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Insights into teaching quantum mechanics in secondary and lower undergraduate education
K. Krijtenburg-Lewerissa,1H. J. Pol,1A. Brinkman,2and W. R. van Joolingen3
1ELAN Institute for Teacher Training, University of Twente, 7500 AE Enschede, Netherlands
2MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, Netherlands
3Freudenthal Institute for Science and Mathematics Education,
University of Utrecht, 3508 AD Utrecht, Netherlands
(Received 9 September 2016; published 17 February 2017)
This study presents a review of the current state of research on teaching quantum mechanics in secondary
and lower undergraduate education. A conceptual approach to quantum mechanics is being implemented in
more and more introductory physics courses around the world. Because of the differences between the
conceptual nature of quantum mechanics and classical physics, research on misconceptions, testing, and
teaching strategies for introductory quantum mechanics is needed. For this review, 74 articles were selected
and analyzed for the misconceptions, research tools, teaching strategies, and multimedia applications
investigated. Outcomes were categorized according to their contribution to the various subtopics of
quantum mechanics. Analysis shows that students have difficulty relating quantum physics to physical
reality. It also shows that the teaching of complex quantum behavior, such as time dependence,
superposition, and the measurement problem, has barely been investigated for the secondary and lower
undergraduate level. At the secondary school level, this article shows a need to investigate student
difficulties concerning wave functions and potential wells. Investigation of research tools shows the
necessity for the development of assessment tools for secondary and lower undergraduate education, which
cover all major topics and are suitable for statistical analysis. Furthermore, this article shows the existence
of very diverse ideas concerning teaching strategies for quantum mechanics and a lack of research into
which strategies promote understanding. This article underlines the need for more empirical research into
student difficulties, teaching strategies, activities, and research tools intended for a conceptual approach
for quantum mechanics.
DOI: 10.1103/PhysRevPhysEducRes.13.010109
I. INTRODUCTION
Quantum mechanics has gained a strong position in
physics research and its applications. Developments in
medical imaging, nanoscience, laser physics, and semi-
conductors are all based on quantum phenomena.
Moreover, quantum mechanics is the foundation of com-
pletely new and promising technologies: quantum com-
puters, quantum encryption, and quantum entanglement.
Quantum mechanics has been an important part of uni-
versity physics and engineering education for a long time,
but the often abstract and mathematical teaching practices
used have been in dispute for several years [1]. Currently,
more emphasis is placed upon visualization and conceptual
understanding [2,3]. This conceptual approach to quantum
mechanics has made it possible to introduce quantum
mechanics at an earlier stage, and therefore it has become
part of the secondary school curriculum in many countries.
Quantum mechanics has been part of the upper secondary
school curriculum in England [4], Germany [5], Italy [6],
and the USA [7] for several years. More recently, quantum
mechanics has been incorporated in the Dutch [8] and the
French [9] secondary school curricula, and in Norway new
teaching modules have been designed and tested in the
ReleQuant project [10].
Because quantum mechanics led to fundamental changes
in the way the physical world is understood and how
physical reality is perceived [11], quantum mechanics
education is faced with several challenges. For instance,
the introduction of probability, uncertainty, and super-
position, which are essential for understanding quantum
mechanics, is highly nontrivial. These concepts are counter-
intuitive and conflict with the classical world view that is
familiar to most students. A radical change in thinking
is needed [12] and ways to instigate conceptual change
[13,14] should be investigated.
Several initiatives have been taken to improve students’
understanding of quantum mechanics and resolve problems
encountered in teaching quantum mechanics, including a
review of misconceptions of upper level undergraduate
students [15]. This review by Singh and Marshman gives
a good overview of students’difficulties on an abstract
and mathematical level. Introductory quantum mechanics
courses mainly focus on the introduction of the main
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concepts and students’conceptual understanding hereof.
Therefore, we reviewed articles covering educational
research on quantum mechanics for the secondary and
lower undergraduate level, aiming to answer the following
question:
What is the current state of research on students’
understanding, teaching strategies, and assessment
methods for the main concepts of quantum mechanics,
aimed at secondary and lower undergraduate education?
More specifically, we researched the following
questions:
(i) What learning difficulties do secondary and lower
undergraduate level students encounter while being
taught quantum mechanics?
(ii) What instruments have been designed and evaluated
to probe students’understanding on a conceptual
level?
(iii) What teaching strategies aimed at the secondary
and lower undergraduate level have been tested,
implemented, and evaluated for their influence on
students’understanding?
The overview presented in this article therefore comprises
(i) students’misconceptions and difficulties, (ii) research-
based tools to analyze student understanding, and
(iii) assessed instructional strategies, activities, and multi-
media applications that improve student understanding.
II. METHOD
For this study three databases were searched: Scopus,
Web of Science, and ERIC. The following query was used
to find appropriate articles, published in journals: “(quan-
tum OR “de Broglie”OR “photoelectric effect”) AND
(student OR instruction) AND (concept OR understanding
OR reasoning OR difficulties).”This search resulted in 471
articles from ERIC, Web of Science, and Scopus, published
between 1997 and the present.
Subsequently, the results were filtered using the follow-
ing criteria: (1) The article addresses the understanding
of quantum concepts for secondary or undergraduate
students in an educational setting, (2) the article includes
an implementation and evaluation of its impact on under-
standing, (3) the article does not expect students to be
familiar with mathematical formalism (e.g., Dirac notation,
Hamiltonians, or complex integrals), and (4) the article
mainly emphasizes physical aspects.
A total of 74 articles matched these criteria. These
articles were analyzed for detected student difficulties,
used research-based tools which measure student under-
standing, and assessed instructional strategies, multimedia
applications, and activities. The following sections present
these difficulties, tools, and teaching approaches, all
categorized and analyzed for content, research methods,
and value for teaching quantum mechanics in secondary
and lower undergraduate education. Where needed, addi-
tional literature has been used to clarify or evaluate the
findings in the selected literature.
III. LEARNING DIFFICULTIES
For the development of effective teaching strategies,
it is important to know what difficulties students have
with quantum mechanics. Therefore this section gives
an overview of findings for the first subquestion: “What
learning difficulties do secondary and lower undergraduate
level students encounter while being taught quantum
mechanics?”To answer this question, the selected articles
were all scanned for misconceptions concerning the topics
shown in Table I. These topics were based on (1) the
learning goals formulated by McKagan et al. [16], which
were based on interviews with faculty members who had
recently taught modern physics; and (2) learning goals
determined in a Delphi study among Dutch experts in
quantum mechanics [17], a method which uses consecutive
questionnaires to explore consensus among experts [18].
The topics in Table Iencapsulate the main topics found
in introductory quantum mechanics curricula around the
world [4–10]. This section gives an overview of miscon-
ceptions and learning difficulties found in the reviewed
articles, organized by the topics in Table I. See the
Appendix for more information concerning the research
methods for articles discussed in this section.
A. Wave-particle duality
The fact that tiny entities show both particle and wave
behavior is called wave-particle duality. This phenomenon
is in conflict with prior, classical reasoning. Several
selected articles addressed the understanding of wave-
particle duality [1,4,5,16,19–34]. Ireson and Ayene et al.
researched existing student views of undergraduate stu-
dents using cluster analysis [20,24,25]. Three clusters
emerged: (1) Classical description, in which students
TABLE I. Quantum topics used for the analysis of the selected articles.
Wave-particle duality Wave function Atoms Complex quantum behavior
Dual behavior of photons
and electrons
Wave functions
and potentials
Quantization and
energy levels
Time dependent
Schrödinger equation
Double slit experiment Probability Atomic models Quantum states
Uncertainty principle Tunneling Pauli principle and spin Superposition
Photoelectric effect Measurement
K. KRIJTENBURG-LEWERISSA et al. PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
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describe quantum objects exclusively as particles or waves;
(2) mixed description, in which students see that wave and
particle behavior coexist, but still describe single quantum
objects in classical terms; and (3) quasiquantum descrip-
tion, in which students understand that quantum objects can
behave as both particles and waves, but still have difficulty
describing events in a nondeterministic way. Similar
categories of understanding were found by Greca and
Freire [22] and Mannila et al. [26]. These clusters all
depend on the extent to which students hold on to classical
thinking and constitute a spectrum from misplaced classical
thinking to correct quantum thinking. Table II gives an
overview of misconceptions and learning difficulties
encountered in the reviewed research, divided into these
three clusters. In the following sections, the listed mis-
conceptions are discussed in more detail.
