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The progression in students' concepts of an electron wave

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Keywords:

Learning

conceptual change, history of science, physics education

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The progression in students’ concepts of an electron wave

ABSTRACT: A debate in physics education is whether to include or exclude classical mechanics (CM) concepts in the teaching of quantum mechanics (QM). The research problem for this paper examines (1) the role of CM – good, bad or neutral – in teaching QM, and (2) how students view and develop physics knowledge, and (3) how students’ old/CM knowledge systems evolve into the new/QM knowledge systems. Two models of conceptual change –

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revolutionary and evolutionary – are considered to explain students’ learning. Three tools – questionnaire, card sort task and interview – were used as the data sources One hundred and

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fifty physics students participated in this research. There are three main findings. First, students’ prior experience is one of the key factors on which students develop their QM mental models.

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Second, students use mental models that keep both CM and QM concepts in their minds without being aware of any conflict. Third, the more experienced students also hold both CM and QM

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concepts in their minds but they can switch between CM and QM knowledge systems to explain phenomena in different contexts. Based on the major findings, this paper proposes new teaching

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and learning frameworks for QM.

w KEYWORDS: conceptual change, history of science, physics education

INTRODUCTION

In order to become a fully-fledged physicist, students go through a series of exams including quantum mechanics (QM). Solving QM problem require strong mathematical skills – such as the ability to operate the momentum operator. However, solving QM problem mathematically 1 ScholarOne, 375 Greenbrier Drive, Charlottesville, VA, 22901

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does not necessarily test students’ understanding of the physical concepts needed to explain the phenomenon. So a common problem in the learning of quantum physics is that students can pass the exams by means of amazing mathematical abilities without much comprehension of the meanings of concepts in the formulas. This emphasis on blind problem solving is described in the research literature under the headings of meaningful learning and differences between novice and expert physicists’ understanding of physics concepts (Reif & Allen, 1992; Reif, 1995). While there is little dispute that in the process of educating physicists it is very important that students obtain and develop mathematical abilities, it does not follow that conceptual

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understandings of core QM concepts should be sacrificed for mathematics.

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So, one might also expect that, owing to the lack of a clear understanding of QM concepts, many students use the QM concepts inappropriately. For example, research has shown how

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some students mix QM with classical mechanic (CM) concepts to interpret quantum mechanical phenomena (e.g. Fischler & Lichtfeldt, 1992). The characteristic of mixed concepts

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has been addressed by Ireson (2000a) that students could hold ‘conflicting quantum thinking’ or

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‘conflicting mechanistic thinking’. For this problem, Fischler & Lichtfeldt (1992) designed a QM teaching programme that omitted all analogies to classical physics. Alternatively, Budde,

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Niedderer, Scott & Leach (2002a) designed a QM teaching programme based on students’ understanding of classical physics. The problem here is that both approaches, including or excluding CM in teaching QM, claim success. Therefore, the research problem for this paper examines (1) the role of CM – good, bad or neutral – in teaching QM, and (2) how students view and develop physics knowledge, and (3) how students’ old/CM knowledge systems evolve into the new/QM knowledge systems.

In learning quantum physics, according to the historical development of QM (see figure 1), the

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concepts developed after de Broglie’s wave theory of particles (e.g. ‘wave function’, ‘probability’ and the ‘uncertainty principle’) are difficult and confusing for they involve a paradigm shift away from the Newtonian influence of classical physics. Therefore, in order to document and understand students’ learning, it is necessary to survey students’ understanding of QM ideas at different educational/conceptual periods. The first research question is:

[Insert Figure 1 about here]

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1. What are the conceptual patterns in the progression of students’ mental models of an electron wave at different conceptual stages?

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After documenting the progression in students’ QM ideas from first year students through to

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Ph.D. students, it will be helpful to examine the differences between them for the purpose of understanding the influence of the classical ‘intermediate’ language – classical mechanics (CM).

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Most students who start to learn quantum physics may be like those physicists working on the

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development of quantum theory: struggling to escape the influence of classical physics on their thinking and conceptualisation of subatomic behaviours. Historically, the concepts in the ‘old’

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quantum theory usefully served as an ‘intermediary’ language (Kaper & Goedhart, 2002) on the road to the ‘new’ quantum theory. A hypothesis for this research, derived from the literature, is that this classical intermediary language might limit and obstruct students’ QM conceptual development. Therefore, in order to understand how students mix and match the concepts of quantum physics with those of classical physics to describe and explain the subatomic behaviour, the second research question is:

2. What are the core concepts and conceptual relationships that constitute students’


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mental models of an electron wave at different conceptual stages?

Another research question has to do with interpreting why students have the ideas they do have. Models of conceptual change from the psychology of learning (e.g. from Piaget’s, diSessa’s and Thagard’s) can be used to inform this research question. The third research question is:

3. What are the key factors that best describe the development of students’ mental models of an electron wave at different conceptual stages?

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This paper is divided into four parts. The beginning of this paper reviews research on learning

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QM and on conceptual change. The second part introduces research methodology employed in this paper. Third, research findings from the result of data analysis are provided. In the final part

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of this paper, the authors suggest a new approach for teaching QM and a new view on meaningful learning based on the research findings.

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REVIEW OF THE LITERATURE

The review of the literature begins with the principal problems in the learning of quantum mechanics (QM). Then the discussion turns to the general issues of learning. The review of learning addresses how people change concept use, beliefs and ideas.

Research on Learning QM

The traditional approach of teaching QM has been criticised (Fischler & Lichtfeldt, 1992; Cuppari, Rinaudo, Robutti & Violino, 1997; Michelini, Ragazzon, Santi & Stefanel, 2000).


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Following the historical development of QM, the traditional approach seems to cause problems by reinforcing classical concepts in students’ QM concepts. For example, the problem in the learning of QM, pointed out by Fischler & Lichtfeldt is that the concepts in the planetary model provide powerful explanations and therefore produce more difficulties when students need to modify the ideas in the planetary model. For students’ QM ideas, Ireson (2000a) reported on the behaviour of the electrons that remain problematic in students’ QM ideas after they have studied quantum phenomena, such as ‘conflicting quantum thinking’ or ‘conflicting mechanistic thinking’. According to Ireson’s investigation, there seems to be conflicts between

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students’ quantum thinking and their classical mechanical thinking. The strange behaviour of the electrons has caused many problems in the learning of quantum mechanics and has given

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rise to a lot of concern.

