Please cite as: Fletcher, P.R., (2004) PhD Thesis - How Tertiary Level Physics Students Learn and Conceptualise Quantum Mechanics (School of Physics, University of Sydney)

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CHAPTER 4

DEVELOPMENT OF FINAL INTERVIEW QUESTIONS : GROUNDED THEORY APPROACH

4.1 INTRODUCTION

This chapter describes the investigation that lead to the development of the

set of interview questions that formed the primary research instrument for this

study. The grounded theory approach with its constant comparison techniques was

adopted, which allowed emerging ideas to guide the study. The final questions and

their sequencing were grounded in the data collected from a range of sources

including concept maps, expert interviews, examination scripts and preliminary

student interviews.

Through the recursive processes of open and axial coding each data source

provided a set of categories which were carried forward to the selective coding

phase. The selective coding phase was adapted to isolate underlying themes,

identify question topics and allow informed judgements to be made on the

appropriate sequencing of the interview questions in the final instrument. For a

detailed account of this stage of the study please refer to Appendix 2. Once a set of

interview questions was agreed upon the analysis shifted from a grounded

approach to a more focused study that employed a phenomenological approach

reported in Chapter 5.

4.2 SOURCES OF GROUNDED DATA

This study was conducted within the Schools of Chemistry and Physics at

the University of Sydney. These Schools contained many potential sources of data

that could be used to ground this study. The researcher drew upon four main data

sources some of which already existed and others that could be readily collected.

1) The first source was a formative assessment task that had been used with

intermediate physics students during a quantum mechanics course. Associate

Professor Ian Johnston had designed a concept mapping task to examine the links

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students made between key concepts in quantum mechanics and 67 student concept

maps were selected as a data source for this study.

2) It was recognised that the academic staff from the Schools of Physics and

Chemistry were a valuable source of knowledge and experience. A large proportion

of academics were involved with research and possessed industry links as well as

curriculum design, course delivery and assessment of student learning in quantum

mechanics or related fields. A number of academics were invited to participate in

expert group discussion/interviews and these became the second data source for the

study.

3) The third data source was based on the strong recommendation of a number

of academics involved with teaching quantum mechanics. A collection of existing

summative assessment tasks on quantum mechanics examinations completed by

junior and intermediate physics students were made available. Of these tasks 137

examination scripts were selected and used in the study.

4) The final source of data consisted of the experiences and knowledge of

students studying quantum mechanics. Students were invited to participate in

preliminary interviews and their observations, responses, discussions and opinions

were used to inform the study. Also the interviews provided a testing platform for

the development of the final instrument.

The analysis phase of these four data sources resulted in a small number of

identified categories, which emerged from the open and axial coding process; to be

fed into the third selective coding step of the grounded approach. The selective

coding process identified three core categories which became the overarching

themes that linked the final set of categories to one another.

4.2.1 Concept Maps In 1999 a concept mapping exercise was distributed to 67 intermediate

University of Sydney physics students who had just completed their second year

quantum mechanics lecture series. The exercise asked the students to draw a

concept map showing how they think the provided list of concepts related to one

another.

The exercise required the construction of a concept map using the nineteen

labels provided. Students were given a general concept mapping instruction sheet

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to assist those who were not familiar with or had not previously drawn a concept

map, and were given 20 minutes to prepare a response.

The exercise was designed to elicit the students’ understanding of the

relationships between the terms used in the context of quantum mechanics. The

nineteen concept labels were presented in alphabetical order and they were: atom,

diffraction, electron, energy, energy level, frequency, intensity, interference, light,

mass, matter, momentum, orbit, particle, photon, probability, uncertainty principle,

wave and wavelength. Figure 4-1 shows an example of a student’s concept map.

Example of Student Concept Map

Figure 4-1 : Copy of Student Concept Map (Student ID 21). This map shows the “wheel linked to another wheel” structural type. Reduced from original A3 with the labels and header instructions cropped (Please refer to Appendix 2, Figures A2-1 through A2-4 for

details of the complete concept mapping exercise)

The analysis of the concept maps focussed on classifying the structure of

each map and identifying nodes. Drawing upon the work of Cronin, Dekkers and

Dunn (1982) and Bailey and Butcher (1997) a set of nine concept map structural

types were identified. The maps were then classified using these types. The nine

types are illustrated and described in Table 4-1 along with the number of maps.

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Concept Map Structure Types Map Type Illustration Description (n) %

String

Three or more concepts are linked in a single chain

(1) 1%

Wheel

A number of single concepts emanate from a single concept

(2) 3%

String with a wheel attached

Four or more concepts are linked in a single chain with a wheel structure attached at one end

(7) 10%

Hierarchy

Concepts are arranged in a simple tree type structure (16)

24%

Complex

Cross-linking between the concepts to form an associative network

(6) 9%

Complex with a wheel attached

An associative structure with an obvious wheel structure attached

(20) 30%

Wheel linked to another wheel

Two concepts with wheel structures which are connected by a number of joining radial links

(12) 18%

Bubble Loops

Several string structures that form closed loops (2)

3%

Disjoint

The concepts are arranged into two or more separate structures

(3) 4%

Table 4-1 : Concept Map Structural Types & Results Summary (n=67 not mutually exclusive)

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The nodal analysis process examined the number of links emanating from

each concept label. The majority of the maps possessed one or more concept label

nodes which had a large number of links to other concept labels. The identification

of these nodes provided information about which concepts the students considered

as key focus ideas that they linked to other concepts.

For a full description of the data collection and analysis please refer to

Appendix 2 (pages A2-2 through A2-12).

Following is a summary of the three categories that emerged from the

concept map analysis, and were carried forward to the selective coding phase.