1. Photons and electrons
In many cases electrons display particle properties, but
that is not the entire picture. Electrons also exhibit wave
properties, such as diffraction and interference. Conversely,
light shows wave and particle behavior. Light diffracts,
refracts, and shows interference, but additionally its energy
is quantized, i.e., transferred in “packages.”The reviewed
literature showed that students have a range of different
visualizations of photons and electrons, and many have
difficulty juxtaposing wave and particle behavior. Research
showed that many secondary and undergraduate students
erroneously see electrons exclusively as particles and
photons as bright spherical balls with a definite location
or trajectory [4,5,22–25,29].
The wavelike behavior of electrons is hard to define, for
electrons appear as bright spots on fluorescent screens in
most of the textbook experiments. The wavelike behavior
of electrons only appears in the distribution of these bright
spots. Quantum mechanics does not describe an electron’s
path, only the probability of finding it at a certain location.
Müller and Wiesner [5] observed that students sometimes
falsely considered this wave behavior to be a cloud of
smeared charge. McKagan et al. [16] and Olsen [29]
reported that several secondary and undergraduate students
considered the wave behavior of electrons to be a pilot
wave, which forces the electron into a sinusoidal path.
Photons are also sometimes considered to move along
sinusoidal paths [30], but Olsen observed that students
showed less difficulty assigning both wave and particle
behavior to light than to electrons [29]. Sen [31] observed
that most students had a more scientific way of describing
photons than electrons and ascribed this to the fact that
photons are introduced later in the curriculum, which he
believes to result in fewer misconceptions of photons at the
start of undergraduate education.
2. Double slit experiment
The double slit experiment is used to illustrate the
wavelike behavior of photons, electrons, buckyballs, and
other small objects. These objects pass through a double
slit, fall onto a detection screen, and cause an interference
pattern. For electrons, this interference pattern appears only
in the distribution of the bright spots. Understanding of the
double slit experiment depends in part on the students’
understanding of the wave and particle behavior of
TABLE II. Misconceptions about wave-particle duality organized into three categories ranging from classical to quantum thinking.
Classical description Mixed description Quasiquantum description
Photons or
electrons
Electrons or photons are depicted as
classical particles [1,4,5,16,20,22–25]
Electrons and photons follow a
definite sinusoidal path [16,29,30]
Electrons are smeared
clouds of charge [5,24,25]
Electrons or photons have definite
trajectories [1,4,5,16,20,22–25]
Electrons are either a particle
or a wave depending on
other factors [21,29]
Electrons or photons are
waves and particles
simultaneously [20,30]
Light always behaves like a wave
[24,25]
Equations of properties of light
also apply to electrons [21]
Double slit
experiment
Light has no momentum [1] There is no relation between
momentum and de Broglie
wavelength [21,34]
There is no relation between
momentum and interfe-
rence pattern [21,34]
Photons and electrons deflect at a slit
and subsequently move in a straight
line [21]
No interference pattern appears
with single photons and
electrons [24–26]
Uncertainty
principle
Uncertainty is due to external effects,
measurement errors or measurement
disturbance [5,20,32]
Photoelectric
effect
Energy is transmitted by wave fronts, more
wave fronts cause more energy [30]
Light collides with
electrons [19,28]
The intensity of light influences the energy
transferred to a single electron [27,28]
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quantum objects. If students see photons as classical
particles with definite trajectories, this influences their
comprehension of this experiment. This can be seen by
the fact that some secondary students considered photons to
deflect at the slit edges and move in straight lines towards
the screen [21]. Another common problem depends on
incomplete understanding of the de Broglie wavelength.
Students do not always understand the influence of velocity
and mass on wavelength and the influence of wavelength
on the interference pattern [21,34].
3. Uncertainty principle
The uncertainty principle states that there are certain
properties that cannot simultaneously be well defined.
An example thereof is the relation between position and
momentum, for which the uncertainty principle is described
as ΔxΔp≥h=4π. This equation shows that when one of
the properties is determined with high precision, the out-
come of a measurement of the other property becomes less
certain. The uncertainty principle for position and momen-
tum can intuitively be related to the wave behavior of small
entities. For example, a strongly localized wave package is
a superposition of many waves with varying wavelength
and momentum. Ayene et al. [20] observed four categories
of depictions of the Heisenberg uncertainty principle:
(i) Uncertainty is erroneously described as a measurement
error due to external effects, (ii) uncertainty is wrongly
described as a measurement error due to error of the
instrument, (iii) uncertainty is falsely thought to be caused
by measurement disturbance, and (iv) uncertainty is cor-
rectly seen as an intrinsic property of quantum systems.
Only a small number of students had views that fell within
the fourth, correct, category. Müller and Wiesner [5] and
Singh [32] also observed that secondary and undergraduate
students attributed uncertainty to external effects. They
reported that some students stated that uncertainty is caused
by the high velocity of quantum particles.
4. Photoelectric effect
The photoelectric effect is the phenomenon by which
materials can emit electrons when irradiated by light of
sufficiently high frequency. This effect is used to show the
particlelike behavior of light. This particlelike behavior
emerges from the observation that the energy of the emitted
electron depends solely on the frequency of the incident
light, whereas the intensity of the light determines only the
number of emitted electrons. For this subject Asikainen
and Hirvonen [19] observed that some students confused
the photoelectric effect with ionization. Their research
also showed that certain students had difficulty with fully
understanding how light and electrons interact, and how
various aspects (work function, kinetic energy, cutoff
frequency, and material properties) together constitute
the photoelectric effect. McKagan et al. [27] observed that
some undergraduate students could not distinguish between
intensity and frequency of light, were unable to explain
why photons are related to the photoelectric effect, falsely
believed that an increase of light intensity will increase
the energy transferred to a single electron, or incorrectly
believed that a voltage isneeded for the photoelectric effect.
This last incorrect believe was also observed with secon-
dary school students by Sokolowski [33]. Özcan [30]
observed that undergraduate students’different models
of light influenced their understanding of the photoelectric
effect. Students who used the wave model falsely described
the energy transfer in terms of vibrations, which were
caused by wave fronts striking the metal. These students
believed an increase in light intensity would lead to an
increase in the number of wave fronts. Oh [28] observed
that some undergraduate students wrongly thought that
light reacts chemically with an electron, and others falsely
believed that the intensity of light could influence if
electrons were ejected or not.
B. Wave functions
In this section the observed misconceptions concerning
wave functions, potential wells, tunneling, and probability
found in the selected articles [35–44] are presented.
Articles matching our search criteria, which addressed
the understanding of wave functions, described difficulties
of undergraduate students only.
1. Wave functions and potential wells
Wave functions represent the state of particles. The wave
function ψis not a physical wave, but a mathematical
construct, which, for a bound electron, is specified by four
quantum numbers, n,l,mand s.ψcontains all information
of a system and predicts how particles will behave given a
specific potential. jψj2can be interpreted as the probability
density. Similar to wave-particle duality, students often
describe the wave function as a sinusoidal particle path
[41]. Table III presents reported misconceptions, divided
into the two categories observed by Singh et al. [42] and
Singh [43]: (1) misunderstanding due to overgeneraliza-
tions of prior concepts, and (2) difficulty distinguishing
between closely related concepts [40–43], which results in
a mix up of energy, wave functions, and probability. The
first category corresponds with the work by Brooks and
Etkina [36], who concluded classical metaphors cause
misconceptions and promote misplaced classical thinking.
This over-literal interpretation of classical metaphors
was also observed by McKagan et al. [38]. These authors
noticed that many students were likely to have difficulties
in understanding the meaning of potential well graphs,
and saw potential wells as external objects. McKagan et al.
also observed that students mixed up wave functions and
energy levels. Domert et al. [40] ascribed this to the use of
diagrams combining energy levels and wave functions as
illustrated in Fig. 1. However, McKagan et al. showed that
K. KRIJTENBURG-LEWERISSA et al. PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
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eliminating these diagrams does not automatically prevent
misconceptions.
