The photoelectric effect is considered an essential topic in the introductory stages of quantum mechanics. In particular, the effect is deemed a crucial step towards an understanding of the wave-particle duality of light. Previous studies reported that attaining a comprehensive understanding of the effect is not an easy task for students[1-6]. Several aspects of students' difficulties gleaned from the studies are summarized as follows.

• Students display a lack of understanding about both the purpose and function of experimental equipment[1,2];

• Students have difficulty in predicting the experimental results of the photoelectric effect[1,2,4,5];

• Students show a lack of understanding about the classical wave model of light[2,5];

• Students have difficulty linking the experimental results of the photoelectric effect with the photon model of light[3-5];

• Students do not infer the particle-like nature of light from the effect’s observations[4,5].

We believe that there are two main goals for the teaching of the photoelectric effect. Firstly, it is important for students to understand the physical processes of the effect including the photon model of light. This helps students predict experimental results. Secondly, students need to obtain a clear picture of the nature of light. In our description of these two learning outcomes we distinguish between two kinds of learning; whereby a quantum-mechanically based (photon) model of light which explains the experimental results is constructed and also one where the nature of light from the experimental results and the photon model is interpreted. Our distinction arises from the following consideration: Firstly, the particle nature of light inferred from the results of the photoelectric effect was a highly controversial issue in the history of physics although physicists submitted that the photon model successfully explained the photoelectric effect[7,8]. And, despite the success of the photon model of light on the photoelectric effect, physicists' attitudes toward the particle-like nature of light, changed affirmatively only after the discovery of the Compton effect. Second, in the study of McKaganet al.[5], the percentage of students who referred to the particle-like nature of light in the photoelectric effect varies from semester to semester, declining from more than 80% in the first semester of their study to less than half in the final semester[5]. On the other hand, a much higher percentage of the students referred to the photon model of light from the effect in their study. Thus, we contend that it is important to clearly distinguish learning about the physical process of photon model of light from interpreting the nature of light. In our distinction, constructing the photon model of light and interpreting the nature of light are respectively related to the first and second learning outcomes described above.

We maintain that teaching should primarily focus on the physical processes, including the photon model of light before interpreting the nature of light to aid meaningful learning about the photoelectric effect. Consequently, this study addressed the following two research questions corresponding to the two learning outcomes:

• What characteristic patterns emerge concerning university students’ understanding of the physical process above all the behavior of light?

• What characteristic patterns emerge concerning their reasoning linking the photoelectric effect and the particle-like nature of light?

Related to the first learning outcome and research question 1), we focused our study on students’ understanding of light's behavior during its interaction with matter in the context of the effect. There have been innumerable studies on student understanding of light (see for example, Ref. 9). Most of the studies pointed to the propagation or perception of light. In contrast, there have been notably fewer studies on students’ understanding about the interaction between light and matter. Moreover, previous studies on the interaction, including studies on the photoelectric effect, focused on the electron's behavior rather than light[1-3,10]. These studies did not query students about the photon's behavior during any interaction with electrons in the photoelectric effect. Thus we need an additional study that focuses on the students’ understanding of behavior of light and its interaction with the electron.

Students' difficulty interpreting correctly the nature of light from the photoelectric effect, has been studied less often than the first learning outcome. Only two previous studies considered the issue, however, the studies did not clearly distinguish between knowing about the physical process of the photon model of light and interpreting the particle-like nature of light in their research[4,5]. Detailed investigation about the pattern of students' interpretation, based on the distinction, is the second research concern in this study. In addition, we also explored the origin of students’ difficulty in logically linking the effect with the particle-like nature of light.

Nineteen university students, all enrolled in a third-year quantum mechanics course for physics education majors, participated in this study. All of them are at Seoul National University, one of the top-ranked universities in Korea. We selected the students from the course with the expectation that their difficulties with the photoelectric effect would be a fair reflection of the whole cohort’s difficulties. All the participants had studied the photoelectric effect in high school, although the treatment there was rather cursory. Among them, thirteen had previously studied the effect in an introductory physics course and seventeen had studied it in a modern physics course. Most had no additional opportunity to study the photoelectric effect in any other course. Hence, all the participants had studied the effect formally on two or three previous occasions. They had all learned photoelectric effect through a series of lectures. During the interview period, they took a traditional quantum mechanics course. One of the authors observed the class to check its relevance to the interview and concluded that the course briefly treated the photoelectric effect and would have little impact on our work.