1. Wave Particle Duality - Concept maps showed a strong separation between

particle and wave. This suggests that the idea of wave/particle duality is a

dominant feature of students’ understanding of quantum mechanics.

2. Uncertainty - A significant variation in where students see uncertainty fitting

into quantum mechanics was evident. For some students their

understanding appeared weak and others associated uncertainty with a

range of different concepts and contexts.

3. Mathematics - For some students mathematics is an integral component of

the structure of quantum mechanics.

4.2.2 Expert Group Discussions/Interviews Several focus group discussions with physics and chemistry lecturers from

the University of Sydney were planned. Whilst I received positive responses to my

discussion invitations, timetabling constraints meant that only one focus group

discussion was conducted. Instead fourteen individual interviews were scheduled

to ensure the views of all the experts were heard.

The Expert Focus Group Discussion The focus group consisted of four lecturers, all of whom were from the

School of Chemistry, and the interviewer. All of the lecturers have taught senior

chemistry options which encompassed components of quantum chemistry. Prior to

commencing the discussion the lecturers were provided with a list of discussion

points which included learning, teaching, difficulties, experiments, analogies,

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mathematics, models and assessment strategies. These were developed from diary

entries from preliminary discussions (Please refer to Appendix 2 - Figure A2-6 for a

copy of the discussion points). A free-flowing group conversation followed with

only a minimum of guidance required to cover the discussion points. The

discussion was at times lively as the lecturers debated their views of various points.

The tape recording of the discussion was immediately transcribed and the

researcher reviewed the data and made a series of reflective notes. These reflective

notes along with selected extracts of the transcript were the basis of presentations

given to the Sydney University Physics Education Research (SUPER) group and the

Science Faculty Education Research (SCIFER) group. The discussions that followed

these SUPER and SCIFER presentations assisted the researcher in developing a

theoretical sensitivity towards this type of data.

The Individual Expert Interviews A total of fourteen individual interviews were conducted. The two

chemistry lecturers have a research background in theoretical chemistry. The

physics lecturers have a variety of research backgrounds including theoretical

physics, applied physics, high energy physics, physical optics, astrophysics and

physics education. All lecturers had previously taught quantum mechanics at

junior, intermediate or senior level. Although the dynamics and debate of a group

discussion was lost, the interviews produced fourteen detailed and rich responses as

a data source.

At the request of the lecturers who wished to provide considered responses

at the interview, a set of guide questions was developed. The questions were

developed from the grounded theory analysis of the existing data sources including

the concept maps, examination scripts, focus group interview and discussions. The

results comprising categories, emerging ideas and themes identified formed the

interview questions topics. These guides were provided to the lecturers at least two

days prior to the scheduled interview. (Please refer to Appendix 2 - Figures A2-7

and A2-8 for copies of the Lecturer Guide Questions)

The interviews were conducted at a time convenient to the lecturer in their

own office. A portable tape recorder was used to record the interview. Interviews

were scheduled for approximately 50 minutes duration, the actual time taken varied

from 40 to 92 minutes. Each interview was slightly different in its tone, pace and

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conversational style. All interviews were relaxed and free-flowing. At times the

researcher prompted and narrowed the conversation to probe specific issues.

The tapes were immediately transcribed so I, the researcher could read

through the transcripts and make reflective notes providing me with cues and

reference points in the data for later perusal. Each transcript was open coded

producing many categories, and then axial coding was employed to reveal the

following eight categories: Teaching Approach, Key Concepts, Assessment, Perceived

Difficulties, Maths, Analogies, Computer Simulations and Experiments. Some of the

categories were attitudinal, others were quantitative (contained a list). For example

Maths contained the lecturers’ views on the importance of mathematics to quantum

mechanics, while Key Concepts contained a list of concepts identified by the lecturer

as important to quantum mechanics.

For a full description of the data collection and analysis associated with these

eight categories please refer to Appendix 2 (pages A2-23 through A2-35).

Upon further analysis, comparison and consolidation the following five

categories emerged from the expert interview analysis, and were carried forward to

the selective coding phase.

1. Real world – Students experience difficulties solving unfamiliar problems

and linking theory, experiment and application. Experts agree the purpose

of quantum mechanics is to understand and explain ‘real world’ phenomena

and students should be able to do this. The experts identified linking

quantum mechanics to the real world as a key concept and as a teaching

approach.

2. Duality – Identified as a key concept in quantum mechanics. Students have

difficulties progressing past a classical view of either a wave or a particle.

The experts feel that teaching does not provide a resolution to the duality

paradox and the concept is not revisited in later years.

3. Uncertainty – Identified as a key concept in quantum mechanics.

4. Analogies – Some experts find analogies to be a useful teaching and learning

tool in quantum mechanics. Others find analogies inadequate and confusing

and prefer to use examples of experiments instead.

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5. Mathematics – Experts feel that students must have the necessary

mathematics skills to succeed at quantum mechanics.

4.2.3 Examination Scripts During the expert interviews, several lecturers referred to students having

difficulties with qualitative or interpretative questions in examinations and

assignments. Three lecturers strongly suggested a review of student examination

scripts might be of use to this study.

A senior academic from the School of Physics who was unconnected with

this study randomly selected 137 junior (first year) and intermediate (second year)

examination scripts. These scripts were then photocopied so there was no student

identification remaining. The scripts were analysed on their contents only, cross-

referencing to other student details was not possible.

In consultation with senior lecturers from the School of Physics who had a

role in setting and marking these examinations, six questions were selected for

analysis. Three from the first year physics examination and three from the second

year physics examination. The six questions had qualitative and quantitative sub-

components and covered a range of key concepts identified by the expert

interviews. These were basically back-of-chapter textbook in style and content and

addressed the following: de Broglie wavelength; quantisation; ground state;

tunnelling; normalisation constant; Heisenberg’s uncertainty principle; Compton

scattering; and the interpretation of graphical and tabulated data.