2. Tunneling and probability
Wave functions are not limited to classically permitted
regions, they can extend past classical boundaries. This
effect causes particles to have a probability of existing at
positions that are classically impossible. An important
result thereof is the phenomenon called tunneling; a small
particle can end up on the other side of a classically
impenetrable barrier. In this phenomenon no energy is lost
and no work is done. In understanding of tunneling, the
false belief that energy is lost during the process is
prominent [37,38,44]. McKagan et al. [38] reported that
students falsely attributed this energy loss to (1) work done
on or by the particle inside the barrier; or to (2) the decrease
of wave function amplitude. The same research also
showed other misconceptions caused by a mix-up of
physical quantities. Several students confused the wave
function and energy. For example, some students erro-
neously believed that a decrease in amplitude causes an
increase in energy, or the energy was partly reflected by the
barrier. McKagan et al. also observed difficulty in under-
standing plane waves, which led to a mix-up of ensemble
and single particle description. Domert et al. [40] observed
that some students believed that only the tops of the waves,
which supposedly were higher than the barrier, could pass
the barrier. They also stated that misunderstanding of
probability is an obstacle to the appropriate understanding
of scattering and tunneling. They reported that many
students had difficulty distinguishing between energy
and probability, which they attributed in part to diagrams
which mix wave functions and energy levels (see Fig. 1).
Bao and Redish [35] and Wittmann et al. [39] observed that
students can have difficulty with the predictability and
stochastic nature of probability. Students falsely believed
that the preceding distribution of outcomes influenced the
subsequent outcome of single events, and tended to use
classical arguments in their reasoning. This tendency was
attributed to the lack of experience students have with
probabilistic interpretations in physical systems.
C. Atoms
The following section describes students learning diffi-
culties related to the understanding of atomic structure,
quantization, and spin, as found in the reviewed articles
[12,24,25,31,45–56].
1. Atomic structure and models
The quantum atomic model describes the probability
of observing the electron at a certain position, but it
does not describe a temporal trajectory of an electron
inside the atom. Research shows that secondary and
undergraduate students hold on to various atom models
[12,24,25,31,45–55] and can develop hybrid models
consisting of combinations of different models [45].
Papageorgiou et al. [56] reported that the use of these
models is influenced by the context of the task. The context
of the question or previous questions influenced students’
TABLE III. Misconceptions about wave functions and potentials, categorized into two categories.
Overgeneralization of prior concepts Mix-up of related concepts
Wave functions and
potentials
Wave functions describe a trajectory [35,41] Change in amplitude causes change
in energy [38]
Potential wells are objects [36,37] The amplitude or equilibrium
of the wave function
is mixed up with energy [38]
Height in potential graphs means
position [35]
There is difficulty to distinguish between energy
and probability [40]
Tunneling and probability The amplitude of wave functions
is a measure of energy [36,38,41]
Only the tops of the waves, which overtop the
barrier, will pass [38,40]
Probability is described with classical
arguments (e.g., velocity) [35,40]
Part of the energy is reflected at a barrier
during tunneling [38,40]
Energy or effort is needed to tunnel
through a barrier [37,38,44]
A single particle is described as an ensemble
of particles [38,39]
FIG. 1. A typical diagram as found in many textbooks,
which simultaneously shows wave functions and energy
levels.
INSIGHTS INTO TEACHING QUANTUM …PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
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descriptions, which was also observed by McKagan et al.
[48]. Based on a questionnaire administered to 140 under-
graduate students, Ke et al. [46] divided the different
atomic models into three different stages: (1) An early,
planetary, quantum model, in which the electron orbits in a
circle of constant radius, (2) a transitional model, in which
the electron moves along a sinusoidal path, and (3) a
probabilistic model, in which the position of the electron is
uncertain. These stages are similar to the categories Ireson
[24] observed. Additionally, Dangur et al. [54] divided the
probabilistic model into a visual conceptual model based
on probability distributions, and a mathematical model, in
which students understand that the state of a particle can
be described by a specific mathematical model. Although
researchers used different classifications, one difficulty
emerged in the majority of articles: Secondary and lower
undergraduate students have difficulty letting go of Bohr’s
planetary atomic model [12,25,45–51,53,55]. Kalkanis
et al. [12] ascribed this to many students believing that
scientific content they learned previously is scientifically
correct. This is in agreement with Stefani and Tsaparlis
[50], who observed that models are sometimes seen as
replicas of reality. Ke et al. [46] and Wang and Barrow [53]
reported that more experienced students understood the
difference between various models and could switch
between them. McKagan et al. [48] claimed the solution
is in comparing and contrasting different models, but also
reported that students had difficulty understanding the
reasons for the development of new atom models, which
Taber [47] also reported in his research related to energy
levels.
2. Energy levels, quantization and spin
To explain atomic spectra, current atomic models include
energy levels. These energy levels cannot be arbitrary, but
they have certain, specified values. These quantized energy
levels can only be explained by considering them as bound
wave functions and taking into account boundary con-
ditions. Taber [47] observed that several secondary students
did not understand the necessity of introducing quantiza-
tion, because they did not see the planetary model as
insufficient. Some students also had difficulty in forming
an adequate concept of orbitals and confused orbitals with
planetary orbits or concentric shells. Didişet al. [55]
reported that some undergraduate students did not under-
stand that energy quantization is a natural phenomenon that
occurs only when boundary conditions apply.
The distribution of electrons over the available energy
levels in a system depends partly on electron spin. Spin is
an intrinsic property of small particles and is a form of
quantum angular momentum. But, in contrast to its
classical counterpart, it is not a factual rotation. With
regard to spin, Zhu and Singh [57], Taber [47], and
Özcan [52] observed that many students falsely believed
that quantum spin is an objects’rotation around its axis or
around the core. Özcan indicated that there seemed to be a
relation between the understanding of atomic models and
spin. Those students who believed that quantum spin is
an actual movement often used the classical atomic model.
For students who described spin correctly, the use of the
quantum atomic model was more dominant.
D. Complex quantum behavior
The concepts discussed in the previous sections all are
reductions from the fundamental principles of quantum
mechanics. A wave function is a solution of the
Schrödinger equation and represents a certain quantum
state, which can be described by a set of quantum numbers.
Little research has been done into misconceptions regard-
ing these more complex subjects, such as quantum states,
superposition and time evolution, for the secondary school
level. Michelini et al. [58] developed and evaluated
materials on quantum states and superposition, and con-
cluded that secondary students’difficulties in accepting
nondeterminism often cause a fall back to classical reason-
ing, and are an obstacle to understanding quantum states.
Passante et al. [59] also researched understanding of
quantum states and observed that undergraduate students
find it hard to distinguish between pure superposition and
mixed states. They also researched student understanding
of time dependence, mainly focusing on upper division
undergraduate level students [60]. One observation that
could be useful for secondary and lower undergraduate
education was that many students believed that for a
time-dependent wave function, the probability of finding
a particle in a region must also be time dependent.
Regarding time dependence, Zhu and Singh [43,61]
observed some students who falsely believed that after
measurement the wave function will remain the same or,
after collapsing, will eventually go back to its initial state.
IV. RESEARCH TOOLS
This section answers the second subquestion: “What
instruments have been designed and evaluated to probe
student understanding on a conceptual level?”and presents
an analysis of the questionnaires and instruments intended
for secondary and lower undergraduate education that were
observed in the 74 reviewed articles. The research tools are
analyzed on how they are designed and evaluated, and on
the topics which they cover. Table IV presents a summary
of this analysis.
A. Multiple-choice concept tests
Several concept tests have been designed and used to
uncover students’difficulties, but a substantial part was
only aimed at the upper undergraduate level and empha-
sized mathematical formalism [43,69–71]; other tests were
not sufficiently evaluated [72]. The selected literature
included three evaluated multiple choice questionnaires
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TABLE IV. Overview of research tools appropriate for probing conceptual understanding of secondary and lower undergraduate level students.
Researchers Year Research tool Level Country Content Design and evaluation
Cataloglu and
Robinet [2]
2002 QMVI Undergraduate
students
US Wave functions, potential wells,
quantization
Content based on existing materials and
commonly used text books. Modified after
student and faculty feedback and item
analysis. Results suggested QMVI scores
may be a reasonable measure of student
understanding
Ireson [24,25] 1999 Multivariate
analysis
Undergraduate
students
UK Wave-particle duality, atomic
structure, quantization
Items based on previous research on students
conceptions [62,63]. Multivariate analysis
resulted in a holistic picture. Findings
were consistent with other research, using
different methodology.
McKagan et al.
[16]
2010 QMCS Undergraduate
students
US Wave-particle duality, wave
functions, potential wells,
atomic structure,
quantization, measurement
Content based on literature, faculty interviews,
textbook reviews and student observations.
