physics and learning of physics is concerned.
It is not only the epistemologies of students that are of importance and consequently have been explored. As interesting as students’ view of what
knowledge and learning involves is teachers’ views of the same issues. Already at the end of the 1980’s Hewson and Hewson (1987, 1988) highlighted
the importance of taking into account teachers’ conceptions and beliefs of
teaching and learning. Several educational researchers have since concluded
that teachers’ beliefs affect their instructional practice (Nespor, 1987; Pajares, 1992; Abd-El-Khalick, Bell & Lederman, 1998; Lederman 1992, 1999).
An important contribution to the research on teachers’ epistemologies was
provided by Linder (1992) who could establish a relationship between teachers’ epistemological beliefs and student learning, arguing that “metaphysical
realism overtones in physics teaching not only affects how we teach but also
affects how students view the learning and understanding of physics”
(Linder, 1992, p. 111). In summary, research which explores the structure of
teachers’ epistemologies and the possible effects of these on teaching practice, students’ epistemologies and student learning has provided important
insights into the relation between epistemology and learning, and has become an important integrated part of epistemological research.
Closely related to the area of epistemology is metacognition, which is a
rather new and emerging area in PER. A good overview of metacognition in
the more general setting of science education is provided by Georghiades
(2004) and an example of a study in the area of physics is Koch (2001),
where students’ understanding of physics texts is explored from a metacognitive perspective. So why is this interest in metacognition emerging? While
epistemology could be seen as students’ views of what knowledge is and
how knowledge should be obtained, metacognition involves a reflection on
and evaluation of one’s own learning process, i.e., metacognition can be seen
as an extension of epistemology. Gunstone (1991) states, drawing on a constructivist perspective, that if “learners ideas and beliefs about learn53
ing/teaching etc. are in conflict with the notion that learners must recognize,
evaluate and reconstruct their existing physics ideas, then little progress is
possible” (Gunstone, 1991, p.135). Thus, metacognitive reflection seems to
be an important factor to take into account in order to achieve as effective
learning as possible.
In summary, epistemology has turned into one of the main research areas in
PER and has provided, and continues to provide, invaluable insights into
students’ learning of physics and how to improve learning outcomes, teaching and curriculum. Paper IV in this thesis, which is described in detail in
Chapter 6, can be seen to contribute to this epistemologically oriented area
of PER, by exploring university students’ epistemological views of what it
means to understand physics equations.
4. Method and methodology
4.1 Introduction
As in any other area of research, PER involves a spectrum of approaches to
deal with the research questions that emerge. Once a research question has
been formulated, finding a suitable way to explore that research question is
the next logical task. In PER this generically involves choosing a method
and a methodology. A method is simply a technique or way of collecting
data, while a methodology is a theoretical lens that is used for the analysis of
the gathered data. In physics such a distinction is generally not made or considered important since physicists in most cases share an implicit view of the
world, as well as a view of what constitutes knowledge and how this knowledge is obtained. This is not the case in PER, where different theoretical
lenses can corresponds to different views of the world as well as different
views of knowledge and learning. In this chapter I will described the methods and methodologies that were used for my studies in Paper II, III and IV.
4.2 Methods of data collection
In all of my educational studies (Paper II, III and IV) the main method for
collecting data has been interviews. The strength of interviews is that they
permit a rather detailed exploration while being able to be tailored according
to the overarching research question(s). The difficulties as a researcher are,
apart from attempting to create a friendly interview atmosphere, to stay away
from pre-conceptions or leading questions, and to know at what stage the
information gathered is sufficient to make a plausible interpretation of what
the interviewee has told me.
In all of the interviews for Paper II, III and IV, interview protocols were
created as a starting point for the interviews. These protocols proved to be
very useful since they kept the interviewer on track and made sure that no
part of the interviews relevant for exploring the research questions was omitted. In most cases the initial interview protocol were tested using pilot inter55
views, where a small number of interviewees were used to assess the usefulness of the protocol. Many times these pilot interviews led to modifications
of the original protocol, reflecting the difficulty of anticipating how the interview discourse could be best established as a research tool.
In all of the interviews for the studies in this thesis, either the entire interview or selected parts of the interviews were transcribed verbatim. In some
cases the interviews were conducted in Swedish, and in those cases the interview excerpts that we chose to use in the publications were translated to
English. The transcribed interviews were used as the main data source for
the analysis process.
There are several possible ways to structure an interview and for Paper II
and III the interviews were semi-structured and open-ended, involving a few
main questions to keep the interviews on track but providing generous space
for the interviewees to express whatever experiences they carried in relation
to the interview questions. In Paper IV, the research focus was much narrower, and in this case, the interviews were much more structured and centred around one main question and associated follow-up questions.
In the empirical study presented in Paper II, 12 intermediate physics students
from two Swedish universities were interviewed. The students were selected
to render a mixture of typical second and third year physics students who
had successfully completed at least one major-level quantum mechanics
course. Students were interviewed while interacting with a specially selected
computer simulation of quantum scattering and tunnelling. The interviews
lasted between 30 and 60 minutes and were semi-structured to explore students’ understanding in a focused, yet open-ended way. A large part of the
interviews asked students to predict what would happen in a particular simulation scenario and then to comment on what actually happened. This allowed us to probe students’ understanding of probability in quantum tunnelling in an efficient way that allowed the students to reflect on many aspects
of probability, which was exactly what we were trying to achieve.
Paper III is based on interviews with thirty undergraduate physics students
from three Swedish universities. These students attended a wide range of
physics courses, such as electromagnetism, classical mechanics and modern
physics. In audio-recorded interviews, the students were asked to discuss
various equations they had been presented with during the courses. To be
able to capture on what students were focusing, we asked the students questions like “What do you see here?”, “What does this equation mean to you?”
and “What does this equation tell you?”. In order to get as wide a variation
as possible, equations of different type and complexity from several different
areas of physics were discussed, ranging from simple equations such as the
definition of angular frequency, = 2ŋf, to the time-dependent Schrödinger
equation. During this interview process we both interviewed students for
whom the equations were relatively novel and students for whom the equations should be familiar to be able to compare and contrast the data for these
different student groups.
In Paper IV, twenty voluntary physics students from three different Swedish
universities were interviewed using a mixed mode semi-structured interviewing strategy involving both face-to-face and e-mail interviews. A good
discussion of the latter type of interviews in qualitative research can be
found in Meho (2006). Seven of the interviewees were first year undergraduate students, nine were second or third year undergraduate students,
and four were PhD students. Each interview lasted approximately twenty
minutes and began with some introductory discussion centred around the
nature of physics and the role of mathematics in physics. The purpose of this
introductory discussion was to set the context to physics and physics equations. After this introductory discussion, the main question that we asked the
students and were interested in exploring was: “When you say feel/say that
you understand an equation, what does that mean?”. Associated follow-up
questions were used for clarification and to allow students to elaborate on
their answers.
4.3 Choosing an appropriate methodology
In any research, it is important to choose an appropriate methodology once
the research questions have been formulated. In other words: what is the best
way to deal with and attempt to answer the research questions? Depending
on the nature of these questions, some research methodologies are often
more adequate and fruitful than others.
In Paper II, we were interested in exploring the variation in students’ understanding of probability in quantum tunnelling. This meant that we were not
interested in describing the individual students, but to map the collective
variation of the group of students as a whole. This focus on the collective
variation made it natural to choose phenomenography (described in detail in
Section 4.4) as our research approach, since phenomenography is explicitly
aimed at capturing the collective variation in how people experience, understand or perceive a phenomenon or situation. The same motivation can be
given for Paper III, where we also were interested in mapping the collective
variation, this time of what students focus on when they are presented with
physics equations.
For Paper IV, where we explore students’ views of what it means to understand equations, we are interested in the variation in this understanding.
While this could have been an appropriate situation to use phenomenography, we decided to conduct a case study to hold focus on the individual (described in Section 45).
The reasons for this choice are twofold. First of all, not only were we interested in capturing the variation in students’ views of what it means to understand an equation, but we were also interested in comparing the understandings of students at various levels in their academic career. This meant that,
apart from looking at the variation, we were also interested in comparing
individual students, which made us question whether phenomenography
would be the best research approach, since phenomenography does not aim
to describe or compare individual experiences or understandings. Surely, this
comparison could be added on top of a phenomenographic study, but there
was another reason for why we felt that a case study might be more appropriate.
In phenomenography, the experiences or understandings are removed from
the context and from the individuals. All the individual experiences or understandings of the phenomenon or situation that are studied are identified,
removed from whatever context that might be surrounding them and put into
a collective “pool” from which the collective variation is discerned. In our
case, we felt that the descriptions of the individual students were too rich to
be cut up and stripped of context. For these reasons we decided to frame the
study in Paper IV as a case study.
4.4 Phenomenography
In the study for Paper II, we were interested in exploring the variation in
students’ understanding of probability in quantum tunnelling which made it
natural to choose phenomenography as our analytical approach. Phenomenography is a research approach which was developed in the Department of Education at Gothenburg University in Sweden in the early 1970’s,
with Ference Marton as one of the principal pioneers. It originated with the
observation that some people learn better than others. This apparently trivial
observation led the research group to consider research questions such as:
What does it mean that some people are better at learning than others? Why
are some people better at learning than others? (see Marton, 1993 for more
details). The attempts to answer these questions paved the way for what
would eventually become phenomenography. Phenomenography is now a
well respected framework and a large number of educational studies that use
phenomenography in different areas can be found. It has, for example, been
used to examine students’ understanding of the fundamentals of force and
motion (Johansson, Marton & Svensson, 1985), sound (Linder & Erickson,
1989; Linder, 1993), central concepts in computer programming (Booth,
1992), concepts in electricity (Millar, Prosser & Sefton, 1989; Prosser,
1994), concepts in special relativity (Bantom, 1999) and displacement, velocity and frames of reference (Bowden et al., 1992).
4.4.1 The main ideas of phenomenography
To describe phenomenography, it is useful to start out by discussing the key
elements of this research approach. These key elements have been neatly
summarised by Trigwell (2000), who states that phenomenography:
…takes a relational (non-dualist), qualitative, second-order perspective, that
it aims to describe the key aspects of the variation of the experience of a phenomenon rather than the richness of individual experiences and that it yields
a limited number of (internally related), hierarchical categories of description
of the variation. (p. 1).
This is indeed a very compact summary of the main ideas of phenomenography. Let us dissect this quote and describe phenomenography in a more detailed manner by taking a closer look at the meaning of some of these different key aspects of phenomenography.
Phenomenography is a relational (non-dualist) perspective
This statement refers to the ontological basis of phenomenography. In some
research perspectives, such as cognitivism, a separation is made between an
individual’s mind and the outside world. In phenomenography there is no
such thing as an independent reality existing without someone perceiving it.
Reality is seen as being constituted as a relation between an individual and
what the individual experiences. This also means that phenomenography
does not separate the learner from what is learnt and that learning takes on a
particular meaning: experience the world in a different way.
Phenomenography is a second-order perspective
A distinction is often made between first-order and second-order research
perspectives. In a first-order perspective, the researcher describes how he or
she experiences the phenomenon or situation that is studied while, in a second-order perspective, the main focus is the experience of the phenomenon
or situation as described by others.