Two possible solutions, different from the traditional approach to teaching QM, have been proposed. First, Fischler & Lichtfeldt (1992) suggested that, if we want to prevent students

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from attempting to understand the quantum phenomena in terms of their pre-concepts of

premises:

1. Reference to classical physics should be avoided.

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classical physics, a new approach for teaching QM should be guided by the following basic

2. The teaching unit should begin with electrons (not with photons when introducing the photoelectric effect). 3. The statistical interpretation of observed phenomena should be used and dualistic descriptions should be avoided. 4. The uncertainty relation of Heisenberg should be introduced at an early stage (formulated for ensembles of quantum objects).


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5. In the treatment of the hydrogen atom, the model of Bohr should be avoided. (p. 183-184)

Fischler & Lichtfeldt (1992) also developed a new teaching unit based on these premises. In an evaluation of the teaching unit, Fischler & Lichtfeldt concluded that large shifts between pre-test and post- test took place within the range of students’ conceptual patterns from the pattern of ‘circular orbit’ (Bohr’s planetary model) to the pattern of ‘localization energy’ (connected with the Heisenberg uncertainty principle).

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In addition to Fischler & Lichtfeldt’s approach, a second solution to the problem of learning

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QM came from the development of a new quantum atomic model – ‘Electronium’ – by the German scholar Friedrich Harrmann in 1990. The idea of Electronium is similar to

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Schrodinger’s interpretation of wave function – charge density. Niedderer & Deylitz (1998) and Budde, Niedderer, Scott & Leach (2002a) have applied Harrman’s ‘Electronium’ model to

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develop a new teaching unit and replace the ‘probability’ model proposed by Max Born in 1926.

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Budde et al. (2002a) argued that the concepts in the ‘probability’ model have no ‘resonance’ with students’ pre-conceptions in classical physics and therefore some learning difficulties

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associated with the quantum atomic ‘probability’ model occur. Different from the ‘probability’ model, the ‘Electronium’ model is developed to ‘resonate’ with students’ pre-concepts in classical physics. In other words, instead of avoiding reference to classical physics suggested by Fischler & Lichtfeldt, the ideas in the Electronium model are developed including students’ classical concepts.

The new teaching model tries to fit between the ‘Electronium’ model and students’ pre-conceptions such as the concept of orbit. However, there are two problems with using the


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Electronium model. First, historically Schrodinger’s idea that electron should smear out in the space is only at the ‘transitional’ wave mechanics stage. The Electronium model is more easily accepted by students because the ideas in the Electronium are closer to classical concepts than the ideas in the ‘probability’ model at probabilistic wave mechanics stage. If we replace the idea of ‘probability’ with the idea of Electronium, it seems that we are moving backwards to classical physics. The ‘Electronium’ could be problem in that it exists as a ‘hybrid model’ (Gilbert & Boulter 1995), not accurately fitting CM or QM conceptual contexts.

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Second, Budde et al. (2002a) discarded the ‘probability’ model because it has no resonance with students’ pre-conceptions. However, since the ideas in QM are so different from classical

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physics, it seems abnormal to expect that learning quantum mechanics should resonate to students’ classical ideas. In addition, cognitive conflict between the new knowledge

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(probability) and students’ old ideas (classical concepts) should be expected if there is to be cognitive growth. A Piagetian model of learning would claim ‘you need to get it wrong to get it

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right’ (accommodation). Taking this stand the challenge is one of designing a beneficial kind of

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‘wrongness’ into learning programmes.

In addition, Budde criticised the traditional teaching model – the ‘probability’ model – by saying that the probability model still results in students retaining a classical, mostly planetary orbit, atomic model. However, it is interesting to note that the development of the Electronium model was based on students’ preconceptions in classical physics. Therefore, it might be expected that students will also retain some classical ideas after they have learnt the Electronium model. Evidence in Budde’s research which shows that students might still hold the classical ideas after they learn the Electronium model is in Budde’s text:


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“…they [students] referred to the concrete charge distribution they tended to emphasize that the charge was more widespread, or distributed further from the nucleus, in higher stages with higher energy. It is therefore concluded that this conception shows strong congruent resonance because it is linked to the preconception that the electron jumps into a higher (in the sense of more distant from the nucleus) orbit, or shell, if energy is added.” (Budde et al. 2002b, p. 208)

In summary, a big problem in the learning of QM is that students still hold classical ideas when they are asked to develop QM concepts. Two completely different solutions to this problem

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have been proposed. One approach suggested by Fischler & Lichtfeldt is that reference to classical physics should be avoided when teaching quantum mechanics. In contrast, the other

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approach is to use the Electronium model which is based on students’ preconceptions in classical physics. The discussion above has pointed to how both approaches still have some

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problems. So neither might be any superior to the traditional historical approach to teaching QM. Therefore, a more detailed investigation of how students’ gradually develop QM concepts

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is still needed. In order to frame this investigation, theories about conceptual change learning

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must be considered.

Research on conceptual change

Students’ prior knowledge is sometimes described as “preconceptions”, “misconceptions”, or “alternative conceptions” (Clement, 1982; McCloskey, 1983; Minstrell, 1982). More recently prior knowledge in physics is called “intuitive knowledge” (Smith, diSessa & Roschelle, 1993; diSessa, 1993) for it is so robust and resistant to change. In order to deal with the preconceptions in the learning of physics, there is a debate between revolutionary and evolutionary assertions of conceptual change that results from two different views on prior


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knowledge. The first group, relevant to the epistemology of Kuhn’s (1970) scientific revolution or Lakatos’ (1978) change of research programme, regards the process of conceptual change as theory change in terms of weak restructuring or radical restructuring (Piaget, 1970; Posner, Strike, Hewson & Gertzog, 1982; Carey, 1985; Thagard, 1992) in that the prior knowledge behaves like an obstacle to the development of new concepts. In contrast, close to Laudan’s (1984) piecemeal view of scientific change, the second group looks at the process of conceptual change as transition of knowledge systems in that the prior knowledge is productive and effective for the development of new concepts (diSessa, 1988; Smith, 1992; Smith et al., 1993;

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diSessa, 1993).