As researcher I was not concerned with the correctness of the responses

instead I was interested in the question “What does the student think is important

for the examiner to see?” and how much variation in responses was present. Each

question was analysed using a phenomenographic approach to reveal aspects of

variation within the student responses. The responses to each section of the

questions were reviewed, coded, categorised and tabulated. (Please refer to

Appendix 2 - Tables A2-9 through A2-34 for the coded datasets, commencing on

page A2-47). The correctness of the student response was tabulated along with

other features that emerged from the analysis. This did not influence the

phenomeographic approach, however it did provide a framework in which to group

and present the finalised tabulated categories.

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Following are a selection of interesting observed features identified in the

responses provided by the first and second year students.

First Year examination Scripts de Broglie

Students were asked to compare the de Broglie wavelength for an electron

and a proton with the same speed, kinetic energy and momentum. The students

demonstrated two ways of presenting their answer:

1. using mathematical formulae and inequality signs to show mathematical

relationships for the electron and proton

2. using a written description to articulate the differences between the electron

and proton.

Many students had difficulties with the relationship between momentum

and kinetic energy. Approximately three quarters of the students successfully

answered in relation to speed and momentum but only one third gave a correct

answer for kinetic energy. Most students had difficulties manipulating the formulae

for de Broglies’s wavelength into a form that allowed them to see a relationship

between kinetic energy and wavelength.

Terminology Approximately one quarter of students did not give a meaning for the terms

quantised and zero point energy. The concept of quantised energy was identified as a

key concept in the expert interviews and from the data it appears that only 43% of

students can correctly define the term either in terms of energy or more generally.

All but two students were able to give a meaning for the terms ground state

and excited state. Students seem to recognise these terms and can successfully define

them.

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Application of Quantised Energy Students were required to use a quantised model of a confined electron to

explain a related example concerning the ability to obtain absolute zero. Just over

half of the students successfully linked electron energy and motion at absolute zero,

but 30% of the students did not respond to this part at all.

Heisenberg’s Uncertainty Principle Students do not seem to know the formulaic representation of Heisenberg’s

Uncertainty Principle, 20% of students did not include a formula in their response

and 65% gave a formula that was incorrect. Most of the mistakes came from the

equality/inequality sign of the formula with students using ≥, ≤, ≈ and =. This

suggests there is some confusion with the relationship between momentum,

position and Planck’s constant.

Regardless of whether the formula was stated or not 76% of students gave

an answer that suggested a connection between momentum and position of a

particle and how this limited the measurement of either quantity. Only two

students suggested that a classical meaning of uncertainty related to an error in

measurement. The terms ‘accurately’ and ‘precisely’ were used by 30% of students

but it was unclear what meaning is given to these terms.

Students were asked to extend the concept of uncertainty to the macroscopic

world and explain it in this context. The student responses suggest that 63% think

that uncertainty relates to all objects regardless of size, while 20% think it only

relates to microscopic objects. In the macroscopic context the proportion of students

relating uncertainty to classical measurement error was 42%. This compares to only

4% when the students were describing the uncertainty formula.

Second Year Examination Scripts de Broglie

All students within the sample attempted the question asking them to

describe de Broglie’s wavelength of a particle. Written descriptions included

references to the wave/particle nature of electrons and the motion of particles and

waves. Some students linked the de Broglie wavelength to electron orbitals. Some

students drew sketches of wave packets and 53% of students included the formula

ph

=λ in their response.

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The second part of the question which asked for a description of an

experiment to measure de Broglie’s wavelength proved to be more troublesome,

29% did not state an experiment at all and 12% described another quantum

mechanics experiment (e.g. photoelectric effect). The most popular group of

experiments described were ones that caused wave interference (e.g. double slit or

single slit diffraction), 30% of students gave this response.

Compton Scattering Analysis of this question revealed that students were not overly familiar

with the Compton scattering experiment, 37% of Normal and 29% of the Advanced

stream students confused Compton Scattering with another experiment (e.g.

photoelectric effect or double slit) others gave a variety of experimental descriptions

including in their responses a range of electromagnetic waves and a range of

targets. When asked to describe the interaction that occurs between photons and

electrons in Compton shifting, 63% of Normal stream students described classical

wave behaviour (reflection, diffraction, scattering etc) then they used this to justify

the particle nature of light. 24% of Advanced stream students described the classical

particle phenomena (collisions transferring momentum and transferring kinetic

energy) and 37% described classical wave phenomena (reflection, diffraction,

scattering etc).

Tunnelling In describing tunnelling students gave somewhat mixed answers. The

majority of responses (65%) described particles as the entity doing the tunnelling a

particle ‘penetrates’, ‘burrows’ or ‘leaks’. Some students (10%) referred to the wave

function tunnelling and 4% described electrons tunnelling. All students stated that

either a well or a barrier was what was tunnelled through.

To accompany their descriptions 50% of the students drew pictures. The

pictures they drew in some cases contradicted their written description, for

example, 28% of students drew pictures of wave functions tunnelling and only 13%

drew pictures of particles tunnelling.

The Advanced Stream were asked to explain the significance of tunnelling to

nuclear reactions in stars. This question required the application of tunnelling to a

real world example. The students’ description or explanation of tunnelling was in

terms of a proton or alpha particle crossing a potential barrier to result in fusion.

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Most students attempted to reconstruct the four-step hydrogen fusion process and

this made up the bulk of their answer.

Wells Students, when provided with graphical and tabulated stimulus material on

wave functions and potential diagrams, could without difficulty interpret the

material and determine the eigenstate and the probability distribution.