Modified after interviews, surveys and
discussions. QMCS is too small to adequately
probe student understanding. Useful as
pretest and post-test for undergraduate
students, but not for graduate students.
Sen [31] 2002 Concept map
strategy
Undergraduate
students
Turkey Wave-particle duality, atomic
structure
Strategy based on Ausubel’s theory on cognitive
and meaningful learning [64,65]. Reliability
and validity were analyzed using
Crohnbach’sαand factor analysis.
Results were consistent with another,
questionnaire-based, study.
Taber [47] 2005 Typology of
learning
impediments
Upper secondary
students
UK Atomic structure Typology based on consideration of the
influence of prior knowledge [66].
Proposed modification: include substantive
learning impediments categorized as
analogical, epistemological, linguistic,
pedagogical, or ontological.
Tsaparlis and
Papaphotis [51]
2009 Questionnaire Upper secondary
students
Greece Atomic structure Content based on questions in an earlier
study [67], which were judged for
content validity by chemistry teachers.
Wuttiprom et al.
[68]
2009 QPCS Undergraduate
students
Australia Wave-particle duality Content based on expert opinions and students
difficulties Modified after trials with students
and experts. Reliability was analyzed
with item analysis, KR21 reliability test,
and Ferguson’s delta.
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[2,16,68] suitable for secondary and lower undergraduate
level students, which will be described in this section.
1. Quantum Mechanics Visualization Inventory
Cataloglu and Robinett [2] designed the Quantum
Mechanics Visualization Inventory (QMVI), based on
existing materials and commonly used text books.
Alterations to the preliminary inventory were made based
on student feedback, comments from faculty colleagues
and an item analysis. The QMVI consists of 25 questions
and focuses on the interpretation of various diagrams.
Although many of the questions require mathematical
reasoning, approximately one-third of the questions
address conceptual understanding of the influence of the
potential energy on probability and the wave function.
These questions can provide useful information on the
student difficulties discussed in Sec. III B. The test was
validated for content by content experts and Ph.D. candi-
dates and analyzed for reliability and item difficulty in
two pilot studies. The test was found to be reliable, but
slightly difficult (α¼0.83, mean item difficulty ¼0.45).
Afterwards, the QMVI was administered to students
ranging from the sophomore level to the graduate level.
Analysis showed there was a large correlation between the
students’confidence in, and correctness of, their answers.
Analysis also showed differences in understanding for
the three different levels of instruction, which matched
expectations. No articles were published on the evaluation
of the QMVI at the secondary school level.
2. Quantum Mechanics Conceptual Survey
The Quantum Mechanics Conceptual Survey (QMCS)
was designed to elicit student difficulties on topics covered
in most courses on quantum mechanics [16]. For the
preliminary version, textbooks were reviewed, students
were observed, and faculty interviews were held to determine
the topics. This preliminary version addressed wave func-
tions, probability, wave-particle duality, the Schrödinger
equation, quantization of states, the uncertainty principle,
superposition, operators and observables, tunneling, and
measurement. Over a period of three years this 25-item
survey was altered, surveys were analyzed, and interviews
were held with students. Finally, 12 questions proved to be
useful for detecting student difficulties. The final question-
naire addresses the conceptual understanding of a broad
range of topics discussed in Sec. III, i.e., wave-particle
duality, wave functions, potential wells, atom structure, and
quantization. Because of the small number of questions,
however, the QMCS is not appropriate for proper statistical
analysis and researchers suggested that more questions
should be developed. The QMCS was tested at different
levels, and the researchers concluded that the QMCS is a
useful post-test for the upper undergraduate level.
Preliminary results indicated it could also be suitable to
investigate learning gains of lower undergraduate level
students, but this needs to be verified in future research.
3. Quantum Physics Conceptual Survey
Wuttiprom et al. [68] developed the Quantum Physics
Conceptual Survey (QPCS) to test student understanding
of basic concepts of quantum mechanics. The researchers
studied syllabi and consulted experts in order to determine
topics and create survey questions. The QPCS addresses
conceptual understanding of the photoelectric effect, wave-
particle duality, the de Broglie wavelength, double slit
interference, and the uncertainty principle, of which student
difficulties were discussed in Sec. III A. The questions were
trialed with different groups of students and each version
of the survey was critiqued by a group of discipline or
teaching experts to establish validity. Subsequently, the
final survey, consisting of 25 items, was administered to
312 lower undergraduate students at the University of
Sydney. The results were statistically analyzed for item
difficulty, discrimination of single items, discrimination of
the entire test and the consistency among the questions.
Analysis showed that two items were likely to be too
difficult and three items too easy (item difficulty
index >0.9or <0.3), five items also turned out to be poor
discriminators (item point biserial coefficient <0.2). Still,
the KR-21 reliability index and Ferguson’s delta were
found to be satisfactory (KR21 ¼0.97,δ¼0.97). The
researchers concluded that even though several items
needed improvement, these results indicated that the
QPCS is a reliable survey.
B. Other tools
Besides multiple choice concept tests, there are other
strategies to investigate students’difficulties. The reviewed
literature included four other evaluated research tools,
which emphasize students’reasoning, mental models, and
underlying causes of misunderstanding [24,25,31,47,51].
1. Multivariate analysis
Ireson [24,25] designed a 40-item Likert-scale question-
naire, of which 29 items tested conceptual understanding
of wave-particle duality, atom structure, and quantization.
This questionnaire was administered to 338 lower
undergraduate students. The analysis was based on the
assumption that understanding can be represented by
clustering the conceptions of a group of students. First,
the responses were subjected to cluster analysis, which
clusters individuals and gives insight into understanding
at the group level. This resulted in three clusters, which
were labeled quantum thinking, intermediate thinking, and
mechanistic thinking. Second, Ireson used multidimen-
sional scaling, which was used to map the response in
multiple dimensions. This resulted in a two-dimensional
model, of which the dimensions represented students’dual
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and nondeterministic thinking. This two-dimensional
model confirmed the existence of three clusters; Ireson
concluded that this method can be used to gain insight in
students thinking and clusters or dimensions in their
understanding.
2. Concept map strategy
Sen [31] used a concept map strategy to evaluate the
learning process, diagnose learning difficulties, and map
the progression of students’cognitive structure. Training
in creating concept maps was provided to 88 undergradu-
ate students, from three different educational levels. At
the end of the semester, the students each individually
constructed a paper and pencil concept map. The concept
map had to contain three main concepts (the atom,
electron, and photon) and students were instructed to
pay attention to the hierarchical order and links among
concepts. Sen scored the concept maps for the number of
valid concepts, relationships, branching, hierarchies, and
cross-links. The scoring of the concept maps was tested
for reliability, Cronbach’sαwas 0.67. Additionally, the
scoring scheme was analyzed for construct validity by
factor analysis. This analysis showed that the five scoring
categories were correlated to separate single factors. The
researcher also observed that the concept maps resembled
results from a questionnaire-based study on the same
subject. Results showed significant differences in the
number of concepts and branches for the three different
educational levels. Sen concluded that the results suggest
that concept mapping can be used to investigate cognitive
structures and the development thereof. However,
the interpretation of the scores needs to be evaluated
empirically [73].
3. Typology of learning impediments
Taber [47] constructed and evaluated a typology of
learning impediments, which he used to analyze underlying
causes for students’difficulties. The typology was based on
the Ausubelian idea that, for meaningful learning, students
need to relate new concepts to prior knowledge. Four types
of learning impediments were defined: (1) Students lack
prerequisite knowledge; (2) students fail to make required
connections; (3) students interpret the material inappropri-
ately, because of their intuitive ideas; and (4) students
interpret the material inappropriately, because of their
cognitive structures. Taber used this typology to analyze
data from an interview-based study on the understanding
of chemical bonding of pre-university students. The
researcher identified all four types of learning impediments
and concluded that the typology is a useful heuristic tool,
which can be used to interpret data on student learning.
Still, Taber also recommended a refinement that takes into
account misconceptions based on analogies or epistemo-
logical assumptions.
4. Questionnaire on atomic structure
Tsaparlis and Papaphotis [51] designed a questionnaire
for a study into the deep understanding and critical thinking
of first-year undergraduates with regard to the quantum
atom model. The questionnaire was based on a preliminary
questionnaire that had been validated for content by
chemistry teachers in a previous study [67]. It consisted
of 14 open-ended questions; 9 of them were designed to test
conceptual understanding, and the other questions were
aimed at algorithmic knowledge. The questionnaire was
administered to 125 students as part of a qualitative study.