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Based on Kuhn’s (1970) and Lakatos’ (1978) epistemological views on the growth of scientific knowledge, two kinds of knowledge change are categorised in science learning research – weak

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restructuring and radical restructuring (e.g. Carey, 1985). By applying this revolutionary view of conceptual change in the growth of scientific knowledge to education, Posner, Strike,

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Hewson & Gertzog (1982) suggest that four conditions must be met when learners change from

1. There must be dissatisfaction with existing conceptions; 2. A new conception must be intelligible;

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existing ideas to new ones:

3. A new conception must appear initially plausible; 4. A new concept should suggest the possibility of a fruitful research program. (p. 214)

The suggestion by Posner et al. is similar to Piaget’s in that, in order to develop new concepts, learners need to be dissatisfied with their old ideas. Two key terms that reflect revolutionary


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change in Piaget’s genetic epistemology are assimilation and accommodation. By assimilation, Piaget refers to the process whereby the learner applies his/her schema to the task unproblematically, and thereby extends via weak restructuring the range of tasks for which the operation of that schema brings about a satisfactory result. In contrast, accommodation refers to a problematic operation of a schema in pursuit of a solution, whereby the disequilibrium or dissatisfaction leads to a radical restructuring of the schema. The schema is transformed rather than augmented (Piaget, 1970).

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The alternative to revolutionary conceptual change is evolutionary conceptual change. diSessa’s model for learning challenge Piaget’s and Posner et al.’s theories. First, diSessa

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challenges the schema theory by presenting the concept of ‘intuitive knowledge’ in physics education. In order to establish an epistemology of physics, diSessa (1983) emphasises the role

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of intuitive knowledge in physics – such as experiential primitives which are relatively independent of context – as the root of many explanations and justifications. diSessa called

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them phenomenological primitives or p-prims for short and claimed that p-prims are the central

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elements of intuition in thinking about the natural world. As diSessa’s formal scheme of intuitive knowledge is domain-specific in the context of physics, it challenges the

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domain- general orientation of schema theory. diSessa derived his p-prims, such as Ohm’s p-prims and springiness, by means of a set of clinical interviews of undergraduate physics students while solving a set of specially designed motion physics problems. For the cognitive mechanism, diSessa (1993) claimed that p-prims occupy ‘mid-levels’ between the sensory schemata at the lower levels and the world of ideas, named concepts and categories at the higher levels. Since p-prims are located at the mid-levels of cognitive mechanism, they are held to be very important elements students use to develop their higher-level concepts of physics. Therefore, p-prims – new or old – are considered to be useful foundations for students to build


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up their concepts of physics.

The second aspect of diSessa’s evolutionary model of conceptual change is the concept of ‘knowledge system’. Instead of replacing the old ideas with new ones, Smith, diSessa & Roschelle (1993) provide an analysis of knowledge ‘in transition’. They propose a constructivist theory of learning about how old ‘knowledge systems’ evolve into new ones. In other words, Smith et al. argue that people could keep both old and new p-prims in their knowledge systems without discarding any ideas, they just learn when to use the p-prim. In the

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process of learning, the knowledge systems are in transition and people may only adjust some relationship between new and old units of knowledge. Similarly, Hammer & Elby (2003) also

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claim that “successful learners tend to see physics as a coherent system of ideas, the formalism as a means for expressing and working with those ideas, and learning as a matter of

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reconstructing and refining one’s current understanding” (p. 54). Therefore, in the process of learning, people may only modify the connections between units of knowledge rather than

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change concepts.

In summary, the difference between diSessa’s p-prims and Piaget’s schemas is that for diSessa

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p-prims address the epistemology of physics and are domain specific. Piaget, on the other hand, developed a domain general account that could be represented with mathematical logic and would cover ALL areas of thought. diSessa’s contribution is that his p-prims provide a useful foundation to realise how beginning and more experienced students explain physics phenomena differently. diSessa’s p-prims also draw our attention to the mid-level of cognitive mechanism which behave as a connection between the low-level schemata of sensori-motor experiences and the high-level concepts in physics. In addition, diSessa also offers a theory of knowledge systems to question if there is actually conceptual change in the process of learning.


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The new knowledge systems might exist with the old ones at the same time rather than replace them.

So, for the problem of learning new/QM concepts caused by students’ old/CM concepts, this research is going to examine the progression in students’ concepts of an electron wave by means of Piaget’s and diSessa’s theories. The goal is to explore how students’ old/CM conceptual systems evolve into new/QM conceptual systems.

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METHOD

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In this research, three tools – questionnaire, card sort task and interview – are used as the data sources. To investigate the differences between physics novices’ and experts’ concepts of

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quantum physics, four different groups of physics students from undergraduate through master on to PhD levels were picked. All the undergraduate and masters level students in this research

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were studying at the same university. First year undergraduate students (U1) would have been

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introduced to concepts of basic quantum physics in their 3rd year in senior high school (equivalent to the 1st year A-level in England). Such instruction would have introduced the

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concepts of energy quantization, photon theory, planetary model and de Broglie’s matter wave. Third year undergraduate students (U3) extend their studies of quantum physics in their Quantum Physics class.

The third group is the 1st year master students (M1) who continued work on quantum physics in the Quantum Mechanics class. For the physics PhD students (PhD), the fourth group, they have passed the PhD entrance exam and some of them have passed the PhD candidate exam. Both of the exams include the subject of Quantum Mechanics and, therefore, these PhD students are


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expected to be more proficient and should possess clearer concepts in quantum physics than students in the other three groups.

Questionnaire design

The wave theory of matter was first proposed by de Broglie in 1923. It expressed the wavelength of a particle by the equation

=

h where p is momentum and h is Planck’s p

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constant (Gasiorowicz 1996, p. 14). For the wave-particle duality of the electron, according to Born’s interpretation, it means that the electron is a particle but its behaviour is described by the

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wave function. In other words, the electron is a particle but its behaviour is described in terms of probability. This is different from that of a classical particle.

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The questionnaire was designed to examine whether students could distinguish the wave

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explanation from the particle explanation of the electron without being influenced by the classical concept of the standing wave (the diagram in figure 2 appears in many physics

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textbooks, such as Fullick 2000, p. 544). The 1st probe directly asked students to describe their

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concepts of an electron wave.

[Insert Figure 2 about here]

The purpose of the 1st probe was to investigate students’ images of the wave behaviour of the electron. The 2nd probe asked students to explain if the electron moves along the shape of the standing wave. Physics students familiar with the details of QM were expected have a clear idea of the meaning of the standing wave here and give a satisfactory explanation of what the wave nature of the electron was. The purpose of the 2nd probe was to test students’ ability to clarify 13 ScholarOne, 375 Greenbrier Drive, Charlottesville, VA, 22901

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the difference between the classical and the quantum concept of wave. The 2nd probe was also used to find if students’ had any mix of classical and quantum concepts of wave.