For a full description of the data collection and analysis please refer to

Appendix 2 (pages A2-36 through A2-72).

The following three categories emerged from the examination script

analysis, and were carried forward to the selective coding phase.

1. Real world – Use of real world examples illustrated gaps, inconsistencies and

misconceptions in student’s understanding of quantum mechanics. These

problems were not noticeable when students were asked similar questions in

a theoretical context. In futher studies, real world examples (e.g.

radioactivity) could be used as a tool to probe student understanding in an

interview.

2. Duality – Students do not seem to match the correct classical behaviour to

waves and particles. Many of them use wave behaviour as evidence of

particle nature. There appears to be no conceptual shift from a wave-or-

particle view to a wave function view.

3. Tunnelling – Students appear to be familiar with the terms, diagrams and

graphs, associated with potential wells and barriers diagrams and wave

functions. However their explanation of tunnelling which brings together all

of these tools, is patchy and expressed in terms of a particle model rather

than a wave function or probability model. Their proficiency with the tools

hides their lack of understanding of the physical situation.

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4.2.4 Preliminary Interviews The preliminary interviews served two purposes; as a source of data for the

grounded theory stage of the study, and an opportunity to trial and refine the

interview protocol leading to the development of the final interview instrument.

In all, 17 preliminary interviews were conducted. These interviews drew on

issues that were emerging from the other data sources (concept maps, examination

scripts and expert discussion/interviews). The initial preliminary interviews were

unstructured or recursive in nature and, as more were completed, they became

semi-structured, with the aim of progressively focusing the interview towards the

final interview instrument.

This section describes and reports the grounded theory analysis and the

identification of categories to be carried forward to the selective coding phase. The

analysis relating to the development and refining of the interview protocol

including trialling question types, question order, selection of opening and closing

questions is reported in Section 4.3 ‘Development of the Final Interview

Instrument’.

4.2.5 Analysis of Data Collected Each interview was transcribed from tape immediately following the

interview and formatted according to the protocol defined in Chapter 3.

The interview transcript was first annotated with reflective notes in the

Personal Log column and then analysed using the grounded theory iterative process

of open and axial coding to reveal categories. Key statements, preliminary ‘in vivo

codes’ and emerging ideas identified during this phase of analysis were recorded in

the Analytical Log Major Point column.

Once the final iteration of axial coding was complete the final set of major

categories was used to re-code the entire transcript. These codes were recorded in

the Analytical Log Category Coding column.

Please refer to Figure 4-2 for and example of a Preliminary Interview

Transcript Cover Page and Figure 4-3 Preliminary Interview Transcript Page.

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Preliminary Interview - Transcript Cover Page

Figure 4-2 : Representative Preliminary Interview Transcript – TED Page 1 Cover Page

Preliminary Interview - Transcript Page

Figure 4-3 : Representative Preliminary Interview Transcript – TED Page 4

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A detailed personal log and analytical log was then written for each interview.

The personal log is an instrument for recording reflective notes and highlights areas

for improvement and development of the interview process. The analytical log is an

instrument which provides a format and forum in which to identify trends and

ideas; then allow for common elements to be identified and condensed, the

identification of relationships between categories and the isolation of core

categories.

An excerpt from a Detailed Analytical Log follows:

Analytical Code Description – SID 06

for Interview 6/10/00 - TED Overview Six primary categories were used to code the transcript – Concept, Personal Comment, Personal Experience, Student Experience, Self Reflection and Time Frame. A secondary set of key-words were selected to provide a greater level of context during this preliminary analysis exercise. The primary categories could be represented in a number of ways and for the purpose of this study it is convenient to adopt a hub structure that is centrally linked to the category of Concept, refer Figure 1.

Figure 1 : Primary categories

This structure although in some sense is arbitrary directly relates to the research project question of conceptual development. Thus the structure provides a useful natural theme without constraining the data-set. … … …

Personal Comment

Personal Experience

Concept

Student Experience

Self Reflection Time Frame

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Please refer to Appendix 2 - Figures A2-15 through A2-23 (page A2-74ff) for

a representative transcript document for one particular preliminary interview.

Each interview was coded and then compared with prior interviews and ten

common categories were identified: Analogies; Assessment; Computer Simulations;

Course Structure; Difficulties; Duality; Mathematics; Potential Diagrams; Real World;

Reflective Thoughts; and Tunnelling. The categories will be briefly discussed.

Analogies Eight students indicated that they found analogies helpful to their learning,

two of these students in particular really liked them and wished they were used in

courses more often. Six students did not like analogies and said they were

confusing. The remaining two students commented that analogies were occasional

useful but often they were inadequate. The students mentioned a limited set of

analogy examples, for example “a ball rolling in some sort of valley” (PrelimSID06)

Assessment The focus of student discussion regarding assessment was the end of

semester examination. Students emphasised the importance of mathematics to

doing well in examinations. Students described their preparation for examinations

in terms of remembering recipes for solving different problems. “To study I try to

learn all of the examples given in lectures” (PrelimSID02). “I memorise the steps so

hopefully I can do it in the exam” (PrelimSID10).

Computer Simulations This category was covered in detail by the physics students as a computational lab

forms part of their course in intermediate physics. The chemistry students referred

briefly to computer generated models of orbitals and molecular shapes. The

majority of students found computer simulations useful for visualising abstract

ideas (e.g. the mathematics of potential diagrams and wave functions). A number

of students felt computer simulations could be more powerful if preceded by a ‘real

experiment’. Three students felt the link between the simulation and the physical

meaning was not made clear enough. “I didn’t understand Schrödinger’s equation

and wells until I saw it in the computational lab…” (PrelimSID04)

Course structure Student comments on course structure were predominantly related to the

integration of lecture and laboratory components of the course. For example one

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student said “the lectures and computer lab got out of sync ...” (PrelimSID11),

another said “it is not clear what the lab has to do with the lecture bit …”

(PrelimSID07).