The researchers only drew conclusions about student
understanding, the questionnaire itself was not evaluated.
V. TEACHING STRATEGIES
This section addresses the subquestion: “What teaching
strategies aimed at the secondary and lower undergraduate
level have been tested, implemented and evaluated
for their influence on student understanding?”and presents
approaches promoting the understanding of quantum
mechanical concepts that have been investigated in the
selected literature. The following section presents the
teaching strategies found in the selected articles, divided
into instructional and multimedia-based strategies. There
are several other activities described in literature, e.g., the
hands-on activities from Visual Quantum Mechanics [74],
the Dutch approach using the particle in a box [8], and the
approach starting with qubits [75], but this review only
discusses strategies which were implemented and evaluated
in an educational setting.
A. Instructional strategies
There are still many questions concerning the teaching of
introductory quantum mechanics. The introduction using
wave-particle duality, for example, is still under discussion.
Several alternative ways to introduce quantum mechanics
have been used [58,76,77], but these alternatives have not
been properly evaluated and compared to the use of wave-
particle duality. However, several articles did describe
investigations into the influence of teaching methods on
student understanding. This section describes implemented
and evaluated instructional strategies that were found
within the selected literature [12,22,36,48,49,54,76,78–89],
organized into four groups.
1. Focus on interpretation
Because of quantum mechanics’indeterminacy, many
interpretations are possible. Today’s quantum experts
do not support one single interpretation, although the
Copenhagen interpretation is often considered to be the
standard interpretation [90]. Baily and Finkelstein [78,79]
researched the influence of addressing interpretations of
quantum mechanics on student interpretations. Results
showed that undergraduate students tended to prefer a
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local and deterministic interpretation if there was no
emphasis on ontology. Baily and Finkelstein also presented
results of the implementation of a new curriculum [76],
which addressed the topic of “physical interpretation”
explicitly. This curriculum included in-class discussions
and experimental evidence, and aimed for understanding
of different perspectives, their advantages, and limitations.
Results of the use of this curriculum showed a clear change
in student interpretation and the researchers concluded this
confirms the importance of emphasis on interpretation.
Greca and Freire [22] also researched the influence of
teaching on undergraduate students’interpretations. For
this purpose an interpretation was chosen that suited their
didactic strategy, which emphasized a phenomenological-
conceptual approach. The researchers used a realistic
interpretation of the Copenhagen interpretation, in which
the probability density function does not predict the
probability of finding a particle, but the probability of
the particle being present at a certain position. Comparison
with a control group showed that in the experimental
groups more students developed reasonable understanding.
These examples showed the importance of an emphasis on
interpretation in the design of new curricula.
2. Focus on models
Research showed that students tend to hold on to Bohr’s
planetary description of the atom [45,46,51,53], because
it corresponds to students’classical worldview. Several
approaches were evaluated to address this problem.
Kalkanis et al. [12] presented an approach that emphasized
the differences between classical and quantum mechanics.
An instructional module focusing on the hydrogen atom
was developed, which contrasted the classical and quantum
models, and used the Heisenberg uncertainty relation as the
basic principle. The module was taught to 98 preservice
teachers and evaluated with pretests and post-tests and
semistructured interviews. Results showed that a vast
majority described the hydrogen atom correctly and could
appropriately apply Heisenberg’s uncertainty principle.
The students had also become more aware of the process
of learning and showed a change in worldview.
Strategies based on the historical development of the
atomic model were evaluated by Unver and Arabacioglu
[88] and McKagan et al. [48]. Unver and Arabacioglu
developed a teaching module focusing on observations
and experiments that led to alterations of the atomic model.
The module was implemented in a course for preservice
teachers (N¼73). Pretests and post-test comparisons
showed a significant change in understanding. McKagan
et al. designed an undergraduate course focusing on model
building and reasoning for each model. Results showed that
emphasis on the analysis of the predictions of each model,
and the explanation of reasoning behind the development
of the model, resulted in an increase in the use of the
Schrödinger model.
Classical analogies are also used to promote under-
standing of the quantum atom model. Budde et al. [80]
developed the Bremen teaching approach for upper sec-
ondary schools, which is based on similarities between the
quantum atom model and liquids. Nine students were
taught that atoms consist of electronium, a liquid substance,
to promote the idea that an atom has a continuous nature, in
which electrons are not moving. Budde et al. observed that
some students described electronium as having a particle
nature, but students still developed the conception that
electrons are not moving. The researchers concluded that its
focus on plausible aspects lead to high acceptance of the
electronium model.
3. Focus on mathematical or conceptual understanding
Lower undergraduate and secondary students do not
have extensive mathematical skills, which are an impor-
tant part of quantum physics. This raises the question to
what extent mathematical skills are needed for good
understanding of quantum concepts. Studies have been
done into the relation between mathematical and con-
ceptual understanding of quantum concepts. Koopman
et al. [84] observed that undergraduate students in a
Quantum Chemistry course lacked mathematical skills,
and they designed a remedial program. This program
consisted of a diagnostic test, a prelecture, and online
mathematics assignments. Students’results were moni-
tored and commented upon. Students could consult a tutor
and, if needed, additional explanation was scheduled.
Koopman et al. observed a positive correlation between
students’scores on the math assignments and the final
exams (N¼29). From a comparison with student’s
grades for calculus, the researchers concluded that math-
ematical skills are necessary, but not sufficient for con-
ceptual understanding. Papaphotis and Tsaparlis [49,86]
researched the relation between algorithmic and concep-
tual understanding in high school chemistry. The study
was conducted on 125 science students at the start of their
first year at university. Students completed a questionnaire
that addressed procedural knowledge and conceptual
understanding. No correlation was found between their
levels of procedural and conceptual performance. To
investigate the effect of a nonmathematical approach on
student understanding of the atomic structure, Dangur,
Avargil, Peskin, and Dori [54,82] developed a teaching
module focusing on real-life applications and visualiza-
tion. This module was used for 122 secondary students
and 65 undergraduate students. Results showed a signifi-
cant improvement of understanding for both secondary
and undergraduate students. Comparison with mathemati-
cally oriented undergraduates showed that the under-
graduate test-group scored significantly higher on
textual and visual understanding. This research suggests
a conceptual, nonmathematical approach for teaching
quantum mechanics can lead to adequate understanding.
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4. Use of activities
Active learning has become increasingly important in
research into student engagement and understanding [91].As
a consequence, several reviewed articles described inves-
tigations into the influence of student activities on conceptual
understanding. One example of active learning is the use of
peer interaction. Shi [87] researched the influence of peer
interaction on student understanding of duality and atomic
models. Peer interaction was used once or twice a week
during an undergraduate course on quantum mechanics.
Students in the experimental group scored significantly
higher than the control group on the post-test. Deslauriers
and Wieman [81] investigated the effect of two different
teaching methods on students’learning. One group (N¼57)
was taught traditionally, while the other (N¼67) experi-
enced interactive engagement methods (quizzes, simula-
tions, clicker questions). The QMCS was used to test
understanding, and comparison of the results for the two
groups showed that the use of interactive engagement
methods resulted in significantly higher scores. Yildiz and
Büyükkasap [89] researched the influence of writing on
understanding of the photoelectric effect. Pre-service teach-
ers (N¼36) had to write a letter to senior high school
students in which they explained the photoelectric effect.
Results showed that these students scored significantly better
on the post-test and exams than the control group. Gunel [83]
explored differences in learning gains for two different
writing tasks on Bohr’s atomic model and the photoelectric
effect (N¼132). The study indicated that secondary stu-
dents who created a PowerPoint presentation had signifi-
cantly higher learning gains than those who completed a
summary report. Muller et al. [85] explored how well
undergraduate students (N¼40) could learn from watching
a video of a student-tutor dialogue on quantum tunneling.
Results were compared to students who watched a traditional
explanation. The students who watched the dialogue per-
formed significantly better on the post-test. These results
all suggest that active learning can contribute to better
understanding of quantum concepts.