Interview and card sort task

After students had completed the questionnaire, their responses were analysed and categorised into different groups by means of the 3-phase criteria from the historical development of quantum theories (see figure 1) – Old Quantum Mechanics stage (OQM), Transitional Wave

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Mechanics stage (TWM), Probabilistic Wave Mechanics stage (PWM). In order to explore more fully students’ concepts of classical and quantum physics, some students from different

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pattern groups were picked to be interviewed. In the process of interview, there are two different tasks for students: (1) give more detailed information for the questions in the

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questionnaire; (2) do a card sort task using concept cards.

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The first task asked students to provide more information about their ideas for the questions in

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the questionnaire. The main part of the interview began with the student being asked to depict the way of an electron wave. In the process of interview, there were two types of elicitation for

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this part. The purpose of the 1st elicitation explored students’ images of an electron wave (the student would be provided with a pen and paper in case s/he wanted to draw a diagram). After the student had given his/her answer to the 1st elicitation, the next elicitation investigated whether the student mixed the concepts of classical and quantum physics to describe the concepts of an electron wave. The purpose of the 2nd elicitation was to test the students’ ability to clarify the difference between the classical and the quantum concept of an electron wave.

After students had given their explanations, the second task asked students to do a card sort task.


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The card sort task was introduced as a result of the pilot and discussions on how to collect further data. Students were asked to organise and connect the concept cards to illustrate their ideas for the concepts of an electron wave. From the historical development of quantum theories (see figure 1), a breakthrough from the ‘old’ quantum theories to the ‘new’ quantum theories was when de Broglie proposed his theory of matter wave (Ireson, 2000b). Therefore, the concepts which appeared before de Broglie’s matter wave are classified as ‘classical’ concept cards while the concepts appeared after de Broglie’s matter wave are categorised as ‘quantum’ concept cards. The 5 classical concept cards (Electron, Energy, Standing Wave,

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Anti-node, Amplitude) and 4 quantum concept cards (Probability, Schrodinger Equation, Uncertainty Principle, Wave Function) were selected based on the fact that they appeared more

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frequently in the pilot study data. The interviewees were also provided with one less relevant card (Force) and two blank cards to add any concept excluded in the concept cards.

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The whole discussion during the card sort task was recorded with a tape recorder. Interviewees

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were asked to name concept cards as they picked. Then they were also invited to connect and

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explain the relationship between pairs of concepts they linked under the certain question – wave behaviour of the electron. In the process of the card sort task, the researcher tried not to give any

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indication of correct/incorrectness by reaction or verbal input. The interviewees were free to pick any concept cards or make any link between pair of concepts. The only interference occurred when the interviewees did not give clear explanations for the links they made. They were then asked to make their explanations as clear as possible. The tape recording of the card sort tasks were to be transcribed and used to reconstruct the concept maps for each interviewee.


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FINDINGS

Students’ mental models of an electron wave

The simplest way of showing progression in students’ mental models is to list all the patterns found in students’ questionnaire responses to the wave behaviour of the electron. By means of the criteria derived from the historical analysis of QM (see figure 1), students’ questionnaire responses are categorised into three conceptual stages – from OQM stage that students have

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almost no idea about the electron wave, through TWM stage in that students mix the concept of classical wave with that of an electron wave, on to PWM stage that students could tell the

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difference between classical mechanical wave and quantum probabilistic wave. The result shows that students at OQM stage work with the planetary model while students with PWM

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thinking begin to operate with the ideas of probability and uncertainty. In addition, students’ questionnaire responses reveal that TWM students’ responses mix classical and quantum

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concepts.

The result in the analysis of students’ questionnaire responses also shows that students in

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different stages of education (U1, U3, M1, PhD) could make the same sorts of errors. The data suggest that the difference between undergraduate and postgraduate students is in terms of number but not in terms of the types of errors. In other words, postgraduate students made the same types of errors as the undergraduate students did. However, more postgraduate students could provide the explanations closer to quantum ideas than the undergraduate students. So, in addition to probing students’ mental models of an electron wave, it might be useful to investigate students’ concept maps of the wave behaviour of the electron at different conceptual stages in more detail to gather information on how students make links between concepts.


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Furthermore, in looking at the connections between individual concepts in students’ concept maps, it is essential to find out the transformation of ‘core concepts’ from students at OQM stage through TWM stage on to PWM stage. This line of investigation is discussed below.

Core concepts and conceptual relationships

After categorising students’ questionnaire responses into three conceptual stages (OQM, TWM, PWM), readers’ attention was drawn to the statistical analysis of links in students’ concept

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maps between the three conceptual groups. The statistical analysis of concept maps shows that students in the OQM group made more links between classical concepts while students in the

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PWM group made more links between quantum concepts. For the concept maps from OQM through TWM on to PWM stages (see appendix), the transition of knowledge systems for an

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electron wave evolves from the concept of a classical wave, through that of mixing mechanical wave with electron wave, on to a probabilistic wave. When talking about the wave behaviour of

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the electron, OQM students seemed to focus more on the classical concepts of mechanical wave

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such as the relationship between energy and amplitude. For the TWM students, similarly to OQM students, they also focused on the classical concepts by making many connections

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between classical concepts. However, TWM students did have more quantum concepts than OQM students. But, TWM students had a problem in that they still did not give up the concept of a classical wave by saying “electron should move along the shape of the standing wave”. For PWM students, they seemed to have given up the concept of a classical wave and focused more on the quantum concepts. Therefore, the novice students (OQM & TWM students) seem to assimilate the concept of ‘an electron wave’ to the concept of ‘a classical wave’ while the expert student (PWM students) has accommodated the wave behaviour of the electron to the concept of ‘a probabilistic wave’.


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Generally, when looking at students’ concept maps from OQM through TWM on to PWM stages (see appendix), it is essential to note that, in the process of conceptual change, some links between concepts get stronger while others fade away. If one only compares the maps at OQM stage with the maps at PWM stage without looking at TWM stage maps, then one might conclude that conceptual change is a revolutionary process where students replace the old conceptual system with a new one. However, when taking the maps at TWM stage into consideration, the evidence shows that conceptual change is a process in that students gradually

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add or delete some links between concepts. Progression then looks more like an evolutionary/transitional process between knowledge systems rather than a

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revolutionary/holistic process between paradigms. Students need time to build new links between quantum concepts and fade away old links between classical concepts.

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Key factors in the development of mental models

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In the process of interview, students were required to compare the difference between the a classical wave (based on classical concepts) and an electron wave (based on quantum concepts).