Some students commented on the teaching approach within the course.

Four students said they enjoyed the historical approach used and three students

appreciated seeing all of the steps described in the lecture examples “if I get down

all the steps I am more confident of figuring it out later” (PrelimSID03)

Difficulties Once a rapport was established between student and interviewer, the

students were more than willing to articulate their difficulties with quantum

mechanics. One student said “it is good to be asked …how long have you got?”

(PrelimSID07). Students were open about their strength and weaknesses:

“I am good at the maths (long pause) but I couldn’t tell you what it all means.”

(PrelimSID05)

“I find the maths overwhelming at times … what is the point, what is it for?”

(PrelimSID11)

“When they want us to explain anything, in assignments, I am stuck….”

(PrelimSID03)

The following list summarises the difficulties identified by students in the

preliminary interviews:

• Conceptual explanations • Duality • Mathematics • Probability • Uncertainty • Unfamiliar problems • Wave functions • Wells

Duality Throughout the preliminary interviews the students used a variety of words

to describe the quantum entity including: wave, particle, wave/particle, wave

packet, smeared particle, wave function and probability density. The students

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appear to view the entity in different ways depending on the situation. “I guess I

don’t think about it, I don’t let it worry me, whatever works.” (PrelimSID17)

One student described how he thinks about a wave function shape for a

particular potential well. “I think of the particle in the well and how it moves for

that potential energy, then I think of where abouts it is going slow or fast and then I

work back to the wave function shape.” (PrelimSID08) This comment demonstrates

how students use multiple entities to solve problems in quantum mechanics and

they need to shift between them. This particular student demonstrated a strong

conceptual understanding of all aspects of quantum mechanics covered in the

interview but it appears from the transcripts that other weaker students have

serious difficulties with multiple entities.

Mathematics The students interviewed split into two distinct groups regarding

mathematics in quantum mechanics. One group (5 students) felt that the

mathematics was “straight-forward” or “easy” once you were shown the steps. The

other group (12 students) found the mathematics “more difficult” or “hard” and at

times “overwhelming”. All students felt you needed mathematics in order to

succeed at quantum mechanics. Four students felt that your understanding

improved with time as your mathematics skills improved. “When you solve

Schrödinger’s equation the first time, its like, ‘oh my god’ … really hard, but in 3rd

year when you do it again its much easier.” (PrelimSID13)

Potential Energy Diagrams The students discussed a variety of potential diagrams used in quantum

mechanics including infinite wells, finite wells, square wells, parabolic wells, ramp

wells, step wells, an array of wells, barriers and humps. Five students recognised

that all of these examples have the same basic structure associated with kinetic and

potential energy and could describe in detail 3 examples. “The wells describes the

energy in the system.” (PrelimSID06).

The remaining students were very familiar with the simple examples (e.g.

square wells) but had difficulties working with and describing other more

complicated diagrams. “The wells steps always confuse me … I get the wave

function shape wrong” (PrelimSID02)

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While most students were familiar with potential diagrams as important

tools in problem solving only three students could clearly explain the relationship

between potential diagrams and physical systems. Most students saw potential

diagrams as useful but isolated tools.

Real World The students were asked to describe three examples of quantum mechanics

applied to the real world. Only two students were able to do so, most other

students could name one but three students could not give a single example. “I

can’t think of any examples … it’s too abstract.” (PrelimSID01).

Reflective Thoughts Throughout the preliminary interviews students made reflective comments

on a range of topics including: high school physics experiences, course structures,

teaching approaches, sequencing of ideas, learning styles and their attitude towards

learning quantum mechanics.

Tunnelling When describing or discussing tunnelling, students use potential diagrams

and wave functions as tools. Ten students drew diagrams of the barrier with a

decaying wave function superimposed. Most students described the wave function

of being in a classically forbidden region, probing this idea revealed a variation in

the depth of understanding. Most students conceptualise a ‘particle’ as the entity

doing the tunnelling but cannot easily link this to their drawing. “I can see how it

works when the wave function overlaps the barrier but what does this mean in

terms of particles?” (PrelimSID10)

Role of Chemistry Student Interviews At this point the categories that were identified from each of the four data

sources were used to develop two final interview instruments for the study. One

interview instrument focused on quantum mechanics learning in chemistry and the

other on learning in physics. At a later date it was decided the learning issues in

chemistry were beyond the scope of this dissertation and so the development of the

chemistry interview instrument, its implementation and subsequent data analysis

are not reported here. The research into learning in quantum chemistry provided

additional theoretical sensitivity for this study.

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For a full description of the data collection and analysis please refer to

Appendix 2 (pages A2-73 through A2-96).

Upon further analysis, comparison and consolidation the following five

categories emerged from the preliminary interview analysis, and were carried

forward to the selective coding phase.

1. Analogies – Some students find analogies useful to their learning of

quantum mechanics, other students dislike analogies and find them

confusing.

2. Tunnelling – This concept links a group of problem solving tools (e.g.

potential diagrams and wave functions) to real world examples of quantum

mechanics. Discussion of this concept can reveal students difficulties with

the tools and how they interpret what the tools do.

3. Difficulties – Students are aware of and can identify the difficulties they

experience in learning quantum mechanics. Difficulties students discussed

included Conceptual Explanations, Duality, Mathematics, Probability,

Uncertainty, Unfamiliar problems, Wave Functions and Potential Wells. It

was recognised that a student’s perception of their strengths and weaknesses

could influence future learning experiences.