Phenomenography describes the key aspects of the variation of the experience
The fundamental aim of phenomenography is to capture and describe the
qualitatively different ways of experiencing a phenomenon, concept or situation. It is important to start out by stressing that in phenomenography, the
word “experience” can be used interchangeably with “understand”, “perceive”, “apprehend”, “conceptualize” etc. There are obviously differences
between these terms, and phenomenography does not try to deny these differences – they are simply not important. As described by Marton (1993):
The point is not to deny that there are differences in what these terms refer
to, but to suggest that the limited number of ways in which a certain phenomenon appears to us can be found, for instance, regardless of whether they
are embedded in immediate experience of the phenomenon or in reflected
thought about the same phenomenon (p. 4427).
So, since phenomenography is aimed towards capturing the variation, we do
not need to worry about whether that experience is a result of unreflected
thought or reflected thought, or about whatever subtle epistemological differences exist between e.g. understanding and apprehending.
The units of analysis of phenomenography are the qualitatively different
ways of experiencing a phenomenon, concept or situation. As a consequence, phenomenography does not produce accounts of the experiences of
single individuals. All the individual experiences are collected in a pool of
experiences, and from this pool the collective variation is extracted.
Something else that merits a discussion is the meaning of qualitatively different ways of experiencing. To answer this question, we need to examine
what phenomenography has to say about what it means to experience something. In phenomenography, to experience something is to be aware of something. We are all the time aware of a lot of things, but we are also aware of
these things to various extents. Some of the things might be in the foreground, i.e., in our focal awareness, while others might be in the background. Marton and Booth (1997) have an elaborate discussion of this in
terms of an anatomy of awareness which involves looking at the structure of
the awareness in terms of the whole and the parts and how these are internally related. An interested reader is referred to Marton and Booth (1997) for
a full description, but the main point is that experiences can be classified as
qualitatively different due to structural differences in the anatomy of awareness.
Phenomenography yields a limited number of categories of description
of the variation
So far, and there is no reason to believe this will change in the future, every
phenomenographic study has yield a limited number of qualitatively different ways of experiencing a phenomenon or situation. These qualitatively
different ways of experiencing are referred to as categories of description.
There is never just one such category and rarely more than five. Of course, if
we simply were to collect individual experiences, we could end up with as
many different experiences as individuals. In phenomenography this is not
the case since we do not look at individual experiences per se but collect all
of the individual experiences in a pool and then map the collective, qualitatively different variation.
4.4.2 Data collection
The data for a phenomenographic study can be gathered in a variety of different ways. For example, there are phenomenographic studies where group
interviews, drawings and written responses have been used. The dominant
way of gathering data in a phenomenographic study is, however, the individual interview and as described in the previous section, this is also the way of
collecting data that I have used in my phenomenographic studies.
Since the phenomenographic researcher is trying to find the variation in how
a concept, phenomenon or situation is experienced, the researcher focuses on
trying to obtain a dialogue where as many aspects of the concept, phenomenon or situation as possible become reflected upon by the interviewee. These
interviews are normally semi-structured, meaning that the researcher knows
what he or she wants the interviewee to discuss but the details or exploratory
direction of the interview have not been determined in advance. Many of the
questions follow from what the interviewee brings up, and the aim is to enrich the dialogue to thematise as many aspects of the concept, phenomenon
or situation as possible, as well as to clarify the interviewee’s meaning.
4.4.3 Data analysis
In phenomenography, all the individual experiences of a concept, situation
or phenomenon are collected into a common “pool of experiences”, from
which the researcher constructs “categories of description” which correspond to the possible qualitatively different ways of experiencing the concept, situation or phenomenon. So how do we get from the pool of data to
the categories?
The first step in the analysis process is to identify overall themes in the interview data and to tentatively group similar pieces of data into categories in
relation to the research question. In the next phase of the analysis these categories are re-examined in relation to the interview data and, if necessary,
modified, replaced, split or merged. In practice, the two steps in the data
analysis process are carried out simultaneously in iterative cycles. This process continues until the categories stabilize into an appealing bigger picture
that gives a satisfactory answer to the research questions and can be supported by illustrative examples from the data.
As far as the categories are concerned, it is important to stress that there is
not a one-to-one relation between these categories and the individuals whose
experiences constitute the data from which the categories are constructed.
An individual may belong to several categories, experiencing the concept,
phenomenon or situation in several different ways. It is exploring the variation in the “collective pool of experiences” that is the aim – not to collect
numerous different individual experiences.
In some situations it may be possible and fruitful, even desirable, to take
these categories a step further by looking for logical relations between the
categories. This is normally done in terms of an inclusive hierarchical ordering of the categories, where categories lower down in the hierarchy are included in categories higher up in the hierarchy.
4.5 Case studies
As described previously, Paper IV in this thesis draws on case study methodology in order to get as full an account as possible of students’ epistemological views of what it means to understand an equation. Case studies have
been used in a multitude of disciplines as early as the 1930s, and have become an important approach in educational research where a detailed investigation is required. A completely comprehensive account of case studies is
beyond the scope of this thesis and an interested reader is advised to take a
look at for example Merriam (1988) or Stake (2005), which provides good
in-depth descriptions of case studies. In this section I limit myself to providing the reader with some key aspects of case studies and the background
necessary to appreciate the methodological basis of Paper IV.
4.5.1 The main ideas of case studies
It is important to begin this description of case studies by stressing that there
is no such thing as a generic blueprint for how a case study should look.
There exists a wide range of various types of case studies, but they all have
at least one thing in common. As described by Merriam (1988), a case study
always involves “a detailed examination of one setting, or a single subject, a
single depository of documents, or one particular event”. The main features
of this description are that a case study allows a detailed, in-depth exploration of a well-defined unit of study.
Data for a case study can be collected in almost any conceivable way. Data
could, for instance, be collected as field notes, observations, questionnaires,
letters, interviews or any combination of these and other methods. The important thing is to choose a method of data collection that allows a detailed
investigation of the unit of study. Similar to the data collection, the analysis
of the data can proceed along many different paths. The main idea is to use
an analytical approach that takes into account the richness of the gathered
data, thereby producing as comprehensive a description of the unit of study
as possible.
The notion of detail in case studies means that the researcher should attempt
to account for as many aspects of the unit of study as possible. This means
that case studies normally involve a thick description, i.e., as complete a
description as possible of the entire research process, i.e., the origin and motivation for the research questions, the data collection process, the analysis of
the data and the interpretation and validation of the results.
When the results have been presented and the process and interpretations
that lead to the results have been scrutinized, the ultimate judge of the value
of case study research is the reader. Since a case study involves a study of a
well-defined unit of study, someone might object that it is difficult to see any
value of this research beyond the particular case. This is where, for case
studies, it is important to have a think description. By providing as detailed
an account as possible, the researcher makes it possible for a reader to understand the particular case in depth, thereby being potentially able to transfer the knowledge acquired from the case to new and foreign contexts. This
is known as “naturalistic generalization” (Stake & Trumbull, 1982). This and
the level of detail of case studies are two of the most attractive features of
case studies which make them an important and most useful research approach.
4.5.2 Conducting a case study
In order to explore the research questions involved in Paper IV in as fruitful
a way as possible, we decided to carry out an exploratory case study. In our
study, we examined university students’ epistemological mindsets towards
the understanding of equations in a group of twenty students. We decided to
frame this study as a case study and to use interviews (described in detail in
Section 4.2) because we wanted a detailed exploration of the experiences of
equations of each individual student participating in the study. This would
enable us to map, characterise and further analyze students’ epistemological
mindsets towards what it means to understand an equation.
To add an additional dimension to the study, we included students at various
levels in their education, ranging from first-year undergraduate students to
PhD students. By having students from various educational stages, a crosssectional case study was created, where we could both characterise students’
epistemological mindsets towards the understanding of equations and analyse whether there is a difference in the mindsets for students at different
stages in their academic career.
The principal aim of the data analysis process was to characterise students’
descriptions of what it means to understand a physics equation when they
feel that they have understood it, and then to take a closer look at these results. For this purpose we used what could be described as a standard qualitative data analysis (data-based inductive analysis). As described by Bogdan
and Biklen (1982, p.145) this involves “working with data, organizing it,
breaking it into manageable units, synthesizing it, searching for patterns,
discovering what is important and what is to be learned, and deciding what
you will tell others”. The process is generally carried out inductively, i.e.,
patterns and themes originate from the pool of data.
The first phase of the analysis process involved identifying overall themes in
the raw data – a process characterised as “open coding” (cf. Strauss and
Corbin, 1990) – and tentatively grouping pieces of data into descriptive
categories corresponding to different characterisations of students’ descriptions of what it means to understand an equation. The categories were given
a descriptive heading and each piece of data was coded with an identification
tag involving the origin of the data and to which of the tentative categories it
was assigned.
In the next phase the characterisations were iteratively compared to modify,
replace, split or merge the characterisations until saturation occurred. During
this phase there was also a continual cross reference to the full transcripts
and the two described steps in the data analysis process were essentially
carried out simultaneously in iterative cycles. This process continued until
the characterisations stabilised into a comprehensive set of outcomes that
well captured the content and richness found in the data. The results of this
analysis are presented in Section 6.6.
4.6 Trustworthiness and value in PER
As in any area of research, it is important to consider the trustworthiness and
value of the research. In qualitative research, Lincoln and Guba (1985) have
described this as being able to answer the question: “How can an inquirer
persuade his or her audiences that the finding of an inquiry are worth paying
attention to, worth taking account of?” (1985, p.301). Lincoln and Guba
(1985) present a number of criteria for judging the value and trustworthiness
of a qualitative research: credibility, transferability, dependability and confirmability. Let us take a closer look at what these criteria involve.
Credibility refers to the truth value of a qualitative study. Is the construction
and interpretation made by the researcher an adequate and believable map of
the experiences and statements made by the interviewees? There is really
only one legitimate judge of this: the interviewees themselves, and there are
several methods to enhance the credibility of a study (for an overview, see
Lincoln and Guba, 1985). One of the most widely used methods is called
member checks, where the researcher presents the interviewees with the data
record, interpretations and findings of the study, asking them to review the
interpretations made by the researcher and to judge whether their perspectives have been adequately mapped by the researcher. This may or may not
be possible to do, depending on the available time of the researcher, the accessibility of the interviewees and the context and aim of the study.
Transferability refers to the extent to which the findings can be transferred to
other contexts and situations that stretch beyond the boundaries of the study.
In qualitative research, the researcher’s role in allowing for transferability is
different from that in quantitative research. In a quantitative study it is the
obligation of the researcher to ensure that findings can be generalised, while
in a qualitative study it is the readers who are the ultimate judges of the level
of transferability. The obligation of the researcher is to ensure and enhance
the possibility of transferability. One way to achieve this is to provide a thick
description. A thick description involves detailed descriptions of the data,
context, analysis and assumptions of the study. By using a thick description,
the researcher makes it possible for a reader to judge the transferability to
their particular context or situation. For all of the studies in this thesis the
aim has been to provide the reader with a thick description.
In quantitative research, reliability is an important criteria that hinges on
assumptions of replicability or repeatability. This basically means that we
should obtain the same results if we observe the same thing twice in the
same setting. However, in qualitative research, it is not possible to repeat a
study in the same way as in quantitative research. Qualitative research deals
with human interactions and “repeating” a study would correspond to inves65
tigating a new context. In qualitative research, we could talk about dependability instead of reliability. The idea of dependability involves accounting
for the consistency of the inquiry processes used over time. This involves
examining whether the researcher has been careless or made mistakes in
framing the study, collecting data, interpreting data or reporting findings.