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When students were invited to interview, students were provided with two different contexts in order to test if they can tell CM from QM concepts. They need to give detailed information about the wave behaviour of the electron in their mental models. Students’ oral responses to the questions are analysed in the order of three conceptual stages. Generally, OQM students can only apply the concept of a classical wave but they do not have much idea about the concept of an electron wave:


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OQM stage

“The electron wave…I think it is like the vibration of a common wave, such as the vibration of a string wave or a water wave. It is very difficult to imagine what the electron is like. Probably the electron will do a little vibration around the orbit. I think the electron wave is similar to a longitudinal wave produced by the vibration of an electron.”…“If the electron should vibrate like a wave…they [electron wave and standing wave] all behave like vibration…Therefore, the orbit for the electron should be like the shape of a standing wave”…“Why should the electron

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vibrate like a wave? It might be that there is a fixed force between the nucleus and the electron…In other words, the electron must be at some fixed places [orbit] around the nucleus.

positive charge.”(M1-02)

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The electron can’t escape from the nucleus in order to keep the negative charge equal to the

From OQM student’s interview transcript, M1-02 describes the electron wave based on her

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sensori-motor experience of displacement with squashing and stretching. M1-02 also points out

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that the electron will vibrate like a wave because of the force between the electron and the nucleus. M1-02’s explanation of an electron wave gives another example of how students make

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improper metaphorical projection from bodily experience of ‘longitudinal wave’ to the abstract concept of an electron wave. In addition, M1-02 seems to successfully assimilate the concept of an electron wave to her old concept of a classical wave by saying: “the electron wave is similar to a longitudinal wave produced by the vibration of an electron”. It is a case of subsuming the problem of the trajectory of the electron under the two weakly held schemas of central force and mechanical wave.

From diSessa’s viewpoint, M1-02 uses the concept of force to explain the creation of an


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electron wave and thinks that the electron should vibrate like a wave because “there is a fixed force between the nucleus and the electron”. In this case, M1-02 is operating a p-prim of ‘Force as a mover’ to explain the creation of an electron wave. Furthermore, M1-02 describes the electron wave based on her physical experience of displacement with squashing and stretching. So, a new p-prim of a ‘longitudinal wave’ could be recognised from M1-02’s description of an electron wave. In this case, there are at lease two p-prims in M1-02’s knowledge system.

TWM stage

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“My idea of a matter wave is that, if we observe microscopically, every object should be

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described by wave function in terms of probability. The problem is that the effect of wave function in the microscopic world is obvious while the effect of wave function in the

er

macroscopic everyday life is not easy to notice. For example, when we pitch a baseball, there is a matter wave with the moving baseball but we can’t observe. If we can go into the microscopic

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world, we will see that the baseball is not moving along a straight line. It will move along the

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curve of a matter wave such as a sine wave.”…“If it is not the standing wave, the model of the hydrogen atom will collapse because of the decreasing energy. That is the reason why the

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electron can move along a fixed circular orbit by the Coulomb force without emitting energy.”…“If we use the standing wave to describe the electron wave, at least, we can be very sure that the hydrogen atom can keep its energy from decaying. So, the electron will not emit energy if it moves by means of a standing wave.”(U3-07)

For the development of the concept of an electron wave, the dependence on the sensori- motor experience of a classical wave is especially evident in U3-07’s explanation. It is important to note that, even though U3-07 applies the concept of probability to interpret wave function,


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U3-07 still sticks to the classical wave model by saying that a baseball “will move along the curve of a matter wave such as a sine wave” if one can observe microscopically. The bodily experience of a classical wave seems to have distorted the conceptual development of an electron wave. In other words, U3-07 tries to employ the concepts of wave function and its probabilistic interpretation but still can not get away from the concept of a classical wave. U3-07 still tries to assimilate the concept of an electron wave to his old concept of a classical wave by saying that the baseball “will move along the curve of a matter wave such as a sine wave”.

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From diSessa’s perspective, even though U3-07 has some ideas of wave function and

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probability, he still relies very much on his physical experience of a classical wave to explain the electron wave. Students at TWM stage demonstrates the possibility to keep both CM and

er

QM concepts at the same time without conflict. In U3-07’s explanation, the physics is wrong. U3-07 thinks that, “if we can go into the microscopic world”, one will see that the electron

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moves along the curve of a wave rather than a straight line. It seems that U3-07 is operating a

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p-prim of ‘a vibration on the proceeding string wave’ or a p-prim of ‘a float on the surface of a water wave’ to interpret the concept of an electron wave. These two physical experiences are

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new p-prims which are not in diSessa’s p-prims (diSessa, 1993). It is suggested that, in order to extend the p-prims for physical experiences with a classical wave, further research needs to be carried out with a set of clinical interviews.

PWM stage

“I think it is a concept of probability. For example, when the electrons pass through a single slit, you can observe the phenomenon of interference fringes. It indicates that the electrons will


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more likely appear at some specific places. In fact, the concept of an electron wave should be close to the concept of probability.”…“I think it is not correct to say that the electron wave is some kind of a standing wave around the orbit. The concept of a standing wave around the orbit only helps us to understand why the electron could only exist on some fixed orbits…I think if we really want to explain the concept of an electron wave, we need to go to Schrodinger wave equation. I can’t accept that the electron should move along the shape of a standing wave.”…“From solving Schrodinger wave equation, you can get wave function which represents the concept of probability. If the concept of a standing wave on the orbit is correct,

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there will be some nodes on the orbit. However, it is not the case.”(PhD-03)

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PhD-03 clearly points out that the concept of an electron wave should be interpreted as ‘appearing’ with indication that “the electrons will more likely appear at some specific places”.

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PhD-03 seems to develop the concept of an electron wave based on the appearing/finding sensori-motor experience – presence-and-absence. Obviously, M1-02’s and U3-07’s wave

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models is quite different from PhD-03’s notion of presence-and-absence which is at the heart of

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the Probabilistic Wave Mechanical (PWM) explanation. The data here does support the idea that different level learners are using different sensori-motor experiences. From the discussion,

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it can be concluded that these students rely on sensori-motor experiences to form the basis of the mental models that are the key factor of their schema. PhD-03 has accommodated the concept of an electron wave to that of probability by indicating that the electron wave should be interpreted as the concept of probability rather than the concept of a classical wave. Therefore, the data here does support the notion of variety in accommodation and assimilation.

By means of diSessa’s perspective, similar to U3-04, PhD-03 also applies the probabilistic p-prim to explain the wave behaviour of the electron. PhD-03 gives an experimental example of


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electron diffraction to explain the concept of an electron wave. PhD-03 clearly points out that the electron wave is a concept of probability which “indicates that the electrons will more likely appear at some specific places”. It seems that PhD-03 also operates the physical experience of presence-and-absence to explain the electron wave. Once more this physical experience of presence-and-absence is a new p-prim which is not in diSessa’s p-prim (diSessa, 1993). Again it suggests that further a survey needs to be done in order to explore and delineate the p-prim of ‘Presence-and-absence’ which lies at the heart of the Probabilistic Wave Mechanical (PWM) explanation.