B. Multimedia
Numerous multimedia applications have been designed
for teaching quantum mechanics, but not all have
been thoroughly evaluated. An overview of useful multi-
media for quantum mechanics education was provided
by Mason et al. [92]. The following section discusses
evaluated multimedia found in the reviewed articles
[5,27,32,33,38,57,58,77,93–100]. First PhET, QuILT, and
QuVis are treated, which are databases covering a large
number of topics. Then other separate simulations and
teaching sequences using simulations will be discussed.
1. PhET
McKagan et al. [98] described 18 simulations on funda-
mental principles, historical experiments, or applications of
quantum mechanics developed in the PhET (Physics
Education Technology) project. Most of them were devel-
oped for use in an undergraduate level course. These
simulations were developed based on previous research,
student interviews, and classroom testing. The interviews
and classroom testing mainly focused on finding problems
in the simulations, but some results of interviews and
exams showed that several simulations (“Davisson-
Germer: Electron Diffraction”and “Photoelectric Effect”)
resulted in better understanding. The researchers also noted
that student interviews on the simulation “Quantum
Tunneling and Wave Packets”suggested that guided
activities could improve students’learning path when using
the simulations. However, more research could still be done
into the learning gains seen with the use of these simu-
lations. The simulations on the photoelectric effect and
tunneling were described more extensively. The simulation
“Photoelectric Effect”was used for curriculum improve-
ment [27]. This curriculum, based on active engagement
techniques, resulted in better understanding of the photo-
electric effect. However, students had difficulty linking this
experiment to the particle behavior of light. The simulation
“Quantum Tunneling and Wave Packets”was also part of
an improved curriculum [38] that led to greater insight into
students’difficulties on tunneling.
2. QuILTs
Singh [32] described the development of QuILT’s
(Quantum Interactive Learning Tutorials) covering a broad
range of subtopics. These tutorials, which were developed
for undergraduate courses, consist of a combination of tasks,
homework, Java applets, and pretests and post-tests. QuILTs
were designed based on knowledge of student difficulties,
and evaluated using pretests, post-tests, and student inter-
views. The multimedia applications used in the QuILT’s
were adapted from different sources (e.g., PhET [98] and
Physlets [101]). Results of the pre-experimental evaluation
of QuILTs on time development, the uncertainty principle,
and the Mach-Zehnder interferometer showed a substantial
change in performance. Zhu and Singh also evaluated a
QuILT regarding the Stern-Gerlach experiment [57] and
quantum measurement [100]. Both resulted in distinct
improvement of understanding. Comparison of the results
for students who went through the tutorial on quantum
measurement with those for a control group showed that the
QuILT resulted in better scores on the post-test.
3. QuVis
Kohnle et al. [96,97] reported on the development of
QuVis, which is a collection of interactive animations and
visualizations for undergraduate students. Student inter-
views and observation sessions were used to optimize the
interface design. Subsequently, the researchers investigated
the influence of two simulations (the potential step and the
finite well) on student understanding in a quasiexperimen-
tal setting. Two groups of students completed a diagnostic
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test: an experimental group, which worked with the
animations, and a control group. Statistical analysis of
the test results showed a significant relation between having
worked with the simulations and performance on questions
covering the corresponding subjects. In more recent work,
Kohnle et al. [95] presented simulations regarding two-
level quantum systems. They evaluated the learning gains
resulting from use of a simulation on superposition states
and mixed states. Results showed a substantial change in
understanding.
4. Simulations on atomic structure
Several simulations were designed to improve under-
standing of the atomic structure. Chen et al. [93] inves-
tigated the different effect of static and dynamic
representations on understanding of atomic orbitals. The
researchers compared two groups of secondary students.
One group completed a learning activity using static 3D
representations, while the second group worked with a
dynamic 3D representation. Analysis of a pretest and post-
test showed that both representations increased conceptual
understanding. However, the researchers concluded that
students who worked with the dynamic representations had
more sophisticated mental models of the atom. Ochterski
[99] used research-quality software (GaussView) and
designed and evaluated two activities (N¼95,N¼71)
to introduce orbitals and molecular shape to high school
students. Pretests and post-tests for both activities showed
an increase in understanding; Ochterski concluded that
research-quality software can be effective, even if students
have little background in chemistry.
5. Teaching sequences using simulations
Other simulations were evaluated within the context of the
design of a course. Malgieri et al. [77] described a teaching
sequence using the Feynman sum over paths method. This
sequence used simulations in GeoGebra, which included the
photoelectric effect and the double-slit experiment. The eight-
hour course was tested on preservice teachers (N¼12)and
evaluated with a pretest and post-test. Results showed a good
level of understanding of the role of measurement and the
single photon interpretation of the double-slit experiment.
However, the understanding of the uncertainty principle was
still not adequate. Müller and Wiesner [5] designed and
implemented a secondary school course using virtual experi-
ments with the Mach-Zehnder interferometer and the double
slit. Interviews and a questionnaire showed that students
(N¼523) who took part in the course developed better
quantum understanding than the control group. Michelini
et al. [58] proposed a secondary school teaching sequence
using prevision experiment comparison (PEC) strategies.
This sequence included simulations on light interaction with
Polaroids and Malus law. Analysis of student worksheets
(N¼300) and a group discussion (N¼17)showedthatthe
approach stimulated learning for atleast 75% of the students.
The researchers concluded that software simulations can help
students in building a phenomenological framework, but are
not sufficient.
6. Quantum computer games
A different way of using multimedia is the use of
quantum computer games. Gordon and Gordon [94]
developed the computer game “Schrödinger cats and
hounds”to teach quantum mechanical concepts in a fun
way. Game-aided lectures were given to 95 undergraduate
students. Analysis of a pretest and post-test showed an
increase in understanding.
VI. CONCLUSIONS
In this paper we presented an overview of existing
knowledge on student difficulties, research tools for
investigation of conceptual understanding, and teaching
strategies. The conclusions of this literature review will be
presented in this section.
A. Student difficulties
Analysis of the selected articles shows that secondary
and undergraduate students have many difficulties when
they learn quantum mechanics. Much research has been
done into misunderstanding of wave-particle duality, wave
functions, and atoms. However, not much research has
been done into student difficulties with complex quantum
behavior, and no research was found concerning secondary
students’understanding of the wave function. Research
into the understanding of wave-particle duality showed that
undergraduate students’understanding can be clustered
according to the extent of classical thinking [20,22,24–26].
Researchers also observed misplaced classical thinking
in understanding of the wave function; several students
displayed an over-literal interpretation of classical meta-
phors [36,38], or used classical reasoning in describing the
process of tunneling [38,44]. Research into students’
understanding of the quantum atomic model also indicated
that both secondary and undergraduate students hold on
to previously learned, semiclassical models [12,25,45–51,
53,55]. From these results we can conclude that many
difficulties that students experience are related to the
inability to connect quantum behavior to the physical
reality as they see it, which results in a mix-up of classical
and quantum concepts. Although this has been researched
mainly for the undergraduate level, the existing research
shows similarities in secondary and undergraduate stu-
dents’understanding of duality and atomic models. This
suggests that the mix up of classical and quantum concepts
is also an important issue at the secondary level.
Researchers have proposed several ideas concerning sol-
utions for the mix up of classical and quantum concepts;
e.g., analogies should be well defined [36], diagrams
should be unambiguous [38,40], and students should have
more knowledge of the use of models in physics [12,48,88].
However, the impact of these proposed solutions remains
to be investigated.
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B. Research tools
The research tools discussed in Sec. IV all include
conceptual questions that could be useful probing the
understanding of secondary and lower undergraduate
level students. The topics addressed in these tools are
wave-particle duality, wave functions, quantization, atomic
structure, and measurement. Table Vgives an overview of
the topics covered by each research tool. As can be seen,
none of the instruments covers the complete spectrum of
quantum mechanics. Furthermore, only the research tools
from Ireson, Taber, and Tsaparlis regarding duality and
atomic structure, are used in secondary school settings. The
QMVI addresses conceptual understanding only in part,
and therefore some questions can be appropriate for the
secondary and lower undergraduate level. The QMCS,
which covers most of the topics, aims to probe conceptual
understanding, but has not been thoroughly evaluated for
secondary and lower undergraduate education. Moreover,
the QMCS includes too few questions for statistical
analysis. These results imply that the development and
evaluation of more questions is needed, not only to cover all
major topics from quantum mechanics, but also to make
statistical analysis possible.