4. Reflection – Given the opportunity students will reflect on their learning in

and experiences in quantum mechanics. Through reflective processes

students come to see relationships and connections in the subject.

5. Duality – Students view the quantum mechanics entity as a wave or particle

or wave function depending on the situation. They often shift between

entities.

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4.3 DEVELOPMENT OF THE FINAL INTERVIEW INSTRUMENT

4.3.1 Categories Brought Forward from the Grounded Study Combining the results from the four data sources, concept maps, expert group

discussions/interviews, examination script and preliminary interviews, eight

categories emerged. They are summarised below.

1. Real world – The experts identified linking quantum mechanics to the real world

as a key concept and as a teaching approach. However they were concerned

that most students were unable to do this. Analysis of student responses in

examinations and interviews indicated students had difficulties with

unfamiliar problems and applications of quantum mechanics to the real

world. Real world examples tended to highlight gaps, inconsistencies and

misconceptions in student’s understanding of quantum mechanics. In future

investigations, real world examples (e.g. radioactivity) could be used as a

tool to probe student understanding in an interview.

2. Duality – Student concept maps suggest that students see a clear separation

between the concepts of particle and wave; however their responses to

examination questions suggest they cannot connect the correct classical

behaviour to waves and particles. There also appears to be no conceptual

shift from a wave-or-particle view to a wave function view following formal

instruction. Instead students view the quantum mechanics entity as a wave

or particle or wave function depending on the situation. The experts feel

that teaching does not provide a resolution to the duality paradox and the

concept is not revisited in senior years.

3. Uncertainty – The experts identify uncertainty as a key concept in quantum

mechanics, however many students appear to have difficulties with it.

Students can use Heisenberg’s Uncertainty Principle to solve mathematical

problems but they cannot link it to other aspects of quantum mechanics or

explain its significance in real word examples. Some students continue to

confuse uncertainty with the common meaning of measurement error.

4. Analogies – Some experts find analogies to be a useful teaching and learning

tool in quantum mechanics. Others find analogies inadequate and confusing

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and prefer to use examples of experiments instead. Students also expressed

a range of attitudes to their use.

5. Tunnelling – Students appear to be familiar with the terms, diagrams and

graphs associated with potential diagrams and wave functions. However

their explanation of tunnelling which brings together all of these tools is

patchy and expressed in terms of a particle model rather than a wave

function or probability model. Their proficiency with the tools appears to

hide their lack of understanding of the conceptual/physical situation.

6. Difficulties - Students are aware of and can identify the difficulties they

experience in learning quantum mechanics. Their perception of their

strengths and weaknesses could influence future learning. Experts also

identify difficulties students in general have with quantum mechanics based

on their teaching and assessment experiences.

7. Reflection – Given the opportunity, students will reflect on their learning

and experiences in quantum mechanics. Through reflective processes

students come to see relationships and connections in the subject.

8. Mathematics – Students and experts both see mathematics as an integral

part of quantum mechanics that must be mastered in order to succeed and

progress in the subject.

As outlined in Chapter 1, the purpose of this investigation is to explore the

teaching and learning processes associated with delivering a tertiary level quantum

mechanics curriculum. The investigation aimed to isolate key concepts, identify

learning difficulties, identify teaching difficulties and so provide both teachers and

curriculum developers with a valuable resource.

The primary focus for the second stage interviews was to explore the

students’ attitude and conceptual understanding of quantum mechanics. It was

found during the preliminary interviews that asking specific mathematics questions

focussed the student’s attention upon that aspect and appeared to put them off

conceptual descriptions. Information was available about the students mathematics

background and was collected, but otherwise it was decided that the interview

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should leave out specific mathematical discussions. If the student brought it up it

was discussed otherwise not.

The seven remaining categories Real World, Duality, Uncertainty, Analogies,

Tunnelling, Difficulties and Reflections would become the topics for the final series of

interviews.

4.3.2 Selective Coding The seven categories were further analysed using selective coding to identify

three underlying core codes. These core codes became the themes that overarched

the grounded theory categories and were carried forward to inform the interview

question sequencing process. (Refer to Table 4-2).

Interview Themes

1. Concepts Basic ideas and definitions used to describe or explain quantum mechanics.

“What they know”

2. Tools Methods, recipes, mathematics, examples and analogies used to solve problems in quantum mechanics.

“What they do”

3. Linking The process of tying together different aspects of quantum mechanics to make a connected and coherent whole.

“How they make sense of it”

Table 4-2 : Interview Themes

4.3.3 Sequencing Topics To provide a workable and logical sequence for the seven interview topics

the format of the interview needed to be considered. The protocol required the

interview to comprise three parts; an opening, a body and a close.

Opening To open the interview we needed a familiar quantum mechanics topic that

the students were comfortable discussing. The ideal topic would be something the

students had previously experienced. It would also be answerable at a number of

levels and consist of a range of aspects that could be discussed. It would have a

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depth of complexity, a range of applications and could re-emerge later in the

interview.

A number of topics were trialled for the opening during the preliminary

interviews. They are listed in Table 4-3 below along with the advantages and

disadvantages revealed in analysis.

Interview Opening Topics Opening Topic Advantages Disadvantages Double Slit Experiment

Familiar topic Students were relaxed and

confident Depth of complexity Later link to analogy

Some students saw it linked to optics but not quantum mechanics

Photoelectric Effect

Familiar topic Depth of complexity

Most students could not recall the details or significance of the photoelectric effect

Did not relax the students Role of Mathematics in Quantum Mechanics

Familiar topic Students were relaxed and

confident Gave good overview of maths

ability

Gave the entire interview a strong maths flavour

Too open and hard to control

Wave/Particle Duality

Familiar topic Depth of complexity Stimulated a range of ideas and

feelings

Too open and hard to control Did not relax students Was unsettling rather than

setting the scene

Table 4-3 : Interview Topics

From this analysis the topic Double Slit Experiment” was selected for the

opening as it best met the stated criteria. To address the possible disadvantage of

this topic the researcher used a follow up question mentioning wave/particle

duality with those students who could not see a link to quantum mechanics.