We used a semi-structured interview and written questionnaire to collect data from the students. The interviews covered topics relating to wave-particle duality. The entire interviews took approximately, ninety minutes and were audio-taped. In this study, only the parts relevant to the photoelectric effect, which were on average forty minutes long, were analyzed from the whole data. Typical properties of the particle in students’ conceptions were additionally explored in the quantum mechanics course about two weeks after the interviews through a questionnaire. The time gap was intended to weaken the interview's influence on the questionnaire; sixteen out of the nineteen students responded to it. Both the interview protocol and written questionnaire had been tested and modified through a preliminary study in which another thirteen undergraduate and five postgraduate students majoring in physics education had been involved. In the main study, some questions in the interview protocol were given to each participant, while some questions were given only to a selected few according to their responses. By and large, students’ initial responses to the questions were too brief or insufficient or both to evaluate their understanding. In those cases, additional questions were used to better resolve their ideas.

By analyzing the transcribed data, students' spontaneously drawn figures, and written responses, we extracted their characteristic patterns of understanding. We constructed several categories by grouping similar responses and made a description statement that reflects the core feature of the grouped responses. We then coded each student's response to the categories. To improve the reliability of the study, another two physics education researchers individually coded each student's responses according to the initially made categorization criterion. Next, the three researchers discussed some of the conflicting results until a consensus was reached. Most disagreements were easily resolved through the discussion. During the process, there was minor modification to the initially constructed categories. Some students changed their initial views during the interview. In those cases, we analyzed the data by considering all the related responses in their entirety.

The interviews were conducted in Korean, thus all the excerpts from students’ responses in this article are translated into English by one author. To avoid distorting the translation, the English statements were retranslated back into the Korean by another physics education researcher. It was then confirmed that no significant misrepresentation occurred.

Here, we report on the result of main study in which nineteen students participated. We note that similar results were obtained from the preliminary study although we do not present those results.

All participants recalled the photoelectric effect as a phenomenon in which electrons in a metal are emitted when exposed to light and most evoked its relevance to the particle-like nature of light. An understanding of the photoelectric effect demands significant reasoning of observed experimental results. In particular, students have to distinguish between observation and inference in their reasoning to understand the photoelectric effect[11]. We verified whether students could characterize observation and inference by asking “Describe only the experimental results of the photoelectric effect without any theoretical explanation of the experiment”. Sixteen (84%) of the 19 students correctly described only the experimental results. Further, through evaluation of the whole interview data set, we concluded that most students had enough reasoning skill to discern observation and inference although the two were not always clearly distinguished in the responses.

1. Students’ inability to elaborate on the physical processes of the photoelectric effect

To investigate student understanding about the interaction between light and electrons, we asked students how electrons were emitted according to the ‘quantum model’ or ‘classical model’. The terms, ‘quantum model’ and ‘classical model’ were used instead of ‘particle model’ and ‘wave model’ in the interview. Since light (photon) has both a particle-like and wave-like property in the ‘quantum model’, it could be misleading to invoke a ‘particle model’ and ‘wave model'. Moreover, as discussed in the introduction, we distinguish between constructing a photon model of light and interpreting the nature of light. Therefore, careful choice of terminology is essential in this study. However, most students were familiar with the terms, ‘particle model’ and ‘wave model’ but unfamiliar with ‘quantum model’ and ‘classical model’. In that case, they were guided in the use of the terminology. Here, in this article, we interchangeably use ‘quantum model’ and ‘photon model’ though ‘quantum model’ was principally used in the interview questions.

In previous studies of the physical process of the photoelectric effect, questions were related to predicting experimental results and focused on the electron’s behavior. This study changes the focus to the behavior of light. This refocusing revealed students’ difficulties which were not reported in the previous studies. By and large, students displayed acute difficulty when they elaborated the physical process of the photoelectric effect, especially, the behavior of light during the interaction. More detailed patterns were revealed in students’ responses as laid out below.

1) Insufficient understanding about the process of energy transfer in the classical model

According to the classical model of light, electromagnetic radiation (electric and magnetic field) causes electrons to be accelerated and emitted from a metal. We asked explicitly “What happens to light in the process of electron emission from a metal in the classical model of light?” to investigate the students’ understanding about this process. Initially fifteen (74%) responded that energy transferred from light to the electron without describing any mechanism for the energy transfer. When asked about the detailed process of the energy transfer, many students appeared not to have thought out its detailed processes. Students merely reiterated the `energy transfer description' but did not consider the mechanism of energy transfer. Students then began to express their ideas in more detail after being pressed further on the issue.