C. Teaching strategies
Various methods and approaches have been designed
and used to promote understanding in introductory courses
on quantum mechanics, at both the secondary and under-
graduate level. Still, only a small selection of these methods
has been evaluated for their impact on students’under-
standing. These evaluations show the following:
(1) emphasis on interpretations influences undergradu-
ate student perspectives, and should be taken into
account in the development of curricula and teaching
sequences;
(2) emphasis on the development of and the differences
between various atomic models can result in better
understanding of undergraduate students;
(3) a nonmathematical, conceptual approach can lead to
adequate understanding for secondary and under-
graduate students;
(4) active learning contributes to the understanding of
quantum mechanical concepts.
However, there is a need for more empirical research into
the teaching of quantum mechanics and teaching strategies
should be researched for both secondary and undergraduate
education.
Furthermore, many multimedia applications have been
designed for teaching quantum mechanics. Table VI shows
that for undergraduate education all quantum topics are
covered by the multimedia applications found in the
reviewed articles. For secondary education there are fewer
applications and most topics are covered. Most of the
applications were evaluated for practical use; only some of
the simulations were also evaluated for their influence on
student understanding. Singh and Zhu [32,57,100] have
made a start with the design and evaluation of tutorials
using multimedia, but more research into how these
applications can be used to promote understanding is
needed.
D. Implications for researchers
This paper shows the current state of research into
learning difficulties and teaching strategies for quantum
physics at the secondary and lower university level.
Analysis of 74 articles showed there are many groups
researching student understanding, teaching strategies or
assessment methods, mostly aiming at undergraduate
education.
1. Lower undergraduate level
For lower undergraduate students, several learning dif-
ficulties were observed in the selected articles, but little
research has been done into the conceptual understanding
of complex quantum behavior. Although these topics are
TABLE V. Topics covered by the research tools.
QMVI QMCS QPCS SenaIreson Taber Tsaparlis
Lower undergraduate education (•) Secondary education (▪)
Wave-particle duality Photons and electrons ••••
/▪▪
Double slit experiment •• •
/▪
Uncertainty principle ••• ▪
Photoelectric effect ••
Wave functions Wave functions and potential wells ••
Tunneling ••
Probability •• ▪
Atoms Atomic structure •••
/▪▪ ▪
Energy levels, quantization, and spin •• ••
/▪▪ ▪
Complex QM behavior Quantum states
Superposition
Time evolution and measurement ••
aDependent on individual student responses.
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also difficult for upper-graduate students, it would be good
to investigate to what extent these topics can be taught
conceptually. More research should also be done into the
underlying difficulties and causes of observed student
difficulties. Several assessment methods have been
designed for the undergraduate level, but there is still need
for tests that cover more topics and are suitable for
statistical analysis. More empirical research is needed for
the further development of lower undergraduate level
courses on quantum mechanics, in which teaching strate-
gies are evaluated and compared using proper assessment
tools. This research should also include investigations into
ways to promote students’understanding using multimedia
applications and experiments.
2. Secondary school level
With regard to quantum mechanics at the secondary
school level, more empirical research into teaching strat-
egies is also needed. But, although many learning diffi-
culties that were found in research at the undergraduate
level were confirmed for secondary school students, several
topics have not yet been thoroughly investigated and more
research into learning difficulties is needed. For the
secondary school level, there is a need for more research
into the understanding of wave functions and potential
wells, topics that are part of several secondary school
curricula. Research into the teaching of quantum states at a
conceptual level is also needed, because this is part of some
secondary school curricula.
To thoroughly investigate teaching strategies, multime-
dia applications, and experiments suitable for secondary
school students, research tools are needed. The existing
concept tests primarily focus on the undergraduate level,
and therefore, it remains to be investigated whether these
assessment tools are also applicable at the secondary
school level.
E. Implications for teachers
Analysis of the current research shows that students
have many difficulties while learning quantum mechanics.
Although most of the research has been conducted at the
undergraduate level, overlapping research shows similar
difficulties at both levels addressed in the studies reviewed.
Therefore, both lower undergraduate and secondary school
teachers can benefit from the research discussed here. This
paper shows that there has been little empirical research
into ways to promote understanding, but teachers should be
aware that students tend to hold on to classical thinking,
which leads to the misinterpretation of unfamiliar quantum
concepts, and the mix up of classical and quantum physics.
It can be helpful to emphasize differences and similarities
between quantum concepts and students’preconceptions,
which has proved to be useful in the teaching of the
quantum atomic model at the undergraduate level. Teachers
should also be aware that it is important to specify the
limitations of metaphors, because they can lead to over-
literal interpretations.
ACKNOWLEDGMENTS
This work was funded by The Netherlands
Organization for Scientific Research (NWO) under
Grant No. 023.003.053.
APPENDIX: OVERVIEW OF RESEARCH INTO
STUDENT DIFFICULTIES`
See Table VII.
TABLE VI. Overview of quantum mechanical topics covered by the multimedia applications.
PhET QuILTaQuVis Malgieri Gordon Chen Ochterski Müller Michelini
Lower undergraduate education (•) Secondary education (▪)
Wave-particle
duality
Photons and electrons •• • • • ▪▪
Double slit experiment •• • ▪
Uncertainty principle •• • • ▪▪
Photoelectric effect •• ▪
Wave functions Wave functions and potential wells •• • ▪
Tunneling •• •
Probability •• • • ▪▪
Atoms Atomic structure •• • ▪▪ ▪
Energy levels, quantization, and spin •• • ▪▪ ▪
Complex
quantum
behavior
Quantum states •• • • • ▪▪
Superposition •• • • • ▪▪
Time evolution and measurement •• •
aTutorials using simulations of other sources.
K. KRIJTENBURG-LEWERISSA et al. PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
010109-14
TABLE VII. Details of the selected articles on student difficulties described in Sec. III.
Part Researchers Topic Level Country Methodology and analysis
A Asikainen and
Hirvonen [19]
Photoelectric effect Undergraduate
students and
physics teachers
Finland A case study, using pretest and post-test and semistructured interviews,
was carried out with preservice (N¼8) and in-service (N¼17)
teachers. Test responses were categorized, interviews were used for
validation.
Ayene et al. [20] Wave-particle duality,
uncertainty principle
Undergraduate
students
Ethiopia Semistructured interviews were conducted with undergraduate
students (N¼25). Responses were categorized.
Dutt [21] Wave-particle duality,
double slit experiment,
photoelectric effect,
quantization
Upper secondary
students
Australia Test and worksheet data from grade 12 students were analyzed and
interviews were held with 6 volunteering students.
Greca and Freire
[22]
Wave-particle duality,
uncertainty principle,
probability
distribution,
superposition
Undergraduate
students
Brazil Concept tests and conceptual problems were used (N¼89), field
notes were collected during classes. Responses were categorized
using hierarchical clustering and multidimensional scaling.
Hubber [23] Light Upper secondary
students
Australia Three semistructured interviews conducted and two questionnaires
were administered (N¼6). Responses were categorized.
A=CIreson [24] Wave-particle duality,
atoms
Undergraduate
students
UK A questionnaire was given to the students (N¼338). Responses were
analyzed with cluster analysis and multidimensional scaling.
A=CIreson [25] Wave-particle duality,
atoms
Undergraduate
students
UK A questionnaire was given to the students (N¼338). Responses were
analyzed using cluster analysis and multidimensional scaling.
Johnston et al. [1] Wave-particle duality Undergraduate
students
Australia Students (N¼33) were given two short-response quizzes. Responses
were categorized and analyzed for correctness.
Mannila et al. [26] Wave-particle duality Undergraduate
students
Finland Intermediate level students (N¼29) answered 8 open-ended
questions. Modified concept maps were created for each response,
compared to a “master map”based on experts’conceptions and
categorized.
Masshadi and
Woolnough [4]
Wave-particle duality Upper secondary
students
UK Students (N¼83) were given a semistructured questionnaire.
Responses were categorized.
McKagan et al.
[27]
Photoelectric effect Undergraduate
student
USA After a reformed course, students’responses to two exam questions
were analyzed (N¼465,N¼188).
McKagan et al.
[16]
Wave-particle duality,
double slit experiment
Undergraduate
students
USA Interviews were conducted (N¼46) during the design and evaluation
of the QMCS.
Müller and
Wiesner [5]
Wave-particle duality,
atoms, uncertainty
principle,
non-determinism
Secondary and
undergraduate
students
Germany A questionnaire was administered to secondary students (N¼523)
and interviews were conducted with secondary students (N¼27)
and undergraduates (N¼37). Responses were categorized.