Close To close the interview we needed a topic that summed up the issues raised

during the body and gave the students a relaxed opportunity to reflect back on their

responses. The interview should end with the student feeling relaxed and

appreciated. A number of closure topics were tested during the preliminary

interviews. These are listed in Table 4-4 below along with the advantages and

disadvantages revealed in analysis.

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Interview Closure Topics Closure Topic Advantages Disadvantages

Reflection on course

Students have considered the topic in part prior to closure

All students have an opinion to offer on the course

A range of issues to discuss

Responses may be destructive or personal

Difficulties Students have considered the topic in part prior to the closure

Allows students to identify their difficulties

A range of issues to discuss

Some students may feel defensive

Interview closes with students focussing on low points

Real world Students have considered the topic in part prior to closure

Links quantum mechanics to useful applications

A range of issues to discuss

Many students may not be able to identify and discuss real world applications

Table 4-4 : Interview Closure Topics

The topic Reflection was chosen as a closure topic as it best fitted the selection

criteria. Care was taken in designing specific questions and prompts for this topic

to address the possibility of destructive or personal criticism emerging.

Body The body of the interview is approximately 45 minutes in length and will

need to include six topics. It is important to sequence the topics and specific

questions in order get the most out of the interview instrument. During the

preliminary interviews a number of questions associated with the topics were

trialled and so we have data to inform the sequencing. In addition the topics can be

classified according to the predominant learning domains1 with which each is

associated. The domains of Skill, Affective and Cognitive were addressed in

addition to the content. The results of the trials appear in Table 4-5.

1 Psychologists distinguish between three kinds of learning or domains based on the type of performance involved.

• Psychomotor or Skill domain (both motor and cognitive skills) • Affective domain (involves feelings and emotions) • Cognitive domain (information and ideas)

For example see Lefrancois, G.R., (1999) Psychology For Teaching, (Wadsworth, Thompson Learning Belmont CA) p118

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Interview Body Topics

Body Topic Advantages Disadvantages Domain Analogies Reveal students’ ability to visualise, and shift context

Helps students understand abstract concepts Students often don’t see limitations of analogies Some students don’t like them Some lecturers don’t use them

Cognitive & Skill

Difficulties Allows students to identify their own difficulties Students describe a range of difficulties

Could make students defensive Affective

Real World In a detailed answer student shows how quantum mechanics is linked to real world

Links between theory, experiment and application Highlights student difficulties

Students often do not see any link between quantum mechanics and the real world

This topic puts off weak students

Cognitive

Tunnelling In a detailed answer students refer to tools such as potential diagrams and wave functions

Tunnelling is a bridging concept between theory and real world examples

Weak students cannot give a detailed response without prompting

Students can get tangled and confused in their answers

Cognitive & Skill

Uncertainty Identified as a key concept of quantum mechanics Students give a range of descriptions Strong links to other topics

Student responses can vary depending on the context used

Cognitive

Wave/Particle Duality

Identified as a key concept in quantum mechanics It can be used to indicate conceptual change (from

wave/particle to wave function) Students give a range of descriptions Strong links to other topics

Very broad topic and students can get off track Concept is not revisited in senior and honours level

courses

Cognitive

Table 4-5 : Interview Body Topics

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The advantages and disadvantages given for each topic suggest preferred

sequencing options. Wave/Particle Duality is best positioned directly following the

opening. Wave/Particle Duality is strongly linked to Double Slit Experiment and

should allow the discussion to broaden after a focussed introduction. In some cases

the interviewer will prompt a connection between Double Slit Experiment and

quantum mechanics by mentioning the idea of wave/particle duality so it naturally

follows the opening.

Students’ answers to the topic Uncertainty will be strongly influenced by the

preceding context wherever it is placed. With this in mind Uncertainty will be

addressed in the interview as two separate questions connected with the topics

Wave/Particle Duality and Analogies.

As many students have difficulties describing tunnelling without prompting

familiar questions on potential diagrams and wave functions will precede any direct

questions on tunnelling. It would be ideal to later ask an application question on

tunnelling in the Real World topic.

The Difficulties topic would sit well in the middle of the interview once

students are relaxed and so it can be reflected upon in the later part of the interview.

Many students have difficulties with the Real World topic so it needs to be placed

between two topics that students have confidence in. Questions associated with

Analogies can be easily imbedded in other topics. Discussion of a specific analogy

should be considered late in the interview in case the student does not provide

adequate information.

The questions selected for each topic came from several sources. Questions

that were trialled and worked well in the preliminary interviews were considered

and usually selected. Some questions were modified and new questions written to

address the advantages and disadvantages that were highlighted by the preliminary

interviews. Questions were reviewed to ensure there was a variety of modes,

learning styles and learning aspects addressed. In addition the questions needed to

address the three themes that tied the grounded data together.

Table 4-6 provides a summary of the final interview instrument. The physics

interview guide follows in Section 4.4 and the complete and detailed questions for

the study appears in Appendix 3.

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Final Interview Instrument Structure

Interview Protocol Timeline (minutes)

Learning Domain

Topic Questions Theme Addressed

Rapport 0 Cognitive Wave/Particle Duality

Double Slit Wave or Particle? Uncertainty

Concept & Linking Concept Concept

Body 10 Evidence of Wave/Particle duality Applications/Examples/Experiments

Linking Linking

Revisit key concepts 15 Cognitive & Skill

Tunnelling Draw a well and a barrier Compare and contrast Discuss terminology

Tools Tools, Linking & Concept Tools & Concept

Different modes of questioning

30 Affective Difficulties Learning difficulties in quantum mechanics

What tools do you need? Analogies and models you use?