Confirmability refers to the extent to which the findings reflect the focus of
the inquiry and not of the biases of the researcher, i.e., the extent to which
the findings could be confirmed or corroborated by others. There are several
ways to increase the likelihood that findings in qualitative research will be
confirmable. Two such methods are to actively search for contradictions of
prior conclusions during the analysis process and to critically examine the
data collection and analysis processes for potential biases or distortions.
Another powerful method is peer debriefing. Peer debriefing involves letting
peer researchers not directly involved in the study scrutinize the data as well
as the interpretations of the data and the conclusions drawn from the data, to
play a “devil’s advocate” and to establish that the interpretations and conclusions are viable and well grounded in the data. This is a process which has
been used for all of the studies presented in this thesis.
To conclude this section, it should be mentioned that the criteria listed above
are not the only ones that could be considered. They may not all be applicable to a particular study and there are several other criteria that could or
should be considered when the trustworthiness and quality of qualitative
research is judged. Examples are meaningfulness (is the study addressing a
meaningful issue?), ethical treatment (have the participant been treated ethically?) and how the study is written (is what the researcher has learnt from
the study communicated clearly enough?).
5. Exploring university students’
understanding of probability in onedimensional quantum tunnelling
5.1 Introduction
Part I of this thesis has described my research in theoretical physics and
since I have always been fascinated by the world of abstract and theoretical
physics, I wanted my research in physics education to involve similar physics. An increasingly growing interest in students’ learning and understanding
of quantum mechanics had begun to emerge at that time in physics education
research and quantum mechanics. This thus constituted a natural starting
point for my physics education journey – eventually leading to the study of
student’s understanding of probability in quantum tunnelling which is presented in Paper II.
This chapter is intended to present and discuss the results of Paper II as well
as to give a reader not familiar with quantum tunnelling some relevant nontechnical background. I will therefore give a short description of some of the
main ideas of quantum mechanics that are highly relevant for Paper II in
Section 5.2 and continue by discussing the particular area of quantum mechanics – quantum tunnelling – that is the topic of Paper II in Section 5.3. In
Section 5.4 I will start to get closer to the actual content of Paper II, by motivating why quantum tunnelling was chosen as a topic of research. Finally,
this chapter is concluded with a summary and discussion of the main results
and implications from Paper II in Section 5.5 and 5.6.
5.2 What is quantum mechanics?
At the end of the 19th century, many physicists believed that physics – apart
from some minor details – was an almost completed science; that we knew
practically all there was to know about physics. This turned out to be far
from the truth. Several observations and experiments at the beginning of the
20th century indicated that the established physical theories were not capable
of accounting for the results of these experiments. Sometimes theories and
experiments gave very different results and sometimes the theory could simply not explain or predict what was going on in the experiments. At this
time, two major innovations in the field of physics were born, leading to a
completely new paradigm in physics. One of them eventually led to Einstein’s special theory of relativity, and the second one to the birth of quantum mechanics. This is neither the time nor the place to discuss the theory of
relativity, but I will spend some time on quantum mechanics, since it is the
area of physics where my research in Paper II is situated.
Quantum mechanics deals with matter and radiation at an atomic level, and
intuitively it seems reasonable to believe that it is difficult to use our everyday knowledge and experiences to fully understand what is going on at such
small scales. Even to someone keeping this in mind, a lot of the conceptual
fundamentals of quantum mechanics still come as a complete surprise to
anyone studying physics. To give quantum mechanics justice I would need a
lot more space than a thesis to describe all of its intriguing properties. Here, I
limit the description of quantum mechanics to a few of the basic properties
of quantum mechanics that are relevant for my research presented in Paper
In quantum mechanics, every particle is associated with a mathematical
function called the wave function. What does the fact that a particle is “associated with a mathematical function called the wave function” really mean?
It is hard to find two things more conceptually different then a wave and a
particle as we know of them in everyday life.
In quantum mechanics – contrary to the case in classical physics - there are
no such things as well-defined paths that particles follow, where we at each
instant in time can measure the position and velocity of the particle. The
picture is far more subtle. In fact, quantum mechanics declares that it is impossible to determine the exact position and velocity of a particle at the same
time. If we try to measure the position with greater and greater precision, the
velocity will become more and more uncertain and vice versa. In quantum
mechanics we can only talk about the probability of finding a particle in a
certain position or with a certain velocity. In quantum mechanics probabilities are the best we can accomplish and classical determinism – in the sense
that from knowing the current properties of a system we can predict how
these properties will evolve – is no longer possible. This is a radical departure from the ideas of classical physics that is hard, not only for learners but
also for physics scholars, to accept. For example, “God does not play dice”
is a famous quote from Einstein, reflecting his concerns about this probabilistic nature of quantum mechanics.
Accepting this bizarre twist in the modelling of reality, the logical question
to pose next is: “Alright, I do believe in this probability stuff, but how do we
get these probabilities?” The answer is that we get the probabilities from the
wave function previously mentioned. If we know the wave function of the
particle, then we just have to take the magnitude of the wave function and
square this magnitude to get the probability density of finding a particle in a
certain position at a certain time. By integrating this probability density over
a certain volume we get the probability of finding a particle in this volume at
a certain time. From the wave function we can also get information about the
velocity, energy, angular momentum and other properties of the particle.
This means that the wave function is the single most important thing to get
hold of in a quantum mechanical system. To get this important and informative wave function we specify in what kind of surroundings the particle is
located, and then we use one of the most important equations of quantum
mechanics – the Schrödinger equation – to calculate the wave function for
the particle and how this wave function evolves with time. Once the wave
function has been found, we theoretically know all there is to know about the
Quantum mechanical ideas, involving wave functions and probabilities, give
rise to many peculiar and counter-intuitive results concerning the behaviour
of particles at atomic scales, and one of them is the tunnelling effect (which
is the topic of Paper II in this thesis and which I therefore intend to discuss
briefly in Section 5.3).
It is hard to fully appreciate the tremendous impact that quantum mechanics
has had on physics and technology. Quantum mechanics has led to discoveries and inventions that have caused a revolution in both these areas. All of
the theories of the fundamental building blocks of matter in modern physics
– such as String Theory – contain quantum mechanics as a crucial ingredient
and a fair portion of our modern technology, such as computer chips and
transistors, is based on the implications and predictions that quantum mechanics has provided. Even though quantum mechanics may seem counterintuitive and sometimes just plain weird, there is currently not a single piece
of evidence that would make us believe that quantum mechanics is not a
good model for the physics that takes place at atomic and sub-atomic scales.
Quantum mechanics has withstood all the tests and challenges it has been
given. It seems like God does play dice after all.
5.3 What is quantum tunnelling?
Quantum tunnelling is a quantum mechanical phenomenon that is a perfect
illustration of just how counter-intuitive the quantum world can be. Imagine
that you ran towards you front door, and instead of crashing into the door as
you would expect, you suddenly find yourself on the other side of the door,
and the door shows no sign of the fact that you passed straight through it. Or
imagine that you rolled a ball towards the end of a table and instead of falling off the table, the ball bounced right back at you. These would be tricks
worthy of applause from even the best magicians.
Even though these are just “generative metaphors”1 (Schön, 1983), type
analogies that involve aspects of our everyday world that we know well but
should not be taken too literally, similar things happen in the microscopic
world. Particles may face obstacles that they do not have sufficient energy to
pass, but somehow they occasionally manage to pass the obstacle anyway. In
a similar fashion, particles may have more than enough energy to pass an
obstacle but sometimes they even so are incapable of passing the obstacle.
The first phenomenon of those described above, where a particle is able to
overcome an obstacle even though the energy according to classical physics
is too low, is called quantum tunnelling. The name stems from the fact that
since the particle, according to classical physics, should not be able to pass
the obstacle, it is as though there is a secret tunnel through the obstacle that
the particle may use. This quantum tunnelling is not just an esoteric phenomenon from quantum mechanical theory, but a very important process in
nature which plays a significant role for such things as chemical bonds, radioactive decay, and behaviour of semiconductors as well as for operation of
the scanning tunnelling microscope (STM), by which we can actually observe individual atoms on surfaces. So once again, no matter how bizarre
and counter-intuitive the quantum world may be, this seems to be the way in
which nature behaves!
5.4 Background to Paper II
In the previous sections I tried to provide the reader with at least a brief nontechnical overview of quantum tunnelling. This next section will move
closer to the actual research presented in Paper II by motivating why we
chose the particular topic for that paper – how university students understand
For Schön a generative metaphor generates ”new perceptions, explanations and inventions”
(Schön, 1983, p. 185)
probability in quantum tunnelling. It can also provide an insightful inside
description of the unpredictable paths that research always travels along.
Quantum tunnelling is a phenomenon within quantum mechanics that is very
interesting to study from a physics education researcher’s perspective. It is a
phenomenon that lies far from people’s everyday experiences – it therefore
belongs to the realm of abstract physics which I am interested in exploring –
and a solid understanding of the phenomenon involves many of the most
fundamental concepts of quantum mechanics, such as wave functions, probability and energy. Together with the fact that no previously published qualitative research could be found on the understanding of this phenomenon,
these provided the main reasons for choosing quantum tunnelling as our
object of research.
Why probability then? Initially our interest in students’ understanding of
quantum tunnelling was aimed towards a completely different goal. We were
interested in exploring the general interplay and relationship between
mathematics and physical concepts, and we believed that quantum tunnelling
could be a suitable conceptual environment for this kind of investigation.
However, it soon became clear to us that the mathematics of this phenomenon and the abstract nature of the phenomenon itself made that kind of study
a very difficult, if not impossible, task.
But this is not the end of the story. From the initially obtained raw data, we
noticed that there were interesting things brought to the fore by the students
that were particularly related to probability and energy, so we changed our
focus to investigate how students’ dealt with these concepts in the context of
one-dimensional quantum scattering and tunnelling. Working with this new
focus and analysing our continuously increasing pool of data, we started to
realize that even though some very interesting issues emerged centred
around the notion of energy, the most interesting concept to explore further
was probability since it was the most problematic aspect to the students. This
led us to pose our main question: What is the variation in students’ understanding of probability in the context of quantum tunnelling?
5.5 Results of Paper II
In Paper II, we carry out a phenomenographic study of how students understand probability in one-dimensional quantum scattering and tunnelling. The
analysis revealed that it was possible to construct four qualitatively different
categories to map the understanding of probability, and that many student71
held views of quantum tunnelling and quantum mechanical concepts can,
from a physics point of view, be classified as erroneous or incomplete. In
Section 5.5.1 and 5.5.2 I present a summary of these results and in Section
5.6 the results and associated pedagogical implications are presented.
5.5.1 Categories of students’ understanding of probability
By analysing the data from our interviews providing the data for Paper II
using a phenomenographic approach, we could identify four different categories, corresponding to four different ways of understanding probability in
quantum tunnelling. These categories are described in detail in Paper II so I
will only briefly discuss the different categories here.
Category 1 – Understanding probability in terms of reflection and
In this category, probability is seen to correspond to the same kind of probability involved in flipping a coin. For a coin, there is a certain probability
for heads and a certain probability for tails. In the scattering process there is
a certain probability for the wave packet to pass the energy barrier and a
certain probability for the wave packet to be reflected. Comparing this category to the phases discussed earlier, it can be seen that this notion of probability is focused on the outcome of analysis of the scattering process – the
transmission and reflection coefficients.