Oh [28] Photoelectric effect Undergraduate
students
South Korea Three groups of students (N¼31,N¼49,N¼49) were given a
pretest and a post-test, which were validated by interviews.
Responses were categorized.
(Table continued)
INSIGHTS INTO TEACHING QUANTUM …PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
010109-15
TABLE VII. (Continued)
Part Researchers Topic Level Country Methodology and analysis
Olsen [29] Wave-particle duality Upper secondary
students
Norway Students from 20 different schools (N¼236) were given a test.
Multiple choice questions were analyzed quantitatively, open-ended
questions were categorized.
Özcan [30] Photoelectric effect,
blackbody radiation,
Compton effect
Undergraduate
students
Turkey Preservice physics teachers (N¼110) were given a questionnaire.
Responses were categorized and analyzed for correctness.
A=CSen [31] Wave-particle duality,
atoms
Undergraduate
students
Turkey Students (N¼88) created a concept map. These maps were analyzed
for number of concepts, relationships, branches, hierarchies, and
cross-links.
Singh [32] Uncertainty principle,
time development,
Mach-Zehnder
interferometer
Undergraduate
students
USA A pretest and post-test were given to students (N¼12) who did the
QuILT. Examples of students’responses were provided.
Sokolowski [33] Photoelectric effect Upper secondary
school
USA A group of students (N¼15) answered one conceptual question
during an assignment. Examples of responses were provided.
Vokos et al. [34] Double slit experiment Undergraduate
students
USA Written problems were given to students (N¼450) in various physics
undergraduate courses and interviews were conducted (N¼14).
Students’reasoning was analyzed and categorized.
B Bao and Redish
[35]
Probability Undergraduate
students
USA Interviews were conducted with physics students (N¼16). The
observations were summarized.
Brookes and
Etkina [36]
Potential wells Undergraduate
students
Students (N¼4) were observed while working on homework
problems. Examples of students’reasoning were shown and
analyzed.
Domert et al. [40] Probability, tunneling Undergraduate
students
Sweden Students (N¼12) were interviewed while working with a computer
simulation. Observations were categorized and examples were
given.
McKagan et al.
[38]
Tunneling Undergraduate
students
USA Data was collected for eight courses, consisting of observations,
responses to essay questions, interviews, and a concept test
(QMCS). Observations were categorized and illustrated, test results
were reported.
Özcan [41] Wave functions,
operators
Undergraduate
students
Turkey Semistructured interviews were held with preservice physics teachers
(N¼34). Observations were categorized.
Özcan et al. [37] Potential wells Undergraduate and
graduate
students
Turkey A concept test was given to undergraduate (N¼95) and graduate
(N¼15) students. Semi-structured interviews were held with 10
students. Student responses were presented.
B=DSingh [43] Wave functions,
probability,
measurement
Undergraduate and
graduate student
USA Surveys were administered to graduate students (N¼202), interviews
were held with graduate and undergraduate students (N¼15).
Results were categorized and examples were given.
Singh et al. [42] Wave functions,
probability,
measurement
Undergraduate and
graduate
students
USA Surveys were administered to graduate (N¼200) and undergraduate
(N¼89) students. Examples of difficulties were presented.
(Table continued)
K. KRIJTENBURG-LEWERISSA et al. PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
010109-16
TABLE VII. (Continued)
Part Researchers Topic Level Country Methodology and analysis
Wittmann et al.
[39]
Probability Undergraduate
students
USA Students (N¼42) were given a pretest and post-test. A series of
questions were given also during the semester. Students’responses
were presented.
Wittmann et al.
[44]
Tunneling Undergraduate
students
USA Written examination questions, ungraded quizzes, surveys, and
interviews were analyzed by content analysis, interpretation of
diagrams, and descriptions of students’actions.
C Dangur et al. [54] Atomic structure Upper secondary
and
undergraduate
students
Israel Pretest and post-test were used to probe secondary (N¼122) and
undergraduate (N¼65) student understanding. A rubric was
designed to analyze the 3-item test.
Didişet al. [55] Light, energy, angular
momentum
Undergraduate
students
Turkey Interviews were conducted, a test was administered and exams were
analyzed (N¼31). The interviews were coded and mental models
were constructed.
Kalkanis et al. [12] Atomic structure, models Undergraduate
students
Greece A concept test was given to the test group (N¼98) and a control
group (N¼102). Semistructured interviews were conducted with a
sample of the test group. Difficulties found during the interviews
were summarized.
Ke et al. [46] Atomic structure Upper secondary
—Ph.D.
students
Taiwan A questionnaire was given to students from high school to Ph.D. level
(N¼140). Responses were categorized. Twenty-eight students
were interviewed using concept cards in order to refine the
categorization.
McKagan et al.
[48]
Atomic structure, models Undergraduate
students
USA One exam question was analyzed for four courses (N¼591).
Responses were categorized.
Özcan [52] Spin Undergraduate
students
Turkey Interviews were conducted with introductory (N¼24) and advanced
(N¼25) students. The results were categorized.
Papageorgiou et al.
[56]
Atomic structure Upper secondary
students
Greece Students (N¼421) were given two cognitive tests measuring field
dependence and reasoning abilities. A third test was used to assess
students’representations of the atom. These representations were
categorized and the influence of student characteristics thereon was
investigated.
Papaphotis and
Tsaparlis [49]
Atomic structure,
uncertainty principle
Undergraduate
students
Greece A questionnaire was given to first-year students (N¼125). Student
difficulties were summarized and illustrated with examples.
Petri and
Niedderer [45]
Atom structure Upper secondary
students
Germany Observations, questionnaires, interviews and written materials were
analyzed to describe the learning pathway of one student within a
course. The data were analyzed for change in conceptions and meta-
cognitive beliefs.
Papaphotis and
Tsaparlis [50]
Atomic structure Undergraduate
students
Greece Interviews were held with 2nd year students (N¼19). The responses
were categorized.
Taber [47] Atomic structure Upper secondary
students
UK Semistructured interviews were conducted with students (N¼15). A
typology of learning impediments was used to categorize the
response.
(Table continued)
INSIGHTS INTO TEACHING QUANTUM …PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
010109-17
TABLE VII. (Continued)
Part Researchers Topic Level Country Methodology and analysis
Tsaparlis and
Papaphotis [51]
Atomic structure,
uncertainty principle
Undergraduate
students
Greece A questionnaire was given to first-year students (N¼125).
Semistructured interviews were conducted with a subsample
(N¼23). Students’discussions were summarized and illustrated
with examples.
Wang and Barrow
[53]
Atomic structure,
chemical bonding
Undergraduate
students
USA Three diagnostic tests were used to analyze student understanding
(N¼159). Interviews, using a think-aloud protocol and interview-
about events, were conducted with a subsample (N¼48).
Representations of conceptual frameworks were created and
analyzed by axial coding.
Zhu and Singh
[57]
Spin, Stern-Gerlach
experiment
Undergraduate and
graduate
students
USA Surveys were administered (n >200) and semistructured interviews
were conducted with a subset of students. Results were used to
design a tutorial.
D Emigh et al. [60] Time dependence Undergraduate USA Four tasks were used to assess student understanding (N1¼416,
N2¼439,N3¼285,N4¼215). The tasks were examined to
identify difficulties, and these difficulties were categorized.
Michelini et al.
[58]
Quantum states,
nonlocality
Upper secondary
students
Italy Students (N¼17) took part in group discussions of worksheets.
Examples of student reasoning and a summary of the discussion
were presented.
Passante et al. [59] Superposition Undergraduate and
graduate
students
USA A multiple choice question was used to explore the understanding of
sophomores, juniors, and graduate students. Juniors (N¼32) were
asked to consider four statements. Results for the multiple choice
question and an overview of student reasoning regarding these
statements were provided.
Zhu and Singh
[61]
Measurement Undergraduate and
graduate
students
USA Concept tests, quizzes, and tests were analyzed over several years.
Interviews and informal discussions were conducted with a subset of
students to investigate students’reasoning. An overview of the
responses and students’reasoning is presented.
K. KRIJTENBURG-LEWERISSA et al. PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
010109-18
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INSIGHTS INTO TEACHING QUANTUM …PHYS. REV. PHYS. EDUC. RES. 13, 010109 (2017)
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