Concept & Tools Tools Tools & Linking

Different contexts 40 Cognitive Real World Explain Electromagnetic shielding or radioactivity in terms of quantum mechanics

Linking

Different styles 45 Cognitive & Skill

Analogies Quantaroo (macroscopic analogy of double slit)

Concept, Tools & Linking

Closure 55

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Affective Reflection Changes in understanding What did you need to understand? Expectation of lecturer Advice to new lecturer

Concept, Tools & Linking Concept, Tools & Linking Concept, Tools & Linking Concept, Tools & Linking

Table 4-6 : Structure of the Final Interview Instrument

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4.4 FINAL INTERVIEW INSTRUMENT

PHYSICS SECOND/THIRD/HONS/POST GRADUATE Wave/particle duality - Double Slit Experiment

- Describe what occurs if you shine the light on the slits? - Draw/describe - What features of this relate to quantum mechanics?

- What separates a wave and a particle? - What is meant by Uncertainty? Duality - Evidence of wave particle duality - Key ideas, experiments Tunnelling - Barriers and Wells

- Compare and contrast - Discuss Wave Function, Eigen Functions/Values, Probability Distribution

Functions TUTORS (*) Difficulties - What difficulties do you anticipate the students might have learning quantum

mechanics? - What tools do you expect students to have? - What Analogies/Models do you use to explain quantum physics concepts? Linking - What links do quantum mechanics concepts have outside, EM shielding,

Radioactive decay - Application - Name three things quantum mechanics has given us? Discussion Question - Imagine you live in a universe in which the value of Planck’s Constant, h, is

much greater than 10-34 – say of order 1000. In this universe you would observe quantum phenomena in everyday life. Now imagine you are a hunter. Every evening a mob of Quantaroos (Quantum kangaroos) bound along a path that passes through a densely packed grove of tall thin trees (River Gums) into a clearing. You would like to capture a Quantaroo as it exits the grove into the clearing. You have a shovel to dig a hole or a trench, a tranquiliser gun and a net.

Epilogue - Sequence of major concepts, changes in understanding. - During the delivery of the course what did you feel you needed to understand? - What do you feel the lecturer wanted you to gain from the course? - What would you advise a new lecturer about teaching the course?

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CHAPTER 4 ........................................................................................................................................62 DEVELOPMENT OF FINAL INTERVIEW QUESTIONS : GROUNDED THEORY APPROACH.62

4.1 INTRODUCTION..................................................................................................................62 4.2 SOURCES OF GROUNDED DATA .....................................................................................62

4.2.1 Concept Maps...............................................................................................................................63 4.2.2 Expert Group Discussions/Interviews ..........................................................................................66

The Expert Focus Group Discussion .........................................................................................................66 The Individual Expert Interviews ..............................................................................................................67

4.2.3 Examination Scripts .....................................................................................................................69 First Year examination Scripts ..................................................................................................................70

de Broglie..............................................................................................................................................70 Terminology..........................................................................................................................................70 Application of Quantised Energy..........................................................................................................71 Heisenberg’s Uncertainty Principle ......................................................................................................71

Second Year Examination Scripts .............................................................................................................71 de Broglie..............................................................................................................................................71 Compton Scattering ..............................................................................................................................72 Tunnelling.............................................................................................................................................72 Wells .....................................................................................................................................................73

4.2.4 Preliminary Interviews .................................................................................................................74 4.2.5 Analysis of Data Collected...........................................................................................................74

Analogies ..............................................................................................................................................77 Assessment............................................................................................................................................77 Computer Simulations...........................................................................................................................77 Course structure ....................................................................................................................................77 Difficulties ............................................................................................................................................78 Duality ..................................................................................................................................................78 Mathematics..........................................................................................................................................79 Potential Energy Diagrams ...................................................................................................................79 Real World............................................................................................................................................80 Reflective Thoughts ..............................................................................................................................80 Tunnelling.............................................................................................................................................80

Role of Chemistry Student Interviews ...........................................................................................................80 4.3 DEVELOPMENT OF THE FINAL INTERVIEW INSTRUMENT.........................................82

4.3.1 Categories Brought Forward from the Grounded Study...............................................................82 4.3.2 Selective Coding ..........................................................................................................................84 4.3.3 Sequencing Topics .......................................................................................................................84

Opening .....................................................................................................................................................84 Close..........................................................................................................................................................85 Body ..........................................................................................................................................................86

4.4 FINAL INTERVIEW INSTRUMENT.....................................................................................90

Figure 4-1 : Copy of Student Concept Map (Student ID 21). This map shows the “wheel linked to

another wheel” structural type. Reduced from original A3 with the labels and header instructions cropped (Please refer to Appendix 2, Figures A2-1 through A2-4 for details of the complete concept mapping exercise) ........................................................................................... 64

Figure 4-2 : Representative Preliminary Interview Transcript – TED Page 1 Cover Page.................. 75 Figure 4-3 : Representative Preliminary Interview Transcript – TED Page 4 ..................................... 75 Table 4-1 : Concept Map Structural Types & Results Summary (n=67 not mutually exclusive)........ 65 Table 4-2 : Interview Themes .............................................................................................................. 84 Table 4-3 : Interview Topics ................................................................................................................ 85 Table 4-4 : Interview Closure Topics................................................................................................... 86 Table 4-5 : Interview Body Topics ...................................................................................................... 87 Table 4-6 : Structure of the Final Interview Instrument....................................................................... 89