Students' mental models of atomic spectra

Abstract

Mental modeling, which is a theory about knowledge organization, has been recently studied by science educators to examine students' understanding of scientific concepts. This qualitative study investigates undergraduate students' mental models of atomic spectra. Nine second-year physics students, who have already taken the basic chemistry and chemistry laboratory application courses, participated in the study. Semi-structured interviews were conducted with the participants. The analysis revealed that students had four types of coherently organized knowledge structures: the Scientific Model of Atomic Spectra (SMAS), the Primitive Scientific Model of Atomic Spectra (PSMAS), No Photon Model (NPM), and the Orbit Model (OM). Identified mental models indicated that students used some fundamental concepts interchangeably, and “electronic transition” and “photon energy” were the threshold concepts for students' scientific understanding of atomic spectra.

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Introduction

Mental modeling is a theory about knowledge organization. For this reason, a mental model can be defined as the representation of an individual's knowledge (Seel, 2006). Mental models are the knowledge structures, which are composed of coherent elements to explain phenomena (Didiş et al., 2014). The function of mental models is similar to computer simulations, because they allow simulations in learners' mind by processing inputs to predict outcomes (Taber, 2013). Thus, they allow individuals' reasoning and facilitate problem solving and as a result they have great importance in everyday reasoning (Gentner, 2002).

Mental modeling research in science education dates back to the 1980s, and two different books about mental models were published in 1983 (Gentner and Stevens, 1983; Johnson-Laird, 1983). The mental modeling framework was recently studied to examine students' understanding of different chemistry concepts such as galvanic cells (Supasorn, 2015), ionic bonding (Coll and Treagust, 2003), chemical bonding (Coll and Treagust, 2001; Coll and Taylor, 2002; Stefani and Tsaparlis, 2009), chemical compounds (Chittleborough et al., 2002), vapor pressure (Tümay, 2014), atomic structure (Harrison and Treagust, 1996; Park and Light, 2009), molecules (Harrison and Treagust, 1996), atomic and molecular orbitals, the Schrödinger equation, and hybridization (Stefani and Tsaparlis, 2009). Chittleborough et al.'s (2002) research investigating the constraints of mental models revealed that eight aspects were in action in students' understanding of chemistry. Out of the eight aspects, four of them were student-centered, such as appreciation of chemical representations, prior chemistry knowledge, lack of any mental model and lack of motivation, and the other four were course-centered, such as teaching and assessment styles, use of chemical representations, the amount of content and the speed of teaching the content. The researchers indicated that for the construction of mental models especially on the submicroscopic level concepts, students' understanding of macroscopic and symbolic representations was very important, because when the reality and the representations are misunderstood, they limit students' development of knowledge organization. In addition, some concepts, which are considered as “threshold” (Meyer and Land, 2003; 2006; Cousin, 2006; Park and Light, 2009; Talanquer, 2015), need to be fully understood by students for the construction of correct mental models and as a result, ‘conceptual troublesomeness’ may associate with these threshold concepts (Park and Light, 2009). Ke et al. (2005) stated that sensory motor experiences also had a role in students' mental models based on the classical interpretation of quantum theory and that other primitive knowledge elements were in the development of mental models of classical and quantum ideas.

Just as atomic structure is an important concept in science from Democritus to today, helping students to understand it is also important in science education (Park and Light, 2009). As science educators, we hope to help students construct coherently and correctly organized knowledge structures, i.e., mental models, about the atomic world, because they may support understanding, reasoning, and predictions (Gentner, 2002) about the other related concepts in quantum theory. However, while learning quantum theory, students may construct incorrect mental models causing incorrect reasoning. Investigation of incorrect models, as well as correct models, is important for revealing the learning process and creating materials that minimize the triggering errors. From the constructivist's perspective of learning, if we could learn more about students' knowledge structures, no matter correct or incorrect, we can help them construct scientific knowledge structures by revising or improving their previous knowledge structures. In light of the existing body of mental model research, in this study we aim to investigate how second-year undergraduate physics students understand atomic spectra, and how they organize fundamental concepts in quantum theory to explain this phenomenon. The research questions of this study are as follows:

• What are the students' mental models of atomic spectra?

• How is the structure of the mental models (i.e. are the concepts used in a model all un/scientific (pure) or a mixture of scientific and unscientific concepts (hybrid))?

• What do the mental models reflect about students' understanding of atomic spectra?

We focused on students' mental models of atomic spectra because this topic contains very important concepts that are used in the upper level science courses that teach the quantum theory, such as the concepts of photons, electronic states in an atom, electronic transitions, and atomic orbitals. In addition, mental models not only indicate whether individual concepts are understood scientifically but also whether these concepts are used in a coherent and correct way. Scientific knowledge organization is also important for future science teachers and professors in their subject matter knowledge when they teach in schools and universities.

Related literature

Pedagogical research about the atomic world

Quantum theory, which can be considered as a paradigm shift from classical to quantum perspective, introduced many new concepts, and these concepts are sometimes too abstract and counter-intuitive to understand. However, learning of these concepts is important for the further development of scientific ideas providing the development of science and technology. For this reason, many science educators aimed to examine upper level secondary, undergraduate, and graduate level students' understanding of the concepts of quantum theory in the context of the atomic world. These were: atomic and molecular orbitals (Stefani and Tsaparlis, 2009; Tsaparlis and Papaphotis, 2009), the Schrödinger equation, physical observables, measurement, wavefunction, Hilbert space (Singh, 2001; 2008a; Zhu and Singh, 2012), the Stern–Gerlach experiment (Zhu and Singh, 2011), atomic spectra (Ivanjek et al., 2015), angular momentum (Zhu and Singh, 2013), wavefunction, probability density (Emigh et al., 2015), energy levels and transition, wave particle duality, uncertainty (Gardner, 2002), probability (Sadaghiani, 2005; Wattanakasiwich, 2005), operators, observables, eigenvalues (Didiş et al., 2010), potential wells (Özcan et al., 2009), and the discrimination of classical and quantum notions (Müller and Wiesner, 2002; Baily and Finkelstein, 2009; 2010). In addition, recent research indicated that the development of software facilitating the learning of the quantum concepts may lower the barrier for students' conceptual understanding. Some of these media are: PhET Simulations for quantum tunneling, wave packets, covalent bonds, the photoelectric effect, the Stern–Gerlach experiment, Rutherford scattering, models of the hydrogen atom, blackbody radiation, etc. (McKagan et al., 2008); Visual Quantum Mechanics: for matter waves, potential energy diagrams, quantum orbitals, wavefuntions, probability, the uncertainty principle, electron diffraction, etc. (Zollman et al., 2002); QuVis Quantum Mechanics Visualization: for spin, photon, waves, etc. (Kohnle et al., 2015).

As far as the literature has found, students' scientific learning of quantum concepts is painful no matter what stage they learn, be it the secondary level to graduate. However, research-based courses and material developments like tutorials (Zollman et al., 2002; McKagan et al., 2008; Singh, 2008b; Baily and Finkelstein, 2009; 2010; Kohnle et al., 2015), revision of quantum theory curricula (Asikainen and Hirvonen, 2009; Carr and McKagan, 2009; Ivanjek et al., 2015; McKagan et al., 2010), exploration of students' conceptual understanding and problem solving by conceptual tests (Wuttiprom et al., 2009; Zhu and Singh, 2012; 2013; Dick-Perez et al., 2016), attitude and motivation (Didiş Körhasan, 2015) towards learning the quantum concepts are in the focus of physics and chemistry educators, who teach the concepts of quantum theory in upper secondary and university courses.

Mental modeling research

In addition to previous research examining students' understanding of the concepts in quantum theory, the following research about how students organized their knowledge of the concepts in quantum theory is also one of the focuses in both physics and chemistry education research, because students' abilities to solve the problems in a quantum mechanics test do not always demonstrate their understanding of the concepts (Ke et al., 2005). If new knowledge is consistent with the previous knowledge, mental models can facilitate learning (Gentner, 2002). For this reason how students construct and organize their knowledge in learning quantum theory was examined by researchers using the mental modeling framework, for example students' understanding of potential energy diagrams and probability (Bao, 1999), quantization of energy, light and angular momentum (Didiş et al., 2014), atomic structure (Ke et al., 2005; Park and Light, 2009), electromagnetic radiation, and Wien's law (Vadnere and Joshi, 2009).

One of the earliest studies using the mental modeling framework is the research of Bao (1999). Bao studied university physics students' mental models of the probability concept for classical and quantum mechanics. He developed his “Model Analysis” tool to make quantitative explanations of students' models. This tool includes two algorithms to examine students' mental models quantitatively. With the Model Analysis tool, by using students' answers in the test with multiple-choice questions, he identified students' model-based responses. In addition, by using these model-based responses, he constructed density matrices, which included the information about students' model states. Five force and motion questions of FCI (Hestenes et al., 1992) were used to determine physical models as “correct”, “incorrect” and “null” models. After the analysis, Bao explained the superiority of this analysis to the score-based (measurement with multiple-choice tests) analysis by indicating the loss of information in score-based analysis. He implemented this analysis to examine students' mental models in quantum mechanics. He first developed tutorials, conducted interviews, conceptual quizzes, and homework/exam questions, and then he developed a multiple-choice test to construct matrices. By the experience of students' difficulties in the classes, he examined students' conceptions of “potential energy diagrams” and “probability” topics. In his study, Bao identified three types of mental models of students for quantum mechanical concepts. These types were: (1) strong classical mechanical models; (2) hybrid models including correct information about quantum mechanical concepts by using classical mechanical reasoning; and (3) mixing models including both quantum mechanical and classical mechanical models at the same time.

Vadnere and Joshi's (2009) study is a considerably new study using Bao's tool (1999) that he developed in his dissertation to examine students' mental model; however, their design differed from that of Bao (1999). The researchers examined students' mental models of electromagnetic radiation, Wien's law, etc. with a weak experimental design. The researchers implemented a pre-test to 119 volunteer Standard 12 (corresponding to K-12) students. Software was then used to teach students these concepts. Afterward, the students were post-tested. In the analysis, the researchers used the “density matrix”, which was used in mental model identification by Bao (1999). They first defined three probabilities for students' mental models such as the expert model (E), the misconception model (M), and the null model (N) by considering students' answers. In the examination of the development of students' mental models, the researchers identified an increase in the probability of triggering the expert state, and a decrease in the null state in the post-test than in the pre-test. This way, they explained that this software was successful in the development of students' mental models. They suggested that the tools of quantum mechanics, which were the mathematical expressions used in the definition of quantum particles such as density matrix, could be used to analyze student learning.

Ke et al. (2005) aimed to investigate variation and progression in students' mental models of quantum mechanics phenomena. A cross sectional survey was carried out and the data were collected first with the implementation of a pencil and paper questionnaire to 140 students in different grades from undergraduate to PhD. Then, two-step interviews were conducted with the selected participants. In the first step of the interviews, participants reexamined their answers in the questionnaire and explained more about their answers. In the next step, they grouped and linked the concepts by using concepts cards with a card-sort task and they discussed the rationale behind their choices. The researchers identified three mental structures about atomic structure with the organization of quantum mechanical concepts such as electron, energy level, orbital, uncertainty, wave function, probability, etc. The researchers named students' mental models by considering the three stages in the history of quantum theory. The results were similar to Bao's (1999) findings about having classical and quantum interpretations. Students having the Early Quantum Mechanics (EQM) model had classical ideas about the electron behavior with fixed orbits; students having the Transitional Wave Mechanics (TWM) model considered the wave nature of the electron with a wave like path; and the last group of students having the Probabilistic Wave Mechanics (PWM) model interpreted the electron behavior by considering probability and uncertainty. The researchers indicated that most students had the EQM model that was based on classical mechanics. The TWM model was considered as a conceptual development with the change in the previous model and the PWM model was mainly observed in experienced students (PhD students). Finally, students had more mental models together, and this result indicated that the development of students' mental models was not linear (i.e. EQM and PWM may exist at the same time) and they improved their mental models.

The study by Park and Light (2009) also aimed to identify students' mental models of atomic structure and troublesome concepts that play a role in the construction of mental models. The researchers implemented a questionnaire to 633 first-year college students taking a general chemistry course. This course focused on fundamental chemical principles and seven of nine chapters were about the atomic structure in the textbook. The researchers not only accepted the quantum mechanical model of an atom as scientific models, they also considered other three models taught in high schools and previously identified by other researchers (Justi and Gilbert, 2000 as cited in Park and Light, 2009) as scientific models. They coded students' explanations in a 13-level hierarchical order by considering the chronological development of the concepts. These levels of understanding were used to classify the mental models of 20 students about the atomic structure such as the Particle model, the Nuclear model, Bohr's model, and the Quantum model. Pre- and Post-interviews were conducted with students. Three high achiever students had a single coherent mental model of the atomic structure in each of the interviews and the others had mixed mental models with fragmented knowledge elements. “Probability” and “energy quantization” were identified as the threshold concepts for understanding atomic structure and they were explained as threshold barriers distinguishing between students who got or did not get the target level of understanding.

Didiş et al. (2014) examined second-year undergraduate students' mental models of the quantization of physical observables and investigated context dependency of mental models in different contexts such as blackbody radiation, photoelectric effect, atom, etc. Researchers collected data during an academic semester (15 weeks) in a modern physics course constructing the basic ideas of quantum theory. Different sources of data were used to identify students' mental models about the quantization of light, energy, and angular momentum. These were basically interviews with 31 students, tests, and examination papers. Six types of context dependent mental models – the Scientific Model (SM), the Primitive Scientific Model (PSM), the Shredding Model (ShM), the Alternating Model (AM), the Integrative Model (IM), and the Evolution Model (EM) were identified. Some of these mental models were hybrid and some students had mixed model states (having more than one model) to explain the quantization phenomenon. Didiş Körhasan et al. (2016) also examined the influence of instructional issues on these mental models. By the analysis of almost 10 week classroom observation data (2760 min) and interviews, students' mental models were reinterpreted in terms of elements emerging from the learning environment. The results revealed that the manner of teaching (instructional methodologies and content specific techniques used by the instructor), the order of the topics, the familiarity with concepts, and peers played roles in the development of mental models.

Research assumptions

Mental models are complex mental structures which cannot be observed directly. For this reason, identification of mental models is not a simple process and it requires in-depth questioning (Taber, 2008). This research had some assumptions about the identification of mental models:

• Since mental models are abstract structures constructed in mind, identification of mental models is based on inferences from participants' explanations.

• Mental models are coherent knowledge structures and for this reason “coherency”, that is having a single conceptual framework (Didiş et al., 2014), among the concepts constructing a mental model is an important feature to specify the model.

• Mental models allow qualitative reasoning (Gentner, 2002) to explain experienced or hypothetical situations as well as physical phenomena.

• Mental models may not have firm boundaries. The elements of one mental model might be confused (Norman, 1983, p. 8).

Methodology

To investigate individuals' knowledge structures, asking for their mental models directly is not enough (Gentner, 2002). For this reason, indirect methods, such as interviews, tests, and questionnaires including conceptual questions and qualitative reasoning problems that reveal students' knowledge patterns, are used to infer students' mental models. Both qualitative and quantitative methodologies can be used to investigate students' mental models with different designs, and each has advantages and disadvantages in terms of the number of participants and the amount of time spent for investigation. This research examined students' mental models of atomic spectra by a qualitative approach, discovered students' scientific and unscientific conceptions related to atomic spectra, and identified the threshold concepts that limit knowledge organization.

Data collection

The participants were selected among the second-year undergraduate students enrolled in the physics program of a university. In total, nine students participated in the study, five of them were males and four of them were females. Participants had already successfully completed the introductory chemistry and chemistry laboratory courses, and they were taking the modern physics course. These nine participants were selected purposively among the 98 students and they participated in the research voluntarily. Because this was a qualitative examination of students' understanding and we did not aim to generalize findings, we delimited the participant selection by selecting almost 10% of the students. The selection criteria were to provide diversities among the participants in terms of students' interest in learning quantum theory as well as their academic performance. The level of students' interest was determined by the first author of this paper who had the opportunities to observe the students' participation in the modern physics course (e.g., how much they asked questions to the instructor in and after class and discussed related topics with their peers).

In order to identify students' mental models of atomic spectra, a test with four questions was developed (see Appendix I) and it was taken by the participants via one-on-one interviews. The questions were designed to reveal the participants' cognitive structures and to encourage them to conceptually explain the phenomenon of atomic spectral lines. Starting from general to specific, visible to invisible, concrete to abstract in the first question, students were asked to interpret the lines in the atomic emission and absorption spectra. In the second question, they explained some basic terms related to atomic spectra: energy with negative values, discrete energy levels, electronic transition, and photon emission. In the third question, they were asked how the transition of an electron between energy levels is related to the emission of a photon. In the fourth and last question, students made qualitative reasoning about the behavior of an electron and the resulting atomic spectra in a hypothetical situation where the electron in the hydrogen atom obeys classical mechanics rather than quantum mechanics. During the interviews, participants were requested to think out loud as much as possible. Each interview took almost one hour and was video recorded.

Data analysis

After the verbal data (i.e. the students' answers to the questions in Appendix I) were transcribed and matched with the visual and mathematical artifacts created during the interviews, the data analysis was conducted by using a code list. All codes emerged from the data. “Mutual exclusiveness” and “appropriateness to explain atomic spectra” were the criteria for determining the codes. A draft code list was presented to two external researchers (one of them is a physics educator and the other is a molecular physicist) for feedback. Both of the authors discussed about the coding, and definitions of the concepts in the code list were clarified and rewritten after resolving the discrepancies. The final code list is presented in Table 1. The first five of the six concepts are scientific and reflects the reality, while the last one, the orbit (O), is unscientific in the context of atomic spectra although it was used in Bohr's atomic model.

Table 1 The code list for the determination of mental models of atomic spectra

Code of concepts	Abbr.	Description
a Unscientific concept for the quantum atom model.
Bound electron	BE	An electron in an atom.
Discrete energy levels	DEL	Energy levels for an electron in the atom.
Spectral lines	SL	The colored lines in an emission spectrum or the dark lines in an absorption spectrum.
Photon energy	PE	Energy of a photon with a certain wavelength.
Electronic Transition	ET	Change of an electron's quantum state from one to another within an atom.
Orbit^a	O	Existence of an electron in a specific orbit.

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The mental models for atomic spectra were identified following the methodology used by Didiş et al. (2014). In the data analysis, participants' coherent use of these codes (i.e. the concepts) to explain the atomic spectra was important. For this aim, we focused on how students used the concepts coherently. If these concepts were meaningfully connected together to have a single conceptual framework, the observed knowledge organization was called a “mental model” (see Appendix II for sample coding). The proposed mental models are not students' definitions of atomic spectra. The students were not asked to answer the question “what is an atomic spectrum”. Instead, the mental models were identified by researchers based on inferences from participants' explanations to the features related to atomic spectra described in the questions listed in Appendix I. In the investigation of mental structures, both of the authors discussed and agreed on four mental models summarized in Fig. 1.

Fig. 1 Composition of mental models about atomic spectra. The triangle outlined by the dotted lines represents the Scientific Model of Atomic Spectra (SMAS); the shaded triangles represent the three unscientific models, and they are: the Primitive Scientific Model of Atomic Spectra (PSMAS), the No Photon Model (NPM), and the Orbit Model (OM). Ellipses on a triangle contain the concepts used coherently (but not necessarily correctly) in a mental model. The unscientific concept, the orbit (O), is in shaded ellipse.

Fig. 1 presents how the codes (concepts) are related to each other and used together to form coherent frameworks, i.e., mental models. For example, the coherent use of the concepts (we call them codes in the data analysis) bound electron (BE), discrete energy levels (DEL), electronic transition (ET), photon energy (PE), and spectral lines (SL) is called the Scientific Model of Atomic Spectra (SMAS), and it is outlined by the triangle with dotted lines.

Results

Students' mental models

Four different mental models of atomic spectra were identified; the Scientific Model of Atomic Spectra (SMAS) and the other three were unscientific: the Primitive Scientific Model of Atomic Spectra (PSMAS), the No Photon Model (NPM), and the Orbit Model (OM). Table 2 lists these mental models with the elements constructing the models.

Table 2 Mental models of atomic spectra

Mental models	Codes of concepts
	Scientific					Unscientific
	BE	DEL	SL	PE	ET	O
a Unscientific mental models of atomic spectra.
Scientific Model of Atomic Spectra (SMAS)	×	×	×	×	×
Primitive Scientific Model of Atomic Spectra (PSMAS)^a		×	×	×
No Photon Model (NPM)^a	×	×	×
Orbit Model (OM)^a	×		×			×

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Scientific Model of Atomic Spectra (SMAS). The Scientific Model of Atomic Spectra (SMAS) was observed in two of the nine students. In this model, students explained atomic spectra with the coherent use of BE, DEL, ET, PE, and SL. More specifically, the spectral lines (SL) arise from the transitions (ET) of an electron in an atom (BE) from a quantum state to another quantum state with a different energy level (DEL). Because a bound electron has discrete energy levels (DEL), the magnitude of the energy of the absorbed or emitted photon (PE) can be determined by the difference between these specific energy values. For this reason, spectral lines correspond to specific wavelengths. One of the two students explained the spectral lines as follows:

Interviewer (I): Let's look at this. [Q1: In an emission spectrum, what do the (colored) lines represent (in the visible region), or in an absorption spectrum what do the dark lines represent? Why do the lines occur? Why do they have different colors (in the visible region) in emission spectra? Why are they dark in absorption spectra?] Do you have any idea?

Student 8 (ST8): Yes I have. This is umm… In an atom, when an electron with a different n is falling down to the other n level, it emits a photon. This has a specific energy, for example passing from n = 3 to n = 2, the photon is emitted with an amount of eV energy between these energy levels. For this reason these lines occur.

I: Ok, then, (pointing to the emission spectrum in the figure) how does this dark area occur?

ST8: This is for example, here, umm…, in this dark area (pointing to the emission spectrum in the figure), a photon cannot be emitted. The magnitude of the energy that does not correspond to difference between two energy levels is dark here. Absorption is the reverse of emission. That means, the absorbed photons correspond to dark lines, and other areas are seen colored.

One of the important observations about the students displaying this model was the relationship between spectral lines and quantized energy levels. Another excerpt given below belongs to the other student who explains spectral lines as follows:

I: Is there any relationship between energy levels and the spectral lines? Is the distance between the spectral lines related to the energy levels?

ST9: Yes.

I: How?

ST9: Energy levels determine the distance between the lines. That means, for example, there are spectral lines. The transition of an electron occurs between the discrete energy levels in an atom and it determines the wavelength of the emitted photon. As a result of this, the energy difference between two levels (pointing to the figure) explains lines, and also the distances among the spectral lines.

—Some conversations are omitted here—

I: (pointing to the figure in Question 2) All right. Could you explain what Part d “each arrow” means here?

ST9: Umm… This is the transition of an excited electron to the ground state. Arrows explain this. It gives the energy as the difference of these energy values. Photon energy, which is hν, equals to the difference between these energy values.

This mental model is the target model for understanding atomic spectra, in which scientific concepts are understood and linked together correctly to explain spectral lines.

Primitive Scientific Model of Atomic Spectra (PSMAS). The Primitive Scientific Model of Atomic Spectra (PSMAS) was another model which was observed in two of the nine students. This model can be considered as a primitive scientific model because it includes the coherent use of three elements of SMAS, which are DEL, SL, and PE.

One of the two students with this model ignored the electronic transitions in an atom in the explanation of spectral lines (SL), and considered radiation independent of the behavior of an electron in an atom. More specifically, the emitted photon (PE) was described as a spontaneous event when energy levels (DEL) change. This student's explanation of the emission and absorption spectra is as follows:

I: Let's examine these spectra. [Q1: In an emission spectrum, what do the (colored) lines represent (in the visible region), or in an absorption spectrum what do the dark lines represent? Why do the lines occur? Why do they have different colors (in the visible region) in emission spectra? Why are they dark in absorption spectra?]

ST1: Umm… I remember that spectra are as the fingerprints of atoms. It is interesting. I like this description… Probably, the colored lines are because of radiation.

I: Why do we see different lines? Why are they lines?

ST1: Line structure is probably because of each color has specific character. For this reason they are different lines. But I have no idea about they are black in absorption spectrum (smiling).

I: Let's look at this (pointing to Question 2). You see “E” here in eV. What does it mean?

ST1: Umm… When an electron is jumping from one energy level to another, umm… For example when jumping from 2 to 1, energy changes due to them. We call “energy levels”…

I: All right! What do these “arrows” (pointing to part d in Question 2) mean?

ST1: Energy level changes. There is energy difference between them. Umm… It is radiation. We define radiation as “hν”. Then, we explain spectrum. Because the energies are different, different colored lights emitted naturally.

The other student who displayed this model also associated spectral lines with just the change of the discrete energy levels rather than connecting it with electronic transitions. Furthermore, this student did not understand the energy levels correctly and associated energies with orbits through mathematical manipulations as shown in Fig. 2.

Fig. 2 Explanation of energy levels in terms of orbits and integrals.

ST2: If we get a small part here (drawing electron orbits and pointing to them), I guess it might be wrong but, anyway, if we consider this small region (on the orbit), we do not see as a “curve” but we see it as a “line”, that is an energy level.

I: Do you mean “the small part is an energy level”?

ST2: Yes (smiling). I think, an energy level is, here (pointing to his drawing), umm, I do not understand that it is the energy of an electron or it is the energy of orbit! I do not know…

I: Ok, I am trying to understand what you mean.

ST2: From the beginning of the semester, it (electron) always changes energy levels and photon emitted or we send photon and it also changes energy level. I remember it. I remember from modern physics, but there were similar explanations in the chemistry course. (Smiling)… Umm… If we get a part from here (pointing to his drawing again)… Don't we integrate small parts in this way? For example, (drawing the integral graph), this is dz (smiling)… I know this… I am not sure about my knowledge. Umm… These lines (pointing to energy levels), they are energy levels of electrons. But there is something missing. I feel I am right.

PSMAS is the closest model to SMAS in terms of recognizing the relationship between photon energy (PE) and discrete energy levels (DEL) of an electron. However, the missing of BE and ET in the model reflects wrong knowledge organization and incorrect understanding of concepts. Students with PSMAS treated spectral lines as a property of an atom that is not connected to the behavior of the electron or any electronic transition.

No Photon Model (NPM). The most common model is the No Photon Model (NPM), since four students displayed this model. All of them were aware of DEL and SL; however, they thought that they were the same concepts. More specifically, students thought that an electron in an atom (BE) has energy levels (DEL) and these different energies correspond to the different spectral lines (SL). No transitions between energy levels or the concept of a photon was mentioned. hν was not considered as the energy for a photon, but the energy level that corresponds to a spectral line. One of the examples below explained students' association of spectral lines with energy levels:

I: Here are the spectra (pointing to the figure in Question 1). First one is an emission spectrum and the other is absorption. You see the colored lines between 400 nm and 700 nm in the emission spectrum. [Q1: In an emission spectrum, what do the (colored) lines represent (in the visible region), or in an absorption spectrum what do the dark lines represent? Why do the lines occur? Why do they have different colors (in the visible region) in emission spectra? Why are they dark in absorption spectra?]

ST7: Umm… They are… These colorful lines are energy levels.

I: Energy levels?

ST7: Yeah…

I: How do you explain, for example, why the dark area between the colored lines is large here and it is small here?

ST7: I am not sure but I think it is because of the difference between energy levels…

Another student gave a similar explanation on atomic spectra.

I: [Q1: In an emission spectrum, what do the (colored) lines represent (in the visible region), or in an absorption spectrum what do the dark lines represent? Why do the lines occur? Why do they have different colors (in the visible region) in emission spectra? Why are they dark in absorption spectra?] Can you explain?

ST5: Yes… It is related to energy levels.

I: How they are related?

ST5: Both are already discrete (smiling).

I: You see lines.

ST5: Yes, these lines have got to be energy levels (pointing to the spectrum).

I: Do you mean “this energy level corresponds to a line in the spectrum (pointing to the figures of Questions 1 and 2) and another energy level is another line in the spectrum”?

ST5: Yes, that's right.

An excerpt from the third student displaying this model showed how energy levels correspond to spectral lines as follows:

ST3: Umm… Energy level is inversely proportional to wavelength. That means, if it has a long wavelength, the energy is so little.

I: Do you mean here (pointing to the spectral lines) “energy levels are different”?

ST3: If a wavelength of this light is different (showing a line in spectrum) probably its energy level is different. This energy is hν (writing the formula in a box).

Orbit Model (OM). The Orbit Model (OM) is similar to NPM, and the main difference is that instead of the discrete energy levels (DEL) that correspond to the different spectral lines, it is the different orbits (O) where an electron resides in an atom (BE) that corresponds to different spectral lines (SL). Similar to NPM, photons were not mentioned in OM. hν was not considered as the energy for a photon, but rather, the student thought hν corresponds to a spectral line. The student's explanation is as follows:

I: [Q4: Suppose the electron in a hydrogen atom obeys classical mechanics rather than quantum mechanics. What would you expect to observe in the spectrum?]

ST6: Umm… I think about that… The electron will fall down to the nucleus…

I: How will they fall down?

ST6: Am I right? I am not sure but this is not a real event so you cannot see such a thing (smiling).

I: What about spectrum?

ST6: If you ask me the spectrum, I think these colors are because of orbits, but I do not know. Spectrum occurs probably due to the distances of the orbits. That means, if it is real, we see these spectral lines here (drawing the discrete spectral lines, which were close to each other at one end of the spectrum around 400 nm, as shown in Fig. 3) because now orbits are being closer to each other at this end. This might happen at the other end (pointing to the opposite end of spectrum for 700 nm).

Fig. 3 A student's drawing of the emission spectrum. A hypothetical electron in the hydrogen atom obeys classical mechanics rather than quantum mechanics. There are discrete spectral lines when the electron is falling down to the nucleus. Lines are getting closer to each other because orbits are getting closer to each other. The student did not consider the difference between the 400 nm end and the 700 nm end of the spectrum, and stated that the closer lines can exist in either end of the spectrum depending on which end the nucleus is. The lines are concentrated at one end because it is approaching the nucleus. “Mor” (in the lower right corner) means the color “violet”.

The student displaying this model had similar knowledge organization to the students with NPM but this student connected spectral lines with orbits that are at different distances from the nucleus. This was also an incorrect model. The student did not have a correct understanding of discrete energy levels, electronic transition, photon energy, or orbitals.

Structure of mental models

Table 3 compares the structures of different mental models in terms of purity (Bao, 1999; Didiş et al., 2014) by considering whether the concepts involved are scientific or unscientific. If all the concepts involved are of the same type (e.g. scientific), it was called a pure model, and if not (e.g. both scientific and unscientific), it was called a hybrid model.

Table 3 Purity of mental models

Mental models	Scientific elements	Unscientific elements	Connection	Purity
(+) corresponds to correct connection. (−) corresponds to wrong connections. (·) corresponds to missing connections.a Wrong understanding with missing concepts.b Wrong understanding with missing/wrong concepts and missing/wrong connections among the concepts.
SMAS	5	0	+	Pure, scientific
PSMAS	3	0	.	Pure, unscientific^a
NPM	3	0	. , −	Pure, unscientific^b
OM	2	1	. , −	Hybrid, unscientific^b

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One of the interesting findings of this study on the structure of mental models reflected in Table 3 is that a pure mental model with all the concepts used being scientific is not necessarily a scientific mental model. Although all the elements of a mental model can be all scientific concepts, whether the mental model is scientific or not depends on the connections among the concepts in the knowledge organization. For example, in the SMAS, students used scientific concepts in a coherent and correct way. However, in mental models PSMAS and NPM, students used scientific concepts with incomplete or incorrect understanding and connected them incorrectly when they organized their knowledge.

The incorrect pure organization of knowledge can be manifested in two ways. First is “without using all the scientific concepts” needed in the construction of a scientific mental model, and the second is “with incorrect links”. For example, the students with PSMAS did not consider electronic transition between energy levels to explain the photon energy and they explained it only in terms of the difference between energy levels. The absence of these concepts in their knowledge organization prevented the construction of correct models.

In another example, students displaying NPM had wrong links among the scientific concepts in addition to missing concepts; they did not fully understand DEL or SL and incorrectly connected these two together.

In addition to the pure mental models, the last type of organization is the hybrid model. In our study, we observed that OM was a hybrid unscientific model which was constructed by the use of scientific and unscientific elements together.

What do the mental models reflect about students' understanding of atomic spectra?

This research indicated some “threshold concepts” explained by researchers (Meyer and Land, 2003; 2006; Cousin, 2006; Park and Light, 2009; Talanquer, 2015). A threshold concept can be described as “opening up a new and previously inaccessible way of thinking about something” (Meyer and Land, 2006, p. 3). They are difficult to understand but when they are understood they provide more connected and deeper understanding (Chandler-Grevatt, 2015). Threshold concepts are assumed to have the following five characteristics: transformative, integrative, irreversible, troublesome, and bounded (Meyer and Land, 2003; 2006; Cousin, 2006; Park and Light, 2009; Talanquer, 2015).

Electronic transition (ET) and photon energy (PE) are identified as threshold concepts in understanding atomic spectra as they demonstrate the five characteristics mentioned above. Firstly, these two concepts are transformative because unscientific mental models emerged due to the missing of one or both of these two concepts as shown in Fig. 4.

Fig. 4 Domains for mental models of atomic spectra identified by two threshold concepts. The visual model of Park and Light (2009) about the threshold concepts was used to indicate their role in the construction of scientific and unscientific models.

Electronic transition (ET) is what differentiates the two models SMAS and PSMAS. Students with PSMAS do not associate the change in energy levels in an atom with the electronic transition between different quantum states. An important thing to point out is that the unscientific model PSMAS is very hard to be identified with common calculation questions in atomic spectra. Students with either SMAS or PSMAS will be able to correctly calculate the wavelengths in the absorption or emission spectra based on that the photon energy equals the difference between two discrete energy levels.

In addition to electronic transition (ET), photon energy (PE) is the other threshold concept: students with PSMAS lack ET and misconnected PE directly to DEL, and students with NPM or OM lack both of these concepts in their mental models.

The lack of the concept of photon energy is easier to be identified than the concept of electronic transition with calculation questions, as the students with NPM or OM will not be able to correctly calculate the wavelengths in atomic spectra based on the wrong connection between spectral lines (SL) and discrete energy levels (DEL) in NPM or orbits (O) in OM.

The transformative characteristic of these two threshold concepts is also demonstrated as they bring a shift from a classical perspective (“common sense”) to the quantum perspective in explaining the atomic spectra. The correct understanding of atomic spectra with electronic transition and photon energy is also a reflection of a scientific understanding of atomic structures and the wave-particle duality of light and matter. As threshold concepts, ET and PE are also integrative because they bring together other important concepts – spectral lines, discrete energy levels, and bound electron – of atomic spectra as shown in Fig. 1.

Given their transformative and integrative roles in explaining the atomic spectra, they are irreversible and unlikely to be forgotten. They are troublesome, that is they require the quantum perspective and are hence counter-intuitive and difficult to understand. Their troublesome characteristic is also why the electronic transition (ET) is missed in all the three unscientific mental models and photon energy (PE) is misused in PSMAS and missed in NPM and OM. Lastly, they are bounded, because they are specific to the disciplines related to quantum theory.

We think that understanding the phenomenon of atomic spectra can be seen as a strategic tool to assess whether the students have a good understanding of the following three aspects shown in Fig. 5: (1) the behavior of an electron in an atom (wave-particle duality of matter), (2) the wave-particle duality of light, and (3) the interaction between an electron and a photon (light–matter interaction). As shown in Fig. 5, the SMAS consists of five scientific concepts concerning the above three aspects.

Fig. 5 The scientific model of atomic spectra (SMAS) as a reflection of the correct understanding of (1) the behavior of an electron in an atom (wave-particle duality of matter), (2) the wave-particle duality of light, and (3) the interaction between an electron and a photon (light–matter interaction).

The behavior of an electron in an atom (BE) can be explained with the wavefunctions (orbitals) associated with the corresponding energies (DEL). In the process of an electronic transition (ET) between different quantum states, the electron absorbs or releases energy that corresponds to the packet of energy carried by a photon (PE), which explains the colors and discreteness of spectral lines (SL). This is how matter and light interact.

The only limitation of using atomic spectra as the assessment tool for the understanding of atomic structures is that SMAS does not require the understanding of orbitals, so students with an incorrect understanding of orbitals can do well in questions on atomic spectra without the misconception on orbitals exposed.

Conclusion, discussion and implications

Mental models are the knowledge structures constructed in an individual's mind, which are composed of coherent elements to explain what is perceived when seeking a deeper understanding (Norman, 1983; Taber, 2013; Didiş et al., 2014). Our research has shown that students' wrong understanding of the concepts results in the construction of unscientific mental models of atomic spectra. Gentner (2002) explained that novices had a tendency to learn “conservatively”. For this reason, students often used locally coherent but inconsistent structures. One of the interesting findings was almost half of the students (4 students) associated discrete energy levels (DEL) with discrete spectral lines (SL) in NPM. Ivanjek et al. (2015) indicated that a large number of Croatian and American students might have such a kind of understanding in atomic spectra. This kind of idea might be considered as a misconception of students about atomic spectra; however, our research findings indicated that students might have this misunderstanding with a coherent wrong organization of the fundamental knowledge elements about the spectral phenomenon. Another model, OM, was the other kind of wrong association of spectral lines. One student having this model explained that the change in orbits caused the collapse of spectral lines toward one end of the spectrum. The findings of Ivanjek et al. (2015) and our findings from different countries reflected how students intended to concretize abstract concepts with observable elements (e.g. spectral lines) with wrong associations.

As Park and Light's (2009) research indicated the importance of threshold concepts to understand the atomic structure well, we also identified that two of the fundamental concepts, electronic transition (ET) and photon energy (PE) were threshold concepts in the construction of a scientific mental model about atomic spectra. There are two important concepts embedded in other concepts such as atomic structure and the nature of light.

Similar to the findings from previous research studies (Bao, 1999; Didiş et al., 2014), we also observed that both scientific and unscientific elements were used together in one of the students' mental models, and that is the hybrid unscientific model OM. One possible explanation of this model is that the students kept the early Bohr's atomic models describing the atom in terms of orbit, but they misused this concept to explain atomic spectra.

Students with NPM and OM incorrectly connected either the individual discrete energy levels or the classical orbits to each of the spectral lines. One possible cause for this misconception is that the students misinterpreted the symbolic representations of the energy levels or the classical electron orbits, and hence drew incorrect connections with the spectral lines (an observable reality). This is consistent with the explanation by Chittleborough et al. (2002) that the misconceptions arise from the wrong interpretation of symbolic representations that are not always clearly defined. In addition, this wrong connection can also be explained with an inappropriate analogy between analog and target. Previous research indicated that mental models were often based on analogies (Duit, 1991; Treagust et al., 1994; Glynn and Takahashi, 1998; Gentner, 2002; Glynn, 2007; 2008; Didiş, 2015). Students' explanations of NPM indicate their wrong analogical reasoning based on the shared feature, i.e. “discreteness”, for both the discrete energy levels and the discrete spectral lines. Didiş et al.'s (2014) research indicated the existence of an analogy in the unscientific mental model of quantization. In the Shredding Model, students explained quantization as a slicing of a cake.

In addition to concretizing difficult abstract concepts by either misinterpreting the symbolic representation or drawing inappropriate analogies, we also observed that students tried to use irrelevant mathematical models to interpret the visual representations. One example is that a student described orbits as an integration process as shown in Fig. 2.

Students' use of the concepts such as “energy level” and “spectral lines”, “orbit” and “orbital” interchangeably in this study is similar to the physics education students' use of some quantum mechanical concepts such as “operator” and “observable” interchangeably (Didiş et al., 2010). In addition, students' use of the concepts “orbit” and “orbital” interchangeably may also indicate students' difficulty in discriminating classical and quantum phenomena (Bao, 1999; Ke et al., 2005). Ke et al.'s (2005) research showed that the experienced students having quantum mechanical mental models knew about the differences between classical and quantum ideas, but students having classical mental models for quantum concepts were not aware of their conflicts. Based on these discussions, we draw attention to the following issues and suggestions for teachers to consider:

(1) As the topic of atomic spectra involves the atomic structure (matter), properties of light, and how light and matter interact, we suggest that teachers start with the introduction of the first two aspects to provide students a solid background, and then explain how a photon and an electron interact through the absorption/emission of a photon during an electronic transition. This may minimize students' mistake in associating electronic energy levels directly with spectral lines.

(2) Developing links among concepts helps identify misconceptions of individual concepts and promotes meaningful learning. A concept may be understood better in relation to other concepts (Taber, 2015). Conceptual linkages made by learners among different concepts help teachers understand how well the students understand them. Teachers should explore and map out the links students hold for concepts taught in the class (Taber, 2015). Atomic spectra is a strategic topic where we can help students understand the fundamentals of both matter and light. The integration of multiple topics has advantages to overcome misconceptions as explained in Tsaparlis (1997) where the integration of atomic orbitals and spectroscopic terms was used to help students clarify the misconceptions around atomic orbitals.

(3) When using quantum terminologies and visual representations to describe quantum phenomena, teachers should make sure that students understand them correctly. Human-like or classical attributions to quantum objects should be carefully used to avoid misconceptions. Mathematical connections should also be emphasized for quantum theory concepts and visuals (Tsaparlis, 1997) to prevent students from applying irrelevant mathematical models to explain quantum concepts.

(4) Conceptual questions on atomic spectra used in the interviews not only help us identify the different mental models students may have, but also help identify students' misconceptions and increase their interest in understanding this phenomenon. Firstly, as noted earlier in this paper, the unscientific model PSMAS is very hard to be identified with common calculation questions in atomic spectra, because students with either SMAS or PSMAS will be able to correctly calculate the wavelengths in the absorption or emission spectra based on that the photon energy equals the difference between two discrete energy levels. Therefore if calculation questions are the only form of assessment of students' understanding of atomic spectra, students with PSMAS can answer them correctly and hence have the illusion that they fully understand the phenomena of atomic spectra. Secondly, we observed in our interview that when the students realized that they cannot fully explain the spectral lines in the absorption and emission spectra, they got really curious and wanted to ask more questions to learn what was really going on. In addition to the conceptual questions listed in Appendix I, another good one is to ask students why lines in the emission and absorption spectral lines appear at the same wavelengths.

(5) We suggest that teachers take time to explain the physical set up for the atomic spectra so that the students understand the difference between absorption and emission spectra instead of inventing or guessing what atomic spectra are.

(6) Knowledge in chemistry and physics can be organized separately (Taber, 2001) by students and this artificial division could be a barrier in students' learning. Greca and Freire (2014) suggested that some physics concepts that are base for quantum mechanics should be modified and taught to all undergraduate students. However, sometimes chemistry students may consider chemistry learning separate from physics learning (Taber, 2001). Vice versa, it is possible that physics students may consider atomic orbitals as a concept in chemistry and fail to transfer the knowledge to understanding atomic spectra taught in a physics class. Teachers should be careful not to promote division between physics and chemistry, but should provide required physics or chemistry background to students so that they can understand the quantum mechanical concepts effectively.

Both qualitative and quantitative methodologies can be used to investigate students' mental models with different designs, and our research used a qualitative approach and identified students' mental models from their answers to a series of conceptual questions. For further research, some quantitative approaches (Bao, 1999; Stamovlasis et al., 2013; Vadnere and Joshi, 2009) may be used with a statistical analysis of data from a larger number of students for generalization of results to different populations.

Appendix I: interview protocol

(1) In an emission spectrum, what do the (colored) lines represent (in the visible region), or in an absorption spectrum what do the dark lines represent? Why do the lines occur? Why do they have different colors (in the visible region) in emission spectra? Why are they dark in absorption spectra?

(2) Consider the figure below, explain (a) negative energy, (b) n, (c) E (in eV), and (d) each arrow.

(3) How would you relate an electron's jump from an outer orbital to an inner orbital to the energy of the emitted photon?

(4) Suppose the electron in a hydrogen atom obeys classical mechanics rather than quantum mechanics. What would you expect to observe in the spectrum?

Appendix II: sample coding

Excerpt	Coding
Student 8 (ST8): Yes I have. This is umm… I̲n̲ a̲t̲o̲m̲, w̲h̲e̲n̲ a̲n̲ e̲l̲e̲c̲t̲r̲o̲n̲ with a different n is f̲a̲l̲l̲i̲n̲g̲ d̲o̲w̲n̲ t̲o̲ t̲h̲e̲ o̲t̲h̲e̲r̲ n̲ l̲e̲v̲e̲l̲, it emits a photon. This has a specific energy, for example p̲a̲s̲s̲i̲n̲g̲ f̲r̲o̲m̲ n̲ = 3̲ t̲o̲ n̲ = 2̲, the p̲h̲o̲t̲o̲n̲ i̲s̲ e̲m̲i̲t̲t̲e̲d̲ w̲i̲t̲h̲ a̲n̲ a̲m̲o̲u̲n̲t̲ o̲f̲ e̲V̲ e̲n̲e̲r̲g̲y̲ b̲e̲t̲w̲e̲e̲n̲ t̲h̲e̲s̲e̲ e̲n̲e̲r̲g̲y̲ l̲e̲v̲e̲l̲s̲. For this reason these l̲i̲n̲e̲s̲ occur.	BE ET ET PE
I: Ok, then, (pointing to the emission spectrum in the figure) how does this dark area occur?	SL
ST8: This is for example, here, umm…, in this dark area (pointing to the emission spectrum in the figure), a photon cannot be emitted. T̲h̲e̲ m̲a̲g̲n̲i̲t̲u̲d̲e̲ o̲f̲ t̲h̲e̲ e̲n̲e̲r̲g̲y̲ that does not correspond to difference between t̲w̲o̲ e̲n̲e̲r̲g̲y̲ l̲e̲v̲e̲l̲s̲ is dark here. Absorption is the reverse of emission. That means, the a̲b̲s̲o̲r̲b̲e̲d̲ p̲h̲o̲t̲o̲n̲s̲ correspond to d̲a̲r̲k̲ l̲i̲n̲e̲s̲, and other areas are seen colored.	PE DEL PE SL

Acknowledgements

We thank Bülent Ecevit University and the Scientific and Technological Research Council of Turkey (TÜBİTAK).

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Footnote

† Current address: Harvard University, Department of Physics, Cambridge, MA, 02138, USA.

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