Category 2 – Understanding probability in terms of energy
In this category, probability is discussed in terms of energy. Either the wave
packet itself is seen as having a probability of having a certain energy or the
discussion is carried out in terms of the energy of the individual waves constituting the wave packet. In the latter case it is common to advocate that
those individual waves that have an energy higher than the maximum of the
energy barrier will be transmitted and the rest of the individual waves, with
an energy lower than this maximum, will be reflected.
Category 3 – Understanding probability in terms of finding a particle
In this category, probability is discussed in terms of the probability of finding the particle represented by the wave packet at a certain place at a certain
time – even though the temporal aspect is often overlooked in comparison to
the spatial aspect.
Category 4 – Understanding probability in terms of an ensemble
In this category the notion of ensembles emerges. Probability is discussed in
terms of repeated experiments with identically prepared systems.
5.5.2 Conceptual difficulties
Apart from identifying the different ways of understanding probability in
quantum tunnelling presented in the previous section, several conceptual
difficulties were identified during the interviews or in relation to the categories presented in Section 5.5.1. One such difficulty was that for some students, tunnelling implies that a particle loses energy when tunnelling, which
is not the case. This conception has also been identified by Wittmann (2003)
and Wittmann et al (2005) and has later been confirmed in two quantitative
surveys (Falk; 2004; McKagan & Wieman, 2006).
We could also identify conceptions where students believe that the wave
packet representing the particle is either completely reflected or transmitted,
and that only particles with “high enough energy” are able to tunnel. Both of
these conceptions have been identified in Falk (2004). Yet another conception identified involves probabilities for particles to have a certain energy as
a determining factor whether tunnelling is possible or not. Once again this
conception has been confirmed by the surveys of Falk (2004) and McKagan
and Wieman (2006)
5.6 Discussion of the results of Paper II
From the discussion of the different phases of the tunnelling process in Paper
II and the different facets of probability that are important to interpret and
analyse the scattering process, these categories described in Section 5.5.1
might look promising. All the important facets of probability involved in
quantum tunnelling can be found in the four categories. However, if one
starts to look at individual students, it can be noted that none of them takes
all these facets of probability into account. We would like our students to
understand and use all these facets, in order to be able to switch between and
link the different facets depending on what phase of the process is being
considered. Lacking one or several of these facets makes it difficult to interpret and analyze the full scattering process.
To improve on the incoherent understanding of probability reported in Paper
II, we postulate that it is important to explicitly raise focal awareness of the
different facets of probability. This could be done in terms of in-class discussions of the different facets of probability and their relation to each other and
the different phases of the process. This implicitly means a need for going
beyond the “standard” treatment of this process, which normally involves
individual waves and snapshots of the process before and after the scattering
complemented by a mathematical analysis.
In light of these incomplete understandings of probability and the conceptual
difficulties identified in Paper II, it seems to be important to go beyond the
“standard” treatment to get a deeper conceptual understanding. From our
study we have come to believe that the use of a computer simulation of onedimensional quantum scattering and tunnelling of a wave packet could be a
way of achieving this. This approach has successfully been carried out for
many areas of quantum mechanics in the Visual Quantum Mechanics Project2. Such a computer simulation could be shown during a lecture accompanied by a discussion with the students or it could be implemented as a tutorial with conceptual questions relating to wave packets, probability and tunnelling. A simulation of this kind naturally cues a discussion of conceptual
issues and could be a fruitful and more alive alternative or complement to
the standard treatment of this process, to help the students to get a firmer
grip of one-dimensional scattering and tunnelling and probability in particular.
This is a project developed by the Physics Education Research group at Kansas State University.
6. Exploring the role of mathematics in
6.1 Introduction
Looking at the research which has been presented thus far in this thesis –
theoretical physics research involving the de Sitter space and educational
research exploring students’ understanding of quantum tunnelling – there is
a common denominator, as promised in the introduction of this thesis. Both
of these studies involve the abstract experienced in physics. If we recall that
the main characteristic of such physics is that it is difficult to relate the physics to everyday situations, this means that it is hard to use everyday analogies as a tool to get a better understanding of the physics and that it is cumbersome to find concrete everyday examples. In such situations, mathematics
becomes the main actor and an understanding of the physics equations involved becomes even more important than in less abstract areas of physics.
Therefore I became interested in exploring how students deal with mathematics in physics, and to explore students’ experiences of physics equations
in particular.
In this chapter I aim to present the reader with results from two empirical
studies (Paper III and Paper IV) which investigate students’ view of what it
means to understand physics equations. Before getting to those results, I
would like to set the stage with a discussion in Section 6.2 of why mathematics plays an important role in physics and in Section 6.3 with a discussion of
how mathematics plays an important role in physics.
6.2 Why does mathematics play an important role in
As any physicist or physics student know, mathematics is a crucial ingredient and an important and useful tool in the field of physics. Mathematics is
often even considered to be “the language of physics”. From a pragmatic
point of view this might seem unproblematic, but if we take a step back and
reflect on the immense importance of mathematics in physics, it starts to
become far from trivial. The basic quandary is that physics is an endeavour
devoted to describing objects and phenomena in the world we live in, while
mathematics on the other hand is a field that deals with abstract, mental constructions and their relations. Why should two such disparate fields have
anything in common and why does mathematics play such an important role
in such a vastly different field as physics?
It is beyond doubt a very subtle question and many well known physicists
have given it a considerable amount of thought. One of these physicists is
Dirac (1939), who claims that there are two basic methods for a physicist to
make progress when studying nature:
a) experiments and observations; and,
b) mathematical reasoning.
The first of these methods involves the process of collecting data from given
experiments and situations, while the latter, for some reason, lets us draw
conclusions about experiments that still haven’t been performed and about
situations that still haven’t been studied. As Dirac states, there is no logical
reason whatsoever for the second method to be possible, but we know from
our experience and from the history of science that it works like a charm in
practice and that it has led to tremendous advances and successes in science.
According to Dirac mathematicians play a game where they set the rules,
while physicists play a game where nature sets the rules. However, as time
goes by, it becomes more and more clear that the rules that mathematicians
consider to be interesting are the same rules that nature has chosen. Dirac
furthermore believes that this has to be attributed to “some mathematical
quality in nature, a quality which the casual observer of nature would not
suspect, but which nevertheless plays an important role in nature’s scheme”.
Wigner (1960) also addresses the question about why mathematics is important in physics in a paper with the expressive title “The unreasonable effectiveness of mathematics in the natural sciences”. In this paper Wigner points
to the fact that mathematical concepts often turn up in completely unforeseen
contexts and then often give rise to an unexpectedly detailed and precise
description of the phenomena. Wigner talks about abstract, advanced mathematical concepts which do not in any natural way derive from the physical
world and describe these constructs as advanced constructs chosen by
mathematicians since they allow them to practice “mental acrobatics” and
perform “brilliant manipulations”. The unreasonable aspect of these, according to Wigner – as indicated by the title of his paper – is that they, for some
reason, turn out to be efficient tools in science and physics in particular. In
the same way as Dirac, he considers this usefulness of mathematics unexpected in the sense that we lack a rational explanation for such usefulness.
A partial “explanation” of why mathematics is important in physics, can be
found in the historical development of mathematics and physics. A large part
of modern mathematics was developed in symbiosis with physics. Such great
mathematicians as Leibniz, Gauss, Laplace, Legendre, and Fourier are also
well known for ground-breaking contributions to the field of physics. A lot
of mathematics has been developed from considerations of physics problems. Let me give some examples to illustrate this:
Calculus was mainly developed by Newton and Leibniz. Leibniz’s
motivation was mainly geometrical problems, while Newton wanted
a mathematical tool to describe the motion of physical bodies. Influenced by Euler, Lagrange, Hamilton and others, mechanics evolved
into analytical mechanics, where we can find the origin of the calculus of variations.
The foundations of Fourier analysis originate in Fourier’s attempts
to solve one of the classical equations in physics: the heat conduction equation.
The Hamilton-Jacobi theory of mechanics was one of the sources of
inspiration for the theory of partial differential equations, which in
turn led Lie to the concept of continuous groups.
This list could easily be extended, and one is led to speculate whether the
mathematical concepts have been constructed in such a way that they suit
physics and the physics problems that currently are studied. This may be true
in certain cases, and the history of science involves several examples of this
as the above list illustrates, but it is not a generally valid statement. As an
example, the mathematics that constitute the backbone of the quantum mechanical theory was developed far earlier. So this historical “crossfertilization” between physics and mathematics can at best only be considered a partial attempt to explain the role of some mathematics in some areas
of physics and does not address the general question of why mathematics is
important in physics.
The surprising usefulness of mathematics in physics has also attracted the
attention of the Danish physicist Niels Bohr. In Bohr’s view, mathematics is
a science of structures that in principle supplies us with all possible structures. Physics is involved with finding structures in nature, and according to
Bohr (1958) it is therefore not surprising that we find the structures that are
adequate for physics in the realm of mathematics. A possible objection is
hidden in the words “in principle” above. Theoretically, mathematics has the
possibility to supply us with all possible structures, but in practice this is not
the case. The structures studied by mathematicians are only a tiny fraction of
all the possible structures, and according to Wigner those structures that are
most useful to physicist are already part of the existing structures.
Where do the thoughts of so many great physicists take us? We can conclude
this section by stating that the question of why mathematics is a successful
and useful tool in physics is subtle and far from simple to answer. Maybe we
have to resort a pragmatist’s point of view and say that we do not know why
mathematics is important in physics, but somehow nature is constructed in
such a way that mathematics turns out to be an important tool to describe
nature, so let us use this fortunate coincidence to our benefit. This is however a banal and, in my opinion, dissatisfying conclusion and the connection
between mathematics and physics and the process of describing nature is
probably far more deep than what such a statement involves.
6.3 What is the role of mathematics in physics?
As discussed in the last section, it is difficult to answer the rather philosophical question why mathematics plays an important role in physics. However, it is still possible to discuss in detail what the role of mathematics is in
physics, which is the topic of this section.
Physics is a branch of science dedicated to describing various features of and
phenomena in nature in order to help us get a better understanding of nature3.
nature is beyond doubt a very complex system so this might seem like a
daunting task. Fortunately, it turns out that nature is not an unstructured
mess. We can, and do, find patterns and regularities in the complexity, and
the primary focus of physics is to identify these patterns and regularities and
organize them into larger, coherent structures.
The identification of these patterns normally involves several stages of approximation and idealization. We need to be able to decide what the important factors in the phenomena we are studying are, and what features of the
situation we can neglect for our particular investigation. Otherwise the pattern may be hidden behind too many different factors and influences. At the
same time, what is important in one situation might be neglected in another
The word “various” is actually more important than it might seem. Physics is not a science
that attempts to answer every imaginable question about nature. While it is well suited to
answer some questions about the world we live in it is for example not very useful when
discussing whether Harold Pinter is a worthy Nobel Laurate, why people have phobias and
how to remedy them or the factors that led to the Second World War.
situation. It all depends on what kind of questions about nature we are posing and trying to answer.
For physicists, patterns and regularities in nature can be found by observations, experiments, theoretical modelling or a combination of some of or all
of these methods. When we, by using these techniques, have gathered
enough information we might be able to construct a physical theory – a condensed set of identified patterns and relations that can describe and unify a
larger set of phenomena in nature.
Such theory can then be used to make predictions about previously uninvestigated phenomena and situations in nature. Part of the ultimate test of a
theory involves checking the predictions of the theory against nature itself:
Does nature behave as the theory predicts? If this is not sufficiently accurately the case, we might have to modify the theory or come up with a new
theory. It is important to stress that we can never prove a theory to be true,
we can only, as Popper emphasized, prove a theory to be wrong by identifying a mismatch between our theory and nature. A theory should always be
viewed as the current best description available of the subset of nature we
are studying.
So far I have only described a physical theory in very general terms – as a
condensed set of identified patterns that can describe a set of phenomena and
observations in nature. It is now time to be a little more detailed and to discuss the building blocks of a physical theory and their relation to nature itself.
A physical theory always involves concepts and constructs – well-defined
properties of the system we are studying. If we, for instance, are studying the
motion of objects, we have concepts such as force, mass, velocity, acceleration. It is important to remember that these concepts do not have a life of
their own within the theory – they always represent various features of the
actual system we are studying.
Apart from the concepts themselves. a physical theory also consists of relations between these concepts – these are the patterns and regularities that we
discussed earlier. Even though it is far from trivial why – as discussed in the
previous section – these relations can be captured very elegantly and compactly using the language of mathematics. Going back to the study of the
motion of objects I mentioned in the paragraph above, I could argue that we
have identified relations between the concepts force, mass and acceleration
that say that the acceleration of the centre of mass of an object is proportional to the resultant external force acting on the object and directed in the
same direction as that resultant force. Moreover, the acceleration is inversely
proportional to the mass of the object – the heavier object the less accelera79
tion for the same resultant force. All these relations can be neatly summarized with a single mathematical formula – known as Newton’s second law:
¦ Fext
ma cm
For physicists familiar with the concepts and the mathematics involved, this
single expression contains all the information about the relations between the
concepts that we earlier described in words. So, in a physical theory, we
define concepts and then describe the found relationships between these
concepts using the language of mathematics.
Some might ask whether the formulation of these relations in mathematical
terms is just a matter of aesthetics and compactness. Do we gain anything
beyond having to write less text, by formulating these relations as mathematical formulas? In fact we do! Once we have established these relations as
mathematical expressions, we can start to use the entire machinery of
mathematics to manipulate these expressions. It is by this mathematical manipulation that we can get to well-defined and testable predictions from our
theory. So, the mathematical expressions are, along with the concepts themselves, crucial ingredients in any physical theory and could be seen as the
condensed products of the process of constructing a physics theory. This
means that if we want to explore students’ learning of physics it is important
to investigate how students experience and deal with physics equations.
6.4 Exploring students experience of equations
As discussed in Section 3.4, a lot of the physics education research as far as
equations are concerned had been focusing on the use of equations or on the
structural relationships of equations. We felt that many aspects of students’
experience of physics equations remained unexplored – such as what students focus on when presented with equations and students’ epistemological
views of what it means to understand equations.
The study presented in Paper III attempts to map out the variation in what
students focus on when presented with physics equations and to discuss the
nature of this variation. We believe this to be of importance for the teaching
and learning of physics since what students focus on generically reflects how
they think about, view and use equations. This is important for a physics
teacher to be aware of and should inform the teaching of physics in terms of
how we structure the way we teach physics and come to know our students
as learners.
Furthermore, apart from problem solving and despite the many researchdriven changes that have taken place in physics education teaching and
learning environments (e.g. Redish, 2003), the most common form of teaching physics remains teacher presentation and/or students reading textbooks.
It is therefore of great interest to explore what students focus on when presented with equations. What aspects of physics equations are brought to the
After investigating what students focus on, we became interested in exploring whether these focuses could have epistemological underpinnings. Epistemologically based research such as that done by Hammer (1994), Roth and
Roychoudhury (1994), Redish, Saul and Steinberg (1998), May and Etkina
(2002), Adams et al. (2006), and Lising and Elby (2004) have indicated that
physics students’ learning is significantly related to their perceptions about
the nature of physics and about physics learning and knowledge. As stated
earlier, a critical ingredient of university physics teaching and learning is the
mathematical representations, i.e., physics equations. The work reported in
Paper IV contributes to the epistemological research as well as to the research on students’ experiences of physics equations by exploring students’
epistemological mindsets (described and defined in Section 6.6.1) towards
what it means to understand physics equations.
6.5 Exploring what students focus on when presented
with equations
6.5.1 Research questions
The main research question that we were interested in finding an answer to
What do university physics students focus on when presented with
an equation?
To answer this question we used a phenomenographic approach (described
in detail in Section 4.4) and in next two sections I will describe and discuss
the results of our exploration of this research question.
6.5.2 What do students focus on? - Results of paper III
From our analysis we came up with five categories of what a teacher could
anticipate students to focus on when presented with equations (these categories are summarized in Table 2 below). Below we will describe the different
categories and provide illustrative examples for some of the categories. For
more illustrative examples from the interview data, the reader is referred to
Paper III. It should be stressed that these categories are collective constructs
and that individual students can focus on several of these facets when presented with equations.
Category A – Name/label
In Category A, students focus on attributing a name (if such a name exists)
to the equation. We suggest that the student’s experience of the formula has
moved from unknown to association with a name, but that there is no focus
on the in-depth meaning of the equation.
Category B – Mathematical aspects
In Category B students focus on the mathematical aspects of an equation,
i.e., what type of object the equation is mathematically (such as an differential equation) and what mathematical manipulations are necessary to be able
to use the equation to attain the desired quantity, i.e., to solve problems.
Category C – Linguistic reading of the equation
In Category C students focus on being able to “read” the equation, i.e., substituting terms for symbols such as “velocity equals frequency times wavelength”. An example of this from our interview data is seen when one of the
students is asked about the meaning of the electromagnetic relation
’ u E 0:
Can you tell me what this means to you?
Um, I think the E is the intensity of an electric field. And the
the curl of E…[quietly to herself] equals zero… I think this is
a conservative vector field – and I know how to calculate it
but I don’t know what it means.
Category D – Understanding of the parts
In Category D, as in Category C, students focus on the parts of the equation.
However, in Category D it is not enough to know the names of the terms that
the symbols represent. Here students focus on understanding what e.g. velocity, frequency and wavelength mean in physics or what a partial deriva-
tive or multivariable function means in mathematics. We differentiate this
from level C since it is far from clear whether saying ‘frequency’, for example, automatically means that the students understand what “frequency”
means4. An example from our data is the excerpt above, where the student
directly associates the fact that the curl of the electric field is zero to the field
being a ‘conservative vector field’ without knowing what such type of field
Category E – Understanding of the whole
In Category E, students move beyond the parts of the equations and focus on
the meaning of the equation as a whole. Here the students try to relate or link
the equation as a whole to a meaning and try to situate the equation in an
appropriate context. In the short excerpt below which is translated from
Swedish, the student is commenting on the Schrödinger equation:
What do you see when you see this equation?
[…] I wasn’t sure about this, this… psi… or what ever it’s
called, like what does it do and what do I need it for? I found
it hard to link it to anything in reality.
This student is focusing on the meaning of the equation itself in “reality” and
feels that something is missing as far as equations is concerned unless there
is a relation to “reality” in one way or another.
These categories are summarised in Table 1.
As diSessa (1993) and others have pointed out, a student can easily learn to express an equation in linguistic terms, repeating it as a slogan without actually understanding what the slogan means.
The name of the equation
Mathematical aspects
The mathematical nature of the equation
Linguistic reading of
the equation
Reading out the symbols in physics/mathematics terms.
D Understanding
of the parts
Understanding the physics or real world
meaning of the symbols in the equation
of the whole
Relating the equation as a whole to appropriate physics/everyday-world situations
Table 1: Students’ focus when presented by a physics equation.
6.5.3 Discussion of the results of Paper III
Looking at the categories summarized in Table 1, it is clear that there is a
range in what students focus on when presented with equations. Some students focus on relating the equation as a whole to appropriate physics or
everyday situations, while others simply focus on being able to read the
equation or understand its mathematical structure.
What struck us as interesting was that, although there is a spread in students’
focuses, many of the students focused on the mathematical aspects of physics equations, especially in their initial encounter with physics equations.
When asked to describe what they saw when presented with an equation or
what the equations meant, a number of students plunged straight into the
mathematics. An example is provided in the excerpt below, where the equation P = Ȍ Ȍ* = |Ȍ|2 is being discussed. To clarify to the reader, the student
talks about the wave function, but is erroneously referring to it as the
“Schrödinger equation”.
What does this tell you? What does this mean to you when
you see it?
Ehh.. this is the big letter psi, right? [Yes]. So it’s… I think it
is the Schrödinger equation and.. the complex conjugate…so
it’s squared and... like that it means that it is always positive,
so… one thing I’ve learned is that it’s always positive when
it’s a square… that’s one of the basic things they print in your
mind, that’s just stuck.
It’s just stuck there? Square means positive?
Yeah (laugh), so… it’s like… the probability function will
always be positive. It will be the, yeah, you know, square of
the Schrödinger equation.
Can you… Is there something else this equation tells you?
You see like x and t, it’s a two-variable… eh… equation or…
yeah. And… it’s just one-dimensional if you could call it…
well, it’s two-dimensional with the time, but we usually work
with it in one dimension, only x.
This example serves as a nice illustration of the fact that many students focus
on the mathematical structure of an equation. There is much that could be
said about the meaning of this equation, yet the student immediately turns to
describe the mathematics involved. Another example is provided below
where a student is asked about what she was thinking when presented with
the same equation:
What were you thinking when the lecturer presented this
I was thinking about psi and the wave function. She said it’s a
complex function so you can’t just write it as a square, but… I
still don’t quite understand what psi is, how you use it.
Once again, the conceptual understanding of what the equation means is
suppressed and the mathematical details of the equation are the focus of the
Others simply looked at the mathematical level of complexity of the equations, thereby dubbing the equations as complex or not, supposedly dependent on whether they could manipulate the equations mathematically or not.
An example is the excerpt below where the student is asked what she believes is difficult to understand about the Schrödinger equation and what she
was thinking of when presented with this equation.
Eh… I don’t know, what one was thinking was that… it’s a
second degree equation and it is a differential equation and
there are partial… derivatives… which makes it more
complicated…it feels very advanced.
Again we have an example were the mathematical aspects tend to dominate
what students are focusing on when presented with an equation.
As another example, for a number of students there was no difference between the equation k=2ʌ/Ȝ (which is a simple definition of the wave number
k in terms of the wavelength Ȝ) and the de Broglie relation p = h/ Ȝ, which
says something fundamental about nature by relating wave and particle
properties. They both had the same simple mathematical structure of an algebraic relation and were therefore seen as understood by the students, even
though a closer investigation of this understanding revealed that in some
cases the students were not even sure what the symbols in the equations represented.
So even though there is a spread in what students focus on, the mathematical
aspects of equations are considered among the most important and seem to
be the determining factor when ruling whether an equation is understood or
not. If the mathematical structure is understood and the students know how
to manipulate the equation mathematically to obtain the desired quantity, the
equation is considered understood.
Similar results to those we found in Paper II have been reported by Redish,
Saul and Steinberg (1998) in a study of students’ expectations in introductory physics. In this study, the authors state that:
An important component of the calculus-based physics course is the development of students’ ability to use abstract and mathematical reasoning in describing and making predictions about the behaviour of real physics systems.
Expert scientists use mathematical equations as concise summaries of complex
relationships among concepts and/or measurements. They can often use equations as a framework on which to construct qualitative arguments. (p. 11)
However, from their study they conclude that the actual treatment of equations as far as students are concerned is far less fruitful and sophisticated:
Many introductory students, however, fail to see the deeper physical relationships present in an equation and instead use the math in a pure arithmetic
sense – as a way to calculate numbers. When students have this expectation
about equations there can be a serious gap between what the instructor intends
and what the students infer. (p.11)
So it seems that students carry expectations that the mathematical aspects of
an equation are important – mainly due to the fact that it allows them to
solve problems. However, as shown in the research in Paper III and by Redish, Saul and Steinberg (1998) this leads to too heavy a focus on the
mathematical aspects, where other important facets of physics equations
never receive an appropriate focus. As described by Redish, Saul and
Steinberg this means that:
…an instructor may go through extensive mathematical derivations in class,
expecting the students to use the elements of the derivation to see the structure
and sources of the relationship in the equation. The students, on the other
hand, may not grasp what the instructor is trying to do and reject it as irrelevant “theory”. Students who fail to understand the derivation and structure of
an equation may be forced to rely on memorization – an especially fallible
procedure if they are weak in coherence and have no way to check what they
A logical question at this stage is then: “Where does this heavy focus on the
mathematical aspects come from?” It is likely that this situation occurs due
to the heavy focus on mathematics and rote problem solving involved in the
traditional teaching that still dominates in many physics departments and
courses. This claim is supported by the results in Redish, Saul and Steinberg
(1998), where no improvement with regard to students’ expectations of the
role of equations can be found after attending a university physics class. In
fact, it is even worse than that since some classes “showed a significant and
substantial deterioration” (p.11).
It seems like we need to go beyond the traditional mathematical focus and
actively discuss and incorporate other important aspects of equations by
exposing the students to a more varied presentation of physics equations.
Otherwise, important aspects of equations besides the mathematics (such as
range of validity and meaning) could go unnoticed and students would continue to have a too strong focus on the mathematical aspects of physics equations and inappropriate expectations of the role of physics equations.
6.6 Exploring university students’ epistemological
mindsets towards the understanding of physics
6.6.1 Mindsets
An ongoing debate exists over how to model a person's epistemology to
better understand and inform student learning. In general, the most common
models are beliefs, traits/styles, and resources (Elby & Hammer, 2001; Hofer
& Pintrich, 1997). Among other things, these models differ as to the form of
the epistemology, whether it is explicit or implicit for the student, and how
context-dependent it is. One additional model makes no claim as to form and
this is the phenomenographic relevance structure of the learning situation
model. Marton & Booth (1997, p.143) describe relevance structure as being
the “persons’ experience of what the situation calls for, what it demands. It is
a sense of aim of direction, in relation to which different aspects of the situation appear more or less relevant”. At the same time the phenomenographic
perspective is firmly anchored in its associated empirical findings that reveal
variation in ways of experiencing as being related to context. Thus for our
research question we would argue that it is most fruitful to draw on the notion of relevance structure and in doing so we are characterizing the notion
of relevance structure as a mindset. We then define mindset as the epistemologically-driven bringing to the fore of perceived critical attributes of a
learning, application, or problem solving situation.
6.6.2 Research questions
In order to begin an investigation of students’ epistemological mindsets towards the understanding of equations, we focus on three main research questions in Paper IV:
When students say that they understand an equation, how do they
describe what that means to them?
How can these descriptions be characterised in terms of epistemological mindsets?
Are similar epistemological mindsets observable for students at
various stages in their academic career?
Using the analytical process described in Section 4.5.2 we arrived at the
results presented and discussed in the next two sections.
6.6.3 Results of Paper IV
In our analysis of the data, it became clear to us that students’ descriptions of
what it means to understand an equation could be seen as epistemological
mindsets involving one or more epistemological components. In this section,
we begin by presenting and describing our characterisations of the components of students’ epistemological mindsets towards the understanding of
physics equations. Illustrative examples of the characterisations from the
interview data can be found in Paper IV.
Throughout the description of these epistemological components, we use the
equation providing the speed of a longitudinal wave in a fluid, v
B/U ,
where v is the speed of the wave, B is the bulk modulus of the fluid through
which the wave propagates and ȡ is the density of the fluid. The use of this
equation is for illustrative and clarification purposes and does not reflect or
represent actual excerpts from the interview data.
Epistemological component A – understanding involves being able to
recognise the symbols in the equation in terms of the corresponding
physics quantities
For this epistemological component, understanding an equation involves
being able to recognise what all the symbols in the equation represent in
terms of corresponding physics quantities. In the case of v
B / U , this
would correspond to identifying v as the speed of the wave, B as the bulk
modulus and ȡ as the density of the fluid through which the wave propagates.
Epistemological component B – understanding an equation involves
being able to recognise the underlying physics of the equation
Here, understanding an equation involves recognising the underlying physics
of the equation. This involves one or several subcomponents such as knowing what the quantities in the equation mean from a disciplinary physics
point of view, what the underlying concepts and principles of the equation
are or being able to know the origin of the equation in terms of how it is
B / U , this could correderived. If we once again look at the equation v
spond to an understanding of what the speed of a wave means, what the bulk
modulus of a substance represents and what density is. It could also involve
knowledge of waves (longitudinal waves in particular) as well as an idea of
how the equation is and can be derived from more fundamental concepts.
Epistemological component C – understanding involves recognising the
structure of the equation
For this epistemological component, the structure of the equation is involved – an understanding of how the different quantities in the equation are
related to each other and the equation as a whole in terms of where the quantities are situated in the equation and what this implies. Using v
B / U to
clarify, this epistemological component would involve considerations of
whether it makes sense to have the bulk modulus B in the numerator and the
density ȡ in the denominator, i.e., does it makes sense that the speed of the
wave increases if we have a larger bulk modulus B and that the speed increases if we decrease the density? It also involves asking questions such as:
What happens to the speed if we have a bulk modulus that is four times larger?
Epistemological component D – understanding involves establishing a
link between the equation and everyday life
For this epistemological component, understanding involves establishing a
link to everyday life. Two main types of links could be identified in the data.
The first type involves situating the equation in an everyday context, by
identifying examples and situations in everyday life where the equation applies. The second type of link between an equation and everyday life consists
of finding analogies from everyday life that help in appreciating the meaning
of the equation. For the first type of link, an example would be realising that
B / U could describe the propagation speed of sound in air. For the second type of everyday linking through analogies, one could compare swimming through water to walking through air in order to appreciate the dependence of the speed on the density.
Epistemological component E – understanding involves knowing how to
use the equation to solve physics problems
In this epistemological component, understanding involves being able to
know how to use the equation, i.e., solving physics problems by using the
mathematical manipulations that are needed to extract the sought information from the equation. This component also involves identifying which information is sought as well as what other information is available or needed.
Once again using v
B / U to clarify this component would involve being
able to use this equation to calculate the speed of longitudinal waves for a
fluid with a given bulk modulus and density, or more generally being able to
calculate any of the three quantities from the equation given the other two.
Epistemological component F – understanding involves being able to
know when to use the equation
For epistemological component F, an understanding of when to apply an
equation and when or when not an equation can be applied is explicitly put
forward and described to be an important part of understanding an equation.
This involves knowledge of the range of validity of the equation, inherent
approximations and idealizations and in some cases also what branch of
physics the equation is supposed to describe. In the case of v
B / U this
would involve knowing for what kind of waves this equation can be used
and for what kind of waves it cannot be used. It would also involve acknowledging factors such as the fact that this equation presumes small amplitudes and linear waves and that the fluid is considered to be a continuum.
After identifying the different components of students’ epistemological
mindsets towards the understanding of an equation which we could identify,
the next step was to map the epistemological mindset of individual students
in terms of which epistemological components we could identify as being
described by the students as an important part of understanding an equation.
How to
to use
Table 2. A mapping of the epistemological components for individual students.
We arrived at Table 2, which shows which epistemological components
(labelled horizontally from A to F, referring to the different components
presented earlier in this section) are present for the individual students in this
study (labelled vertically from 1 to 20). Students 1-7 are first year undergraduate students, students 8-16 are second or third year undergraduate stu91
dents and students 17-20 are PhD students. The last row involves a count of
the number of occurrences of the components corresponding to each column.
6.6.4 Discussion of the results of Paper IV
The results presented in the previous section contain several interesting features. Let us look at these results in terms of the original research questions
we posed. The first two questions of the research investigation were:
When students say that they understand an equation, how do they
describe what that means to them?
How can these descriptions be characterised in terms of epistemological mindsets?
First of all, we found that students’ descriptions of what it means to understand physics equations could be seen as epistemological mindsets, composed of one or several components. Looking at our characterisations of the
components, it can be concluded that to understand an equation could mean
a range of things for the students. For some students understanding simply
involves being able to recognise the symbols and being able to use the equation to solve physics problems. For others, understanding has to involve an
appreciation of the underlying physics as well as an ability to link the equation to an everyday life situation. This range in what it means to understand
an equation becomes even more apparent when looking at Table 2. The most
frequently occurring components are how to use the equation and being able
to recognise the symbols, followed by understanding the underlying physics
and relating the equation to everyday life. Thus, the most striking structure
of Table 2 is that there is no discernable structure; there are no clear correlations between different components and no obvious pattern can be found. So
in conclusion, students’ views of what it means to understand an equation
could be characterised as mindsets involving one or more components, with
a large spread in the composition of these mindsets.
The third question we were interested in answering was:
Are similar epistemological mindsets observable for students at
various stages in their academic career?
Looking again at Table 2, no clear differences could be identified between
students at various academic levels. The only thing that could be said, although the low number of students makes such a statement highly tentative,
is that all of the higher level students find an understanding of the underlying
physics important. So, apart from this, there does not seem to be any temporal convergence to a particular view of what it means to understand an equation or a particular set of components emerging in students’ mindsets.
Taken together, these results indicate that, apart from being able to use equations, students should also be given an opportunity to reflect on what it
means to understand physics equations. Taking previous epistemological
research into account, it seems plausible that students’ views of what it
means to understand an equation affects how they approach equations. As an
example, take a student with an epistemological mindset where understanding an equation means being able to use it. It is likely that this student’s efforts go into being able to manipulate the equation mathematically, thereby
being at risk of missing other important attributes of physics equations. So
we suggest that epistemological discussions should be a natural and important ingredient in introductory physics courses. Questions such as “What is
physics?” and “What does it mean to understand a physics equation?” should
be addressed and students should be given an opportunity to reflect on these
matters. Due to the importance of epistemology shown by educational research, such an intervention is likely to have a positive effect on students’
learning of physics, especially in the light of a successful implementation of
such an approach by Linder and Marshall (1998).
7. Conclusions and outlook
7.1 Concluding remarks
Being close to the end of this thesis, I hope that I have been able to provide
the reader with an accessible account of my research and associated pedagogical implications. I also hope that at this stage the reader, after being
presented the thesis with my research questions and findings originating
from these questions, agrees with what I claimed in the introduction – that
this thesis can be seen as making research contributions in three main areas:
understanding and investigating the nature of some potentially puzzling aspects of de Sitter space in relation to inflationary cosmological models;
a contribution to the exploration of university students’ understanding of quantum mechanics in the area of quantum tunnelling; and,
an exploration of university students’ experience of physics equations in terms of what they focus on when presented with equations
and how they epistemologically view what it means to understand
physics equations.
Looking at this list, a natural follow-up question that emerges is: “Where do
we go next?” I attempt to provide the reader with an answer to this question
in the next (and last) section of this thesis.
7.2 Future research
Looking at the research presented in this thesis, many ideas comes to my
mind as I reflect on possible future research directions inspired by the research reported here.
My research in theoretical physics has involved what is known as the de
Sitter space. This has always been a favourite laboratory for many ideas and
toy models in theoretical physics. With the discovery that our Universe is not
only expanding, but accelerating and approaching a de Sitter space, many
questions arise that pose difficulties for theoretical physics and that need to
be resolved. One such difficulty is obtaining a consistent string theory in de
Sitter space, which turns out to be a highly non-trivial task. Another difficulty associated with the discovery that our Universe evolves towards a de
Sitter space is that Einstein’s famous cosmological constant has “returned
from the dead”. An understanding of the origin and nature of the cosmological constant is currently one of the most extensively researched areas in
theoretical physics, with a very large potential impact on our view of nature.
So aspects of de Sitter space are likely to play an important role in future
theoretical physics research.
As far as the conceptual understanding of quantum mechanics is concerned,
this is an area where I believe that many things remain to be done. It is not
easy to find concepts which are unexplored in for example mechanics, while
the situation is very different for quantum mechanics. There still exist many
unexplored areas of quantum mechanics where we do not have an understanding of students’ ideas, conceptions and understandings. The surveys by
Falk (2004) and McKagan and Wieman (2006) are important steps towards a
fuller understanding of students’ conceptual understanding in quantum mechanics, but further development of such surveys hinges on data acquired
from explorations of previously unexplored students’ conceptions.
Apart from mapping student conceptions, it would also be interesting to explore the origins of these conceptions. Unlike mechanics, where many conceptions and conceptual hurdles stem from prior everyday life experiences,
this is due to the nature of quantum mechanics, not the case for conceptions
of quantum phenomena. Thus, many of these conceptions are born at the
same time quantum mechanics is first encountered and learnt, and it would
be interesting to explore the origin and development of these conceptions
and which factors affect their formation and how.
Another area which my research presented in this thesis has explored is students’ experience of physics equations. In my research, I have investigated
what students focus on when presented with equations as well as students’
epistemological views of what it means to understand an equation. I have
suggested that the results imply that it would be valuable to incorporate epistemological reflections of what the role of physics equations are and what it
means to understand physics equations. It would be interesting to make a
coherent implementation of this suggestion and to evaluate the impact as far
as student’s view of what it means to understand an equation and actions
stemming from this view is concerned. “Coherent” refers to the fact that
such an implementation would need to involve all aspects of a course. There
is no point in making such an intervention if the motivation is unclear to the
students or if such things as the assessment or problem solving on that
course are not modified in accordance with this introduction of, and focus
on, epistemological considerations.
Besides trying out such an implementation, I can see other things related to
students’ experience and understanding of equations that would be interesting to explore. One such thing would be to make similar explorations for
physics teachers and/or researchers and comparing and relating their views
to those of the students. Another interesting thing would be to investigate
what physicists consider to be the key ingredients of a fruitful and appropriate understanding of physics equations. This could then be followed by empirical studies mapping students’ understanding of these key ingredients.
In Paper IV we introduce the notion of epistemological mindsets. From a
theoretical point of view it would be interesting to expand the idea of mindsets in terms of the phenomenographic construct of relevance structure in
relation to how students frame their learning.
In summary, I see many possible research questions and directions building
on the research presented in this thesis, and I hope that future research will
be able to explore these areas further, thereby expanding and adding to our
understanding of physics as well as the teaching and learning of physics, in
the same way as I believe that my research presented in this thesis has made
such a contribution.
8. Svensk sammanfattning
8.1 Introduktion
Denna avhandling innefattar forskning inom två skilda områden: teoretisk
fysik och fysikens didaktik. Vid en första anblick kan det tyckas att dessa
båda områden har väldigt lite gemensamt. Det finns dock en explicit gemensam nämnare mellan min forskning i teoretisk fysik och i fysikens didaktik: i
båda fallen rör det sig om ett utforskande av abstrakt fysik och som en följd
av detta även av de matematiska representationerna av fysik. I det ena fallet
”från insidan” innefattandes mina egna erfarenheter och i det andra fallet
”från utsidan” via en kartläggning av studenters erfarenheter. Från en inlärningssynvinkel finns många frågeställningar och erfarenheter som är desamma oavsett om du är forskare eller student och jag är övertygad om att
mina egna erfarenheter av forskning inom abstrakt fysik har varit ovärderliga
när jag har utforskat studenters sätt att erfara abstrakt fysik.
Jag har dock ännu inte definierat vad jag avser när jag säger abstrakt fysik.
Med abstrakt fysik syftar jag på fysik som inte kan relateras till vårt vardagliga liv eller till tidigare erfarenheter. Detta är självfallet något subjektivt och
många grenar av fysik innehåller begrepp eller fenomen som skulle kunna
klassificeras som exempel på abstrakt fysik. Min egen forskning har innefattat såväl strängteori som kosmologi och kvantmekanik och dessa områden
innehåller många aspekter som tveklöst kvalificerar sig som abstrakta och
som innefattar begreppsmässiga utmaningar när det gäller inlärandet av dessa aspekter.
Jag hoppas att jag i avhandlingen lyckas övertyga läsaren om att min forskning bidrar till förståelsen av abstrakt fysik samt till förståelsen av studenters
erfarenheter av densamma. Mer specifikt bidrar denna avhandling med relevant forskning inom tre huvudområden:
Förståelse och utforskning av några potentiellt problematiska aspekter hos de Sitter-rummet i relation till kosmologiska modeller som
innefattar inflation.
En undersökning av fysikstudenters begreppsmässiga förståelse av
kvantmekanisk tunnling och i synnerhet av sannolikhetens roll för
detta fenomen.
En utforskning av studenters erfarenhet av ekvationer inom fysik i
termer av vad de fokuserar på när de möter ekvationer och hur de
från en epistemologisk synvinkel betraktar vad det innebär att förstå
8.2 Forskningsfrågor
Den första delen av denna avhandling innefattar forskning inom teoretisk
fysik relaterad till kosmologi och förståelse av de Sitter-rummet. De flesta
moderna kosmologiska modeller involverar en epok som går under benämningen inflation. Under denna era genomgår vårt universum en hastig, accelererande expansion. Inflationsmodeller förekommer i många varianter, men
gemensamt är att de uppvisar lovande resultat när det gäller att förstå och
komma till rätta med en del problematiska aspekter av kosmologier som
saknar en inflationsfas.
I Artikel I i denna avhandling konstruerar vi ett kosmologiskt scenario som
kan ses som en leksaksmodell för inflation, och i detta scenario formulerade
vi och utforskade en intressant fråga:
I ett kosmologiskt scenario där en de Sitter-fas följs av en fas med en
skalningsfaktor som utvecklas som tq, där 1/3<q<1, finns det en möjlighet att information kan dupliceras i strid med kvantmekanikens
Den andra delen av denna avhandling omfattar forskning inom fysikens didaktik. Med ett genuint intresse i teoretisk fysik, var det naturligt att från ett
lärandeperspektiv studera ett område där teoretisk abstraktion spelar en viktig roll: kvantmekanik. Artikel II undersöker studenters förståelse av kvantmekanisk tunnling. Detta är ett utmärkt didaktiskt laboratorium eftersom
tunnling är ett fenomen som dels innefattar aspekter där vi inte har några
vardagliga erfarenheter att luta oss mot och som dels kräver en förståelse av
ett flertal centrala kvantmekaniska begrepp. Den huvudsakliga forskningsfrågan för denna studie var:
Hur ser fysikstudenters förståelse av sannolikhet för kvantmekanisk
tunnling ut, och vilken variation i denna förståelse är möjlig att påvisa?
Från denna studie blev det uppenbart att det kan vara besvärligt att förstå
abstrakt fysik och att matematiska representationer (ekvationer) tenderar att
spela en väldigt viktig och dominerande roll för studenter när det gäller förståelsen av denna typ av fysik. Detta ledde till ett intresse av att försöka kartlägga studenters erfarenhet och förståelse av fysikaliska ekvationer. Artikel
III och IV är båda produkter av detta intresse. I Artikel III var vi intresserade
av att utforska studenters fokus när det gäller ekvationer inom fysik och den
centrala frågan var:
Vad fokuserar fysikstudenter på när fysikaliska ekvationer presenteras för dem?
De svar på denna fråga som vår undersökning fann, fick oss att börja spekulera i om studenters fokus kan vara relaterat till deras syn på vad det innebär
att förstå fysikaliska ekvationer. Detta ledde till studien i Artikel IV där vi
var intresserade av att utforska studenters epistemologiska syn på vad det
innebär att förstå fysikaliska ekvationer. För denna studie formulerade vi tre
forskningsfrågor som vi var intresserade av att besvara:
När studenter säger att de förstår en fysikalisk ekvation, hur beskriver de vad detta innebär för dem?
Hur kan dessa beskrivningar kategoriseras i termer av epistemologiska attityder?
Ser dessa epistemologiska attityder likadana ut för studenter på olika
nivåer i deras utbildning?
8.3 Metod och metodologi
Denna rubrik är i första hand relevant för Artikel II, III och IV som utgör
forskning inom fysikens didaktik. Som i all forskning är inom fysikens didaktik av största vikt att välja ett lämplig sätt att besvara de uppställda forskningsfrågorna. För alla dessa artiklar har det huvudsakliga datamaterialet
insamlats genom intervjuer avsedda att utforska studenters erfarenheter och
tankar kring det ämne som forskningsfrågan varit avsedd att undersöka. Dessa intervjuer har utformats på ett sätt som tillät studenterna att reflektera över
frågeställningar på ett så mångfacetterat sätt som möjligt för att öka sannolikheten att få en adekvat representation av mångfalden i studenternas erfarenheter. Genomgående har intervjuerna även genomsyrats av en ambition
att få till stånd ett intervjuklimat där studenterna känner att de kan uttrycka
sina erfarenheter på ett tryggt och öppet sätt.
Beroende på frågeställningen har data från dessa intervjuer därefter i iterativa cykler analyserats och tolkats i relation till forskningsfrågorna. I såväl
Artikel II som III var vi intresserade i att kartlägga variationen i studenters
förståelse av sannolikhet i relation till kvantmekanisk tunnling respektive
studenters fokus när det gäller fysikaliska ekvationer. För dessa två studier
utnyttjade vi därför fenomenografi, ett teoretiskt ramverk som har för avsikt
att kartlägga variationen i personers erfarenhet av ett fenomen, ett begrepp
eller en situation. När det gäller Artikel IV var vi inte enbart intresserade av
variationen i studenters beskrivningar av vad det innebär att förstå en ekvation, utan även att karaktärisera dessa beskrivningar i termer av epistemologiska attityder samt att jämföra studenter som befinner sig på olika nivå i sin
utbildning. Av detta skäl valde vi att utföra en fallstudie av de i projektet
ingående studenterna, för att få en så rik beskrivning av studenternas epistemologiska attityder som möjligt, där bredden och den mångfacetterade naturen av studenternas erfarenheter kunde representeras på ett tillfredsställande
8.4 Resultat
8.4.1 Att undvika en potentiell informationsparadox
I Artikel I presenterar vi ett kosmologiskt scenario där möjligheten till en
informationsparadox uppstår. Huvudidén bakom denna potentiella paradox
är att ett de Sitter-rum ersätts av en fas med långsammare expansion. I detta
scenario uppstår nu möjligheten att ett föremål som inte längre kan kommunicera med en observatör under de Sitter-fasen (det vill säga all eventuell
information associerad med föremålet är otillgänglig) plötsligt får möjlighet
till kommunikation i den efterföljande fasen, vilket nu gör informationen
tillgänglig. Om vi nu antar att informationen från föremålet även når observatören under de Sitter-fasen i form av strålning från den horisont som omger varje observatör har vi ett scenario där informationen kan fördubblas
trots att fundamentala kvantmekaniska principer förbjuder detta.
Genom att beräkna den tid det skulle ta för föremålet som försvunnit bortom
de Sitter-horisonten att återigen bli tillgängligt för kommunikation visar vi i
Artikel I att ett sådant scenario kan undvikas. Det visar sig att denna tid är av
samma storleksordning som rekurrenstiden för de Sitter-rummet, vilket innebär att vi betraktar ett experiment som saknar mening eftersom en eventuell detektor hos observatören (med kortare rekurrenstid) bokstavligen tappat
minnet efter denna tid, vilket omöjliggör insamlandet av information från
strålningen från horisonten i de Sitter-fasen och hotet om duplicerad information är således undanröjt.
8.4.2 Studenters begreppsmässiga förståelse av
kvantmekanisk tunnling
I Artikel II undersökte vi hur fysikstudenter förstår kvantmekanisk tunnling
av ett vågpaket för olika typer av barriärer. Studenterna intervjuades och i
samband med intervjuerna fick de även interagera med datorsimuleringar av
kvantmekanisk tunnling. En fenomenografisk analys av de data som intervjuerna genererade visade att det fanns fyra kvalitativt skilda sätt att förstå
sannolikhet i samband med kvantmekanisk tunnling och under denna analys
fann vi även ett antal exempel på svårigheter i förståelsen av tunnling och
kvantmekaniska begrepp hos studenter.
För vissa studenter innefattade förståelsen av sannolikhet i samband med
tunnling en syn på sannolikhet som liknar den när man singlar slant. För en
slant finns en viss sannolikhet för krona och en viss sannolikhet för klave. På
samma sätt finns vid tunnling en viss sannolikhet att passera barriären och en
viss sannolikhet att reflekteras.
Andra studenter såg sannolikheten i relation till energi – antingen som en
sannolikhet för ett vågpaket att ha en viss energi eller i termer av energifördelningen för de enskilda vågorna som tillsammans bildar vågpaketet. I det
senare fallet argumenterade ett flertal studenter att de enskilda vågor som har
en tillräckligt hög energi passerar barriären medan de övriga reflekteras.
Ytterligare ett sätt att se på och förstå sannolikhet i samband med kvantmekanisk tunnling som vi kunde finna hos studenterna, var att diskutera sannolikhet i termer av sannolikheten att hitta partikeln som representeras av vågpaketet på en viss plats vid en viss tidpunkt. Slutligen kunde vi även (om än
för endast en student) identifiera en förståelse av sannolikhet i termer av
upprepade experiment på identiska system.
Förutom vissa av dessa förståelser, som från ett fysikaliskt perspektiv representerar felaktiga uppfattningar om sannolikhet i samband med tunnling, så
finns de centrala aspekterna av sannolikhet viktiga för en fullständig förståelse av tunnling med bland kategorierna över studenternas förståelse. Problemet är dock att om vi tittar på enskilda studenter så är det ingen av studenternas förståelse som innefattar alla dessa aspekter. Vi skulle önska att
våra studenter förstod och utnyttjade alla dessa komplementära förståelser av
sannolikhet för att få en helhetsbild och djupare förståelse av tunnlingsprocessen. I Artikel II presenterar vi ett antal pedagogiska förslag för att öka
och förbättra studenters förståelse av tunnling och även andra konceptuella
svårigheter vid sidan av sannolikhet som vi kunde identifiera i studien.
8.4.3 Fysikaliska ekvationer och studenters fokus
Vårt huvudsakliga intresse i Artikel III var att kartlägga vad studenter fokusera på i samband med fysikaliska ekvationer. Intervjuer i kombination med
en fenomenografisk analys visade att det finns ett flertal olika saker som
studenter fokuserar på:
Huruvida ekvationen har ett namn och i sådana fall vilket
Att förstå ekvationens matematiska aspekter, det vill säga vilken typ
av objekt ekvationen är rent matematiskt och vilka matematiska manipulationer som krävs för att använda ekvationen
Att kunna ”läsa” ekvationen, det vill säga ersätta symboler med ord
Att förstå ekvationens olika delar, det vill säga vad de storheter som
representeras av symbolerna i ekvationen har för fysikalisk innebörd
Att förstå ekvationen som helhet i termer av dess innebörd och att
länka ekvationen till ett lämpligt sammanhang
Vid sidan av kartläggandet av denna variation, var den mest intressanta upptäckten att en stor andel av studenterna hade ett väldigt starkt fokus på en av
dessa kategorier – de matematiska aspekterna. Många studenter besvarade
frågorna ”Vad tänker du när du ser den här ekvationen?” eller ”Vad har den
här ekvationen för innebörd för dig?” med utläggningar om den matematiska
strukturen hos ekvationen. Andra verkade bedöma huruvida de hade förstått
en ekvation eller inte utifrån om de kunde förstå och manipulera den matematiskt. Detta starka fokus på matematik är en smula oroväckande eftersom
det innebär att andra viktiga aspekter av ekvationer riskerar att aldrig bli
föremål för studenternas fokus. Studenterna behöver erfara en mer varierad
presentation och diskussion av ekvationer där de matematiska aspekterna
utgör en del av förståelsen av en ekvation.
8.4.4 Fysikaliska ekvationer och epistemologiska attityder
I Artikel IV undersökte vi studenters epistemologiska beskrivningar av innebörden av att förstå fysikaliska ekvationer. Från vår analys kunde vi konstatera att dessa beskrivningar kan beskrivas i termer av epistemologiska attityder sammansatta av en eller flera komponenter som sågs som viktiga ingredienser i förståelsen en ekvation. De komponenter vi kunde identifiera kan
kortfattat summeras av nedanstående lista:
Att känna igen de symboler som ingår i ekvationen i termer av motsvarande fysikaliska storheter
Att ha kännedom om ekvationens underliggande fysik
Att ha en förståelse för strukturen hos en ekvation
Att etablera en länk mellan ekvationen och det vardagliga livet
Att förstå hur ekvationen används för att lösa problem
Att förstå när ekvationen kan (och bör) användas
För de enskilda studenterna bestod deras epistemologiska attityder av en
eller flera av dessa komponenter (ingen uttryckte samtliga av dessa) och en
av de mest tydliga slutsatserna vi kunde dra var att förståelse av en ekvation
kan ha väldigt olika innebörd för olika studenter. För vissa innebar förståelse
ett igenkännande av de symboler som ingår i ekvationen samt att kunna använda ekvationen för att lösa problem, medan det för andra krävdes ingredienser såsom att kunna länka ekvationen till en vardagssituation eller att veta
när ekvationen kan tillämpas för att en förståelse skulle anses som uppnådd.
En relaterad slutsats var att studenternas epistemologiska attityd inte uppvisade någon tydlig struktur. Inga otvetydiga korrelationer mellan olika komponenter eller kunde påvisas. Slutsatsen är således att studenters syn på vad
det innebär att förstå en ekvation kan karaktäriseras som epistemologiska
attityder innefattande en eller flera komponenter.
Vi var också intresserade av att undersöka om det fanns några märkbara
skillnader i studenters syn på vad det innebär att förstå en ekvation, när studenter på olika nivå i deras utbildning jämfördes. Några tydliga sådana skillnader, vid sidan av att alla studenter i ett senare skede av sin utbildning såg
förståelse av den underliggande fysiken som en viktig komponent, kunde
inte påvisas.
Resultaten från denna studie indikerar att det finns en kraftig spännvidd i
studenters syn på vad det innebär att förstå en ekvation och i ljuset av tidigare epistemologisk forskning är det inte orimligt att anta att denna syn på vad
det innebär att förstå en ekvation påverkar hur de hanterar ekvationer. Detta
föranleder oss att föreslå ett större fokus på epistemologiska frågeställningar
när det gäller fysikundervisningen samt en mer varierad presentation och
hantering av ekvationer. En sådan presentation bör göra multipla aspekter av
vad det innebär att förstå en ekvation tydliga samt föremål för reflektion och
diskussion hos studenterna.
8.5 Några avslutande ord
Avslutningsvis vill jag understryka att det dessvärre inte finns några ”mirakelkurer” när det gäller att förbättra fysikundervisningen. Att få en helhets105
bild och förståelse av in- och utlärning av fysik är att lägga ett tämligen
komplicerat pussel. Dock så har uppkomsten och utvecklingen av fysikens
didaktik som ett forskningsfält försett oss med flera viktiga pusselbitar och
denna avhandling utgör ytterligare ett bidrag i en strävan efter att få en så
komplett bild som möjligt. Ju mer komplett bild vi kan få, desto närmare kan
vi komma till vad som rimligtvis måste vara målet - att skapa en kreativ,
framgångsrik och stimulerande miljö för våra fysikstudenter.
Vi bör ta chansen att utnyttja de insikter fysikens didaktik har att bidraga
med för att vi skall lyckas med att framställa fysik på ett sätt som får våra
studenter att inse skönheten och kraftfullheten hos fysik. Min egen resa som
doktorand har utan tvekan gett ytterligare näring åt min med största sannolikhet livslånga kärlek till fysik!
Being at the end of this thesis, it is the appropriate time to express my gratitude to many people without whom this thesis would not have existed. First
and foremost I would like to thank my supervisors: Cedric Linder and Ulf
Danielsson. They have patiently guided me through the landscapes of physics education research and theoretical physics in a way which has made me
appreciate the beauty of these research areas as well as their ways of contributing to an understanding of the world we live in.
Moreover, I would like to thank all my fellow colleagues and co-authors
which I have had the privilege to work with during these years. You have all
contributed to this thesis in various ways and I have enjoyed many happy
moments and stimulating discussions in your company. In particular, I
would like to mention Martin Olsson, Anna Danielsson, Liselott Dominicus
and John Airey who have been my closest companions on this journey, with
whom I share many unforgettable memories.
For the process of writing up this thesis I have had invaluable support from
Cedric Linder and Rebecca Kung. Anna Danielsson, Johan Falk, John Airey
and Peter Kung should also be mentioned among those who have helped me
finalise the thesis you are currently reading.
I would also like to thank Olof Karis, who made me aware of that there was
a possibility to do research in physics education and Hans Karlsson, my first
physics teacher, who made me realise the beauty of physics. I am also grateful to the department of physics at Uppsala University for making it possible
for me to pursue one of my passions while being a PhD student – to teach.
Last, but certainly not the least, I would like to thank my wonderful girlfriend Ingela, my family and all my friends for all the support and encouragement. You have always been, and continue to be, the rocks that I can lean
“In my end is my beginning”
Mary – Queen of Scots
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