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It is also important to note that PhD-03 points out “it is not correct to say that the electron wave

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is some kind of a standing wave around the orbit. The concept of a standing wave around the orbit only helps us to understand why the electron could only exist on some fixed orbits”. It is

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evidenced in this transcript in that PhD-03 has both classical and quantum p-prims in her QM knowledge system. PhD-03 is capable of switching between them which depends on what

vie

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situation she meets.

According to the interview transcripts analysed by using Piagetian and diSessa’s viewpoints,

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three claims are made from this research:

1. Students rely on sensory experiences to form the basis of the mental models that are the key factor in their schema. The sensory factor of the mental models in the schema derive from one or more of a limited set of physical experiences. 2. Students who hold on to use different mental models operate different physical intuition – p-prims. New/QM and old/CM p-prims can exist in students’ knowledge systems at the same time.


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3. The less experienced students (OQM or TWM stage) keep both CM and QM concepts without being aware of conflict while the more experienced students (PWM stage) demonstrate the ability to switch between CM and QM knowledge systems to explain phenomena in different physics contexts.

SUGGESTIONS

Recall the problem in the learning of QM that students usually mix classical with quantum

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concepts to interpret the subatomic behaviour and that two different teaching approaches have been proposed in order to help students understand QM concepts. The first approach asserts that

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reference to the classical physics should be avoided when teaching QM (Fischler & Lichtfeldt, 1992). The second approach (Budde et al., 2002a) advocates that QM teaching should include

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students’ understanding of classical physics and propose a hybrid atomic model – Electronium. However, the major findings of this research suggest that the real problem is one of learning

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rather than the large problem mentioned above – “don’t use CM” or “do use CM”. The learning

w

vie

problem approach suggests that the problems in the learning of QM should be re-orientated to:

1. some students develop QM concepts based on improper sensori-motor experiences/p-prims;

2. some students keep both classical and quantum concepts in their minds without being aware of any conflict.

Therefore, any recommendations for teaching frameworks of QM need to attend to the evidence of learning. Based on the major findings discussed, a new teaching framework for QM is proposed which focuses on (1) how to help students develop QM concepts based on proper


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sensori-motor experiences/p-prims, and (2) how to facilitate and develop the ability of students to progress from the old/classical to new/quantum knowledge systems (Ke, 2003). The structure of the teaching framework is organised along three factors. Each of the three factors is discussed below, in turn.

Experiments and experiences with quantum phenomena

The analysis of students’ interview transcripts suggests that students develop their QM mental

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models based on sensori-motor experiences/intuitive knowledge. An important principle which seems to dominate students’ understanding of quantum mechanics is the primacy of first hand

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experience – Piaget’s sensori-motor experience or diSessa’s intuitive knowledge. Less experienced and more experienced students employ different sensori-motor experiences to

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develop their QM ideas. In addition, the research findings also suggest students at different conceptual stages (OQM, TWM, PWM) explain subatomic behaviour through different

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p-prims. Therefore, the kind of physical experiences teachers provide and the range of

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opportunities that are made available to discuss, interpret or otherwise share experiences can decide whether students can accommodate their old ideas to the new concepts of QM.

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Physics students in the traditional approach of teaching QM, which emphasises practising mathematical formulas to solve problems, lack exposure to QM sensori-motor experiences. Hence, they are at a loss to develop the QM mental models based on their sensori-motor experience in classical physics. Therefore, the first recommendation for teaching QM is to provide students with experiences of QM phenomena. The initial emphasis should be on performing and/or examining experiments rather than going straight into teaching on the symbolic Schrodinger equation and wave function.


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Applying both CM and QM models to explain QM phenomena

In this research, students whose mental models were categorised at the OQM stage made more links between classical concepts while those students whose mental models were categorised as at PWM stage tended to make more links between quantum concepts in the card sort tasks. In other words, from OQM through TWM on to PWM stages, CM links gradually fade away while QM links get stronger. When taking the maps at TWM stage into consideration, the

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intermediate stage shows that conceptual change is a painstaking process in that students gradually add or delete some links between concepts. Progression then looks more like an

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evolutionary step by step process. The evidence does not support holistic change. Students need time to build new links between quantum concepts and fade away old links between classical

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concepts.

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The evidence from this research study challenges the view of science education scholars (e.g.

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Fischler & Lichtfeldt,1992) who have suggested that references to classical physics should be avoided when teaching the concepts of QM. Note that the U1 students who have not yet been

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exposed to much schooling in QM made many links between classical concepts. In other words, even though physics teachers might avoid mentioning the classical concepts in their expositions on QM, the links between classical concepts in students’ minds are already there before they come into the QM classroom. In order to achieve the goal of meaningful learning, Ausubel reminds us of the role of prior knowledge and asserts that teaching should start from what students have known. Furthermore, diSessa also highlights the influence of intuitive knowledge on the learning of physics. Therefore, I would argue that the point is not to avoid CM concepts – prior/intuitive knowledge – in the QM classrooms. Rather, physics teachers need to find a way


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to help their students gradually, but efficiently, switch between CM and QM knowledge systems. That is, knowledge system is flexible and the individual should know when to use links in classical contexts and when to use links in the quantum contexts.

Therefore, I suggest that a possible method is to provide instructional contexts that push students into cognitive conflict. For example, students would be asked to apply both CM and QM models to explain quantum phenomena. As a result, they will find that some of the old links between classical concepts can no longer account for the evidence and thus are no longer useful

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for explaining the subatomic behaviour. They will need to construct and employ new links using quantum concepts to account for the evidence. Here the use of QM phenomena to

paradigm change.

er

Pe

facilitate students’ conceptual change is similar to Kuhn’s (1970) use of anomalies to cause

Looking at the history of conceptual change in science, the ability of successful explanations to

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account for phenomena seems to be the key to building up the strengths of links in the scientific

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context (Thagard, 1992). In other words, a new system must be frequently and successfully applied to explain phenomena in order to strengthen the links in this new system and therefore

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replace the old system. In the context of learning the concepts of quantum physics, it is similar to the conceptual change in science that students should be given the chance to be dissatisfied with their prior/CM knowledge when they use the CM models to explain subatomic behaviour. After students meet difficulties in using CM knowledge to explain QM phenomena in the laboratory, they should be encouraged to frequently apply the QM rules explain quantum phenomena in order to build up and increase the strength of the rules that provides links between QM concepts. By doing so, student could gradually replace in their minds the conceptual system at OQM stage with the conceptual system at PWM stage.


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Discussing the relationship between CM and QM physics models

The evidence in students’ concept maps shows that students could possibly keep both CM and QM knowledge at the same time without perceiving any conflict between them. However, even though physics students could hold both CM and QM ideas in their minds, the analysis of students’ interview transcripts suggests that less experienced students only stick to CM models to explain subatomic behaviour but more experienced students can switch between CM and

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QM models in different situations. Therefore, the recommendation is that after applying CM and QM models to explain subatomic behaviour and experiencing cognitive conflict, students

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should be encouraged to discuss and critically examine the relationship between CM and QM models. Instead of avoiding CM concepts, the old/classical knowledge system could be useful

er

and productive for students to develop a new/quantum knowledge system (Smith, diSessa & Roschelle, 1993). The suggestion is that some elements or some relationship between the

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elements remain unchanged when the old/classical knowledge systems evolve into the

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vie

new/quantum knowledge systems.

By asking students to compare the different contexts between CM and QM, the realist view that some students use could be modified. It would seem that conceptual change learning rooted to flexible knowledge systems would encourage students to be more instrumentalist in their thinking so that they can employ different models in different situations. For this point, Giere (1988) has emphasised that successful understanding lies in the ability to deploy different models in different given contexts. Giere reminds us of the need to develop the ability to employ models in different contexts. In this research, it is important for students to develop effective QM knowledge system – including concepts, math, evidences and reasoning – at their


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disposal and develop the ability to flexibly switch between CM and QM knowledge systems in different contexts.

In summary, then, there are several recommendations to make about the teaching and learning of QM. First, it would be more meaningful if students have direct practical experiences with phenomena that relate to wave mechanics, such as the phenomenon of electron diffraction. Through processes of relating evidence to explain QM phenomena, students can have the chance to test their intuitions against QM phenomena. If students build their QM mental models

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with the classical idea of orbits or waves, they will have difficulties when they are asked to explain the phenomena or predict what will happen next. Such difficulties are not to be deplored

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or lamented for they are the essence of learning. As a result, students can find out that some of their intuitions need to be modified in order to interpret the QM phenomena.

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Second, physics teachers need to provide more QM phenomena which are pivotal cases (Linn,

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2003) for students in the process of conceptual change. The concept of pivotal cases is similar

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to that of Kuhn’s anomalies that cause paradigm shift. In addition to the phenomenon of electron diffraction, there are many pivotal cases for the learning of QM. For example, the

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concept of an electron wave – quantum probability – shows up very clearly in radioactive decay that occurs when an unstable nucleus spits out particles and transforms itself into other nuclei. Another important experiment is the discovery by Stern and his collaborators (see Born, 1965) that molecular rays (of H2 and He) also show diffraction phenomena when they are reflected at the surfaces of crystals. Students’ CM knowledge systems can gradually evolve into QM knowledge systems when more and more pivotal cases come into their minds.

Third, with the many CM and QM physics phenomena, physics teachers can provide students


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with the chance to develop the ability of switching between CM and QM knowledge systems. One strategy is to help students be more instrumentalist so that they can recognise which knowledge systems (CM or QM) can be used to deal with the physics phenomena under certain situations. One way of achieving this goal is to get students to discuss the similarities and differences between CM and QM concepts in different physics contexts. As suggested by Smith et al. (1993), there is continuity between more experienced and less experienced students’ knowledge systems. In order to develop the ability to switch between knowledge systems in different contexts, physics teachers should help students reflect on their present CM ideas and

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find CM contexts for students’ existing knowledge. Then provide students with QM contexts and ask students to refine parts of their CM knowledge in order to explain QM phenomena. The

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goal is to provide a learning context that is supportive of the processes of evolution of knowledge systems and help students recognise the continuity – similarities and differences –

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between old/CM and new/QM knowledge systems.

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For meaningful learning, Ausubel (1968) has proposed a famous principle for educational

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psychology in that teaching should start from what students already know. In this research, I suggest that ascertaining what the learner already knows is only the first step. Based on that, it

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is important for students to have the chance to learn by switching between knowledge systems to explain phenomena in different contexts then the learner can learn to flexibly apply the appropriate theories to explain phenomena. For the learning of QM, some researchers argue “learning by avoiding” the old/CM concepts while other researchers claim “learning by mixing” the old/CM knowledge. Between them, I suggest a third way “learning by switching” between old/CM and new/QM knowledge systems and emphasise the necessity to provide students with different experiences in different contexts.


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REFERENCES

Ausubel, D. P. (1968) Educational Psychology: A Cognitive View. New York: Holt, Rinehart & Winston. Born, M. (1965) Atomic Physics. London: Blackie & Son Limited. Budde, M., Niedderer, H., Scott, P. & Leach, J. (2002a) ‘Electronium’: a quantum atomic teaching model. Physics Education, 37(3), 197-203. Budde, M., Niedderer, H., Scott, P. & Leach, J. (2002b) The quantum atomic model

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‘Electronium’: a successful teaching tool. Physics Education, 37(3), 204-210. Carey, S. (1985) Conceptual Change in Childhood. Cambridge, MA: Bradford Books.

Physics, 50, 66-71.

er

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Clement, J. (1982) Students’ preconceptions in introductory mechanics. American Journal of

Cuppari, A., Rinaudo, G., Robutti, O. & Violino, P. (1997) Gradual introduction of some aspects of quantum mechanics in a high school curriculum. Physics Education, 32(5), 302-308.

Re

diSessa, A. (1988) Knowledge in pieces. In G. Forman and P. Pufall (eds) Constructivism in The

vie

Computer Age (pp. 49-70). Hillsdale, NJ: Erlbaum.

diSessa, A. (1993) Toward an epistemology of physics. Cognition and Instruction, 10(2 & 3),

w

105-225.

Fischler, H. & Lichtfeldt, M. (1992). Modern physics and students’ concepts. International Journal of Science Education, 14(2), 181-190. Fullick, P. (2000) Physics. Oxford: Heinemann Educational Publishers. Gasiorowicz, S. (1996) Quantum physics. New York: John Wiley & Sons, Inc. Giere, R. (1988) Explaining Science – A Cognitive Approach. London: The University of Chicago Press. Gilbert, J. K. & Boulter, C. J. (1995) Stretching models too far. Paper presented at the annual


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meeting of the American Educational Research Association (San Francisco). Hammer, D. & Elby, A. (2003) Tapping epistemological resources for learning physics. The Journal of the Learning Science, 12(1), 53-90. Ireson, G. (2000a) The quantum understanding of pre-university physics students. Physics Education, 35(1), 15-21. Ireson, G. (2000b) A brief history of quantum phenomena. Physics Education, 35(6), 381-386. Kaper, W. & Goedhart, M. (2002) ‘Forms of energy’, an intermediary language on the road to thermodynamics? Part

. International Journal of Science Education, 24(2), 119-137.

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Ke, J-L. (2003) The progression in students’ concepts in quantum physics. Unpublished PhD thesis. King’s College, University of London, United Kingdom.

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Kuhn, T. (1970) The Structure of Scientific Revolutions. Second edition. Chicago: University of Chicago Press. Originally published 1962.

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Lakatos, I. (1978) The Methodology of Scientific Research Programmes. Cambridge:

Re

Cambridge University Press.

Laudan, L. (1984) Science and Values: The Aims of Science and Their Role in Scientific Debate.

vie

Berkeley: University of California Press.

Linn, M. (2003) WISE design for lifelong learning – pivotal cases. In Gardenfors, P. &

w

Johansson, P. (Eds.), Cognition, Education and Communication Technology. Mahwah, New Jersey: Lawrence Erlbaum Associates. McCloskey, M. (1983) Naïve theories of motion. In D. Dentner & A. Stevens (Eds.), Mental Models (pp. 299-324). Hilsdale, NJ: Lawrence Erlbaum Associates, Inc. Michelini, M., Ragazzon, R., Santi, R. & Stefanel, A. (2000) Proposal for quantum physics in secondary school. Physics Education, 35(6), 406-410. Minstrell, J. (1982) Explaining the “at rest” condition of an object. The Physics Teacher, 20, 10-14.


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Niedderer, H. & Deylitz, S. (1998) Introduction to atomic physics – a concept based on the Schrodinger equation. http: // didaktik . physik . uni-bremen . de / niedderer / projects / quanten / index . html # dow. Piaget, J. & Inhelder, B. (1966) The Psychology of The Child. London: Routledge & Kegan Paul. Piaget, J. (1970) Genetic Epistemology. New York: Columbia University Press. Posner, G., Strike, K., Hewson, P. & Gertzog, W. (1982) Accommodation of a scientific conception: towards a theory of conceptual change. Science Education, 66, 211-227.

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Reif, F. & Allen, S. (1992) Cognition for interpreting scientific concepts: a study of acceleration. Cognition and Instruction, 9(1), 1 -44.

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Reif, F. (1995) Millikan lecture 1994: understanding and teaching important scientific thought processes. American Journal of Physics, 63(1), 17-32.

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Smith, J. (1992) Old Learning Mechanisms Die Hard: The Problem of “Replacing” Students’ Conceptions. Paper presented at the Annual Meeting of American Education Research

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Association, San Francisco, CA.

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Smith, J., diSessa, A. & Roschelle, J. (1993) Misconceptions reconceived: a constructivist analalysis of knowledge in transition. The Journal of The Learning Science, 3(2), 115-163.

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Thagard, P. (1992) Conceptual Revolution. Princeton: Princeton University Press.


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Figure 1. The historical development of quantum theories 1900

1905

1913

1923

1926

1930

8.Born

10.Dirac

J. Thomson 1.Planck

2.Einstein

4.Bohr

5.de Broglie

(pumpkin

(Photon

(Planetary

(Wave Theory (Wave Mechanics) (Probability

(Energy

atom model) Quantisation) Theory)

Model)

7.Schrodinger

1927

of Particles)

Interpretation) Theory of QM)

3.Rutherford

6.Heisenberg

9.Heisenberg

(Nuclear Theory)

(Matrix Mechanics)

(Uncertainty

Classical Physics

(Relativistic

Principle) Old Quantum Mechanics

Transitional Wave Mechanics

Probabilistic Wave Mechanics

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Figure 2. de Broglie’s associated wave around the orbit

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Re APPENDIX: Concept maps from OQM through TWM to PWM stages


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Concept map by the OQM pattern student(U1-01)

1.Electron

Classical

Quantum

2.Energy

7.Probability C

C

C

C

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C

3.Standing Wave

C C

8.Schrodinger Equation

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4.Anti-node

5.Amplitude

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C

9.Uncertainty Principle

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10.Wave Function

6.Force

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N: not within any model; C: within classical model; Q: within quantum model C & C (n=6) (2-3) The energy of standing wave is stable. (2-4) Electron’s energy at anti-node is the largest. (2-5) Energy is proportional to the square of amplitude. (3-4) The energy at anti-node of the standing wave is the largest. (3-5) The amplitude of standing wave represents energy. (4-5) The amplitude at anti-node is the largest.

C & Q (n=2) (2-10) Wave function is used to describe energy. (3-7) Standing wave is the most probable explanation of electron’s wave. Q & Q (n=0)


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Concept map by the TWM pattern student(M1-12)

1.Electron

Classical

Quantum

2.Energy

7.Probability N

C N

r Fo 3.Standing Wave

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4.Anti-node

9.Uncertainty Principle

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5.Amplitude

Q


Q

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10.Wave Function

6.Force

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N: not within any model; C: within classical model; Q: within quantum model C & C (n=1) (2-3) Standing wave represents the conservation of energy.

Q & Q (n=2) (7-10) Wave function depicts the distribution of probability. (8-10) Wave function is derived from Schrodinger equation.

C & Q (n=2) (2-8) Schrodinger equation is used to calculate the relation between energy and position. (2-9) There is uncertainty in the measurement of energy.


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Concept map by the PWM pattern student(PhD-09)

1.Electron

Classical

Quantum

2.Energy

7.Probability

r Fo 3.Standing Wave

Q

Q

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4.Anti-node

9.Uncertainty Principle Q

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5.Amplitude

Q


Q

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10.Wave Function

6.Force

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N: not within any model; C: within classical model; Q: within quantum model C & C (n=0)

Q & Q (n=5) (7-10) Wave function depicts the probability distribution of finding the electron. (8-9) Operators in Schrodinger equation should follow the rule of uncertainty principle. (8-10) Wave function is derived from Schrodinger equation. (9-10) The calculation of wave function can derive the relation of uncertainty principle.

C & Q (n=0) (5-7) The amplitude of an electron wave represents the concept of probability. (5-10) The amplitude of wave function represents the probability of finding the electron.