Their final responses can be categorized into six groups as shown in Table 1. Only four (21%) of nineteen students correctly mentioned that electrons were driven by electromagnetic radiation. Another four students (21%) stated ‘energy transfer through the medium’, mentioning that electrons vibrate as does the medium but without further elaborating on the process. One student (5%) mentioned collisions between waves and electrons. Another student (5%) recognized that light is electromagnetic radiation but could not explain the interaction between radiation and electrons. Eight students (42%) explicitly expressed their lack of ideas about the process of energy transfer or just repeated the ‘energy transfer description’ without further remarking about the mechanism. On the other hand, one student (5%) thought that energy is not transferred from light to an electron in the classical model.

Table 1

Student ideas of the process of energy transfer from light to electron.

	Idea about the process of energy transfer	Number of students
For the classical model	Electrons were driven by electromagnetic radiation.	4 (21%)
	Electrons vibrate as does the medium.	4 (21%)
	Light wave collides with electrons.	1 (5%)
	No idea about the interaction between electrons and light except recognition that light is electromagnetic radiation.	1 (5%)
	No clear idea about the energy transfer from light to electron.	8 (42%)
	Energy is not transferred from light to an electron in the classical model.	1 (5%)
For the quantum model	Photon loses all its energy and is annihilated.	7 (37%)
	Photon loses its energy partially and rebounds off electron.	9 (47%)
	Photon can be annihilated or rebounds off electron.	1 (5%)
	No detailed idea about the energy transfer from light to electron.	2 (11%)

Many students mentioned that energy is continuously transferred in the classical model. Some recalled from their personal experiences that an object becomes hot when exposed to light. These students seemed to relate the experience to the classical model of light, because they considered the energy transfer in the classical model as a continuous process. In contrast, they regarded the energy transfer of the quantum model as a transitory event.

Four students seemed to be conflicted between the idea of energy transfer through the medium and that light has no medium. For instance, Moun (pseudonym) claims thus.

Hence, more than two-thirds of students could not recognize that light is considered electromagnetic radiation in the classical model when they considered energy transfer in the photoelectric effect. Only four of the five students, who correctly mentioned electromagnetic radiation described the physical process of the energy transfer. By and large, students showed a lack of understanding concerning the interaction between light and an electron in the classical model. While students' lack of understanding of the other aspects in the classical model of light were seen in the previous studies[2,5], their inability to elucidate the mechanism of energy transfer in the classical model of light is a new finding.

2) Understanding about the drawbacks of the classical model

Most students recalled that the classical model of light has difficulty in explaining the experimental aspects of the photoelectric effect. When asked about experimental results being hard to explain by way of the classical model, fifteen students (79%) mentioned the problem of threshold frequency. Only four students added other experimental results such as instant emission of electrons regardless the light intensity and the independence between the maximum kinetic energy of electrons and light intensity. When asked why the threshold frequency was difficult to explain through the classical model, most students responded that electrons must be emitted irrespective of the frequency of light because light energy is continuously transmitted or that the radiation of high-energy light causes electron emission. In their responses, most did not differentiate between the propagated energy of light and absorbed energy in the electron, except for one case whereby it was explicitly distinguished. Students' responses seem to stem from the fact that the textbook usually does not distinguish between the two. On the other hand, four (21%) of nineteen students could not suggest which experimental results were relevant to the drawbacks of the classical model.

In summary, many students confidently mentioned the drawbacks of the classical model through the brief energy transfer argument without demonstrating any detailed mechanism for energy transfer. These students’ pattern might revert to instruction or textbooks or both. Many textbooks do not explicitly distinguish the propagated energy of light from the absorbed energy in the electrons in their explanation[12,13,14]. They also provide energy transfer arguments against the classical model without explaining the detailed process of energy transfer.

3) Inability to correctly elaborate on the mechanism of energy transfer in the quantum model

In the quantum model of light, absorption (annihilation) and emission (creation) of a photon are the basic interaction processes between light and matter. Roughly speaking, an absorbed photon does not exist, and an emitted photon is created through the interaction. In the Compton effect, a photon and an electron appear to collide, but in fact, one photon is absorbed and another photon is emitted in the effect. We investigated students’ understanding about these processes in the context of the photoelectric effect. When asked “What happens in the energy transfer from light to electron in the quantum model?” fifteen students (79%) expressed the idea in terms of ‘collision’, ‘hit’, or ‘impact’. To resolve what students meant by those terms, we asked what happened to light during the energy transfer. The question seemed to be an unfamiliar one to students. For instance, one student responded: “I have not learned about that”. Many students expressed their ideas only after further questioning and some even changed their view during the response phase.

Their final responses are shown in Table 1. Only seven students (37%) correctly mentioned that a photon lost its whole energy and disappeared. On the other hand, nine students (47%) responded that the photon lost its energy partially and bounced off the electron. Six of the nine students mentioned that the deflected photon had a longer wavelength than the initial photon. One student (5%) mentioned both of the above two possibilities. Two students (11%) did not clearly expand upon their ideas for energy transfer.

Many students who initially had no idea about the detailed interaction between light and an electron surmised that the interaction is similar to a typical two-particle collision. They seemed to regard a photon as a particle without giving enough attention to the interactions of the photoelectric effect. There could be two possible reasons for these ideas. First, it could be that students apply their knowledge of the Compton Effect to the photoelectric effect. This conjecture is supported by the students’ responses that the wavelength of the bounded photon lengthens. Another possibility is that students believe that a photon is a kind of classical particle. This naïve interpretation could lead to the idea that the interaction is similar to the typical two-particle collision. The concept of a photon was developed from the analogue of a classical particle. A photon, however, is not a classical particle. Students seemed not to distinguish the classical particle analogue and the photon concept developed from it and seemed to commit inappropriate analogical reasoning about the behavior of the photon.

Student responses might have been affected by previous instruction or textbooks or both. In many textbooks, the interaction between photon and electron is described only in the frame of ‘energy transfer’ without focusing on the behavior of the photon. This limited description could be problematic since students’ proper understanding of the physical process is requisite to an appropriate interpretation of the nature of light.

4) Difficulty in explaining experimental results using the quantum (photon) model of light

The quantum model of light explains the experimental results of the photoelectric effect through several assumptions. The key idea of the quantum model is that an electron is emitted through a one-time energy transfer due to single photon absorption rather than energy accumulation on the electron. To investigate students’ understanding about the explanation, we asked “How does the quantum model explain the experimental results of the photoelectric effects?”

During the interview, eleven students (58%) explicitly mentioned that one photon has to be related to one electron emission in order to explain the effect. On the other hand, six students (31%) explicitly mentioned the possibility of multi-photon involvement in electron emission. In the following interview, only two of the six students accurately inferred that the possibility of multi-photon involvement in one electron emission could be ignored because the occurrence is rare. The other four students could not resolve the problem of energy accumulation due to multi-photon involvement. For instance, with regard to light with lower than threshold frequency, Chae (pseudonym) claimed thus.

For some students, energy accumulation on the electron is more plausible to their common sense than the single energy transfer, as shown thus.

Students whose concern was the possibility of multi-photon involvement tried to resolve the problem of threshold frequency during the interview. For instance, four students applied the concept of discrete energy levels as suggested in Bohr’s atomic model. However, their explanations were flawed, as shown in the following excerpt:

The response seemed to reflect their confusion due to the possibility of multi-photon involvement.

In summary, many students had significant trouble in explaining the phenomena using the quantum model by considering energy accumulation due to multiple photons. Energy accumulation could appeal to them more than single energy transfer. Students’ anxiety about the possibility of multi-photon involvement was previously reported as one of the unexpected questions asked by students during instruction utilizing computer simulations of the effect[5]. Our finding shows that the problem needs to be more carefully treated during instruction. Participants in this study might not recognize the problem prior to the interview since they seemed to have an under-developed idea concerning the physical process between a photon and an electron during the interview’s initial stage. It can be suspected that interview questions focusing on the physical process themselves might have prompted students’ consideration of the multi-photon involvement. However, the issue of multi-photon involvement was not, directly or indirectly, provided to students through any interview questions and they voluntarily mentioned about its possibility.

In our opinion, proper consideration of the physical processes is required for a meaningful learning of the photoelectric effect to take place. If teaching focuses on the physical processes, the problem of energy accumulation due to multiple photons could be a serious obstacle to students’ achievement in learning outcomes, as this study illustrates. Thus, instructors should be prepared for the problem as many textbooks merely mentioned a single energy transfer without further reasoning. Most instructors might not have noticed this obstacle as they did not unpack students’ understanding of the photoelectric effect and, therefore, this problem did not arise.

2. Students' reasoning linking the photoelectric effect and the particle-like nature of light

The majority of the students recalled that the photoelectric effect is relevant to the particle-like nature of light, though some could not explain the effect with the quantum model. To see how students argued the relationship between the photoelectric effect and particle-like nature of light, we asked the following questions.

Question 1: Does the photoelectric effect support the particle model of light? If so, please explain why.

Question 2: Why do the experimental results of the photoelectric effect, such as the existence of threshold frequency, support the particle-like nature of light?

Question 3: Does light have a particle-like nature in the photoelectric effect? If so, which properties (or features) of the particle does light have in the context of the photoelectric effect?

For Question 1, many students merely mentioned that experimental results support the particle model of light. Question 2 was used additionally in such cases to more vividly unfold students’ ideas. Questions 1 and 2 were used to get students’ reasoning about how the particle-like nature of light could explain the experimental results, while Question 3 was used to glean students’ reasoning about the particle-like property of photon. We emphasized during the interview that the purpose is to get students' own ideas and the reason for their conclusion. However, in many cases their responses seemed to be a mix of their own ideas and accepted knowledge by authority. The detailed patterns of students’ responses are discussed thus.

5) Difficulty in logically linking the photoelectric effect and the particle-like nature of light

Questions 1 and 2 were used to probe students’ reasoning linking the effect and the particle-like nature of light. Thirteen students (68%) mentioned that the classical model (in their expression the wave model) cannot explain the photoelectric effect. Only six students (32%) provided direct reasons for the particle model although we additionally queried for reasons to support the particle model in lieu of refutation of the wave model. Some students mentioned the experimental results of the photoelectric effect such as the existence of the threshold frequency yet, could not answer to the Question 2. We concluded that these students could not provide direct reasons for the particle model. Direct reasons provided by students were by and large similar to the responses to the Question 3, discussed in detail below.

In brief, more than two-thirds of students did not directly provide reasons for the particle model as related to Questions 1 and 2. Many students regarded the photoelectric effect as evidence to reject the classical model of light but had difficulty clearly linking the effect and the particle-like nature of light.

For the first part of Question 3, fourteen students agreed, while five responded that they were unconvinced. Student responses to the second part of the question are summarized in Table 2. Three students (16%) mentioned the properties related to energy quantization such as photon individuality or the ability to count photons. Eight students (42%) cited collision as a particle-like property of light. One student (5%) mentioned that the photon has energy as a particle-like property and two students (11%) that the photon has momentum or mass. Conversely, six students (32%) did not provide any specific particle-like property of light in their responses.

Table 2

Particle-like properties of light in the photoelectric effect as mentioned by students.

Number of Students	Particle-like property of light (photon)
3 (16%)	Photon is individual or countable.
8 (42%)	Photon collides with electron. (Photon transfers energy and momentum to electron.)
1 (5%)	Photon has energy.
1 (5%)	Photon has momentum.
1 (5%)	Photon has mass.
6 (32%)	No clear idea for the particle-like property of photon.

N.B. The sum of the students in Table 2 is twenty, which is more than the nineteen-total number of participants in this study as one student mentioned more than one particle-like property in our categorization. In some cases, students mentioned both words of `energy' and `collision' as shown thus.

For the cases, we concluded that the students referred to collision in our categorization as students mentioned ‘energy transfer’ as a companion explanation for the collision rather raising that light has energy as a particle-like property. We made a similar judgment for the case that students mentioned momentum conservation additional to collision.

Some students exhibited difficulty about particle properties. Three students asked the interviewer what the actual particle-like property is. Naive interpretation for duality added to student difficulty, as shown thus.

To further discuss students' difficulties in providing particle-like properties of light in the photoelectric effect, we need to evaluate the correctness of reasoning provided by the students. We can say that the student who mentioned a photon’s `individuality' and `countability' as a particle-like property of light is correct. As to ‘collision’, the physical process between a photon and an electron in the photoelectric effect significantly differ from a typical two-particle collision in that momentum is not considered and a photon is absorbed after elastic interaction. Thus, it is unclear whether the interaction between photon and electron can be regarded as a collision between two particles and whether ‘collision’ is a particle-like property of a photon in the context of the photoelectric effect. In fact, only eight students provided ‘collision’ as a particle-like property of the photon in the photoelectric effect while fifteen students mentioned ‘collision’ or something similar such as ‘hit’ and ‘impact’ for the interaction between photon and electron. Further, four of the eight students showed wrong reasoning that the photon deflects after collision. Thus, we cannot simply say that students who mentioned collision as a particle-like property of light are correct. On the other hand, for the response of ‘energy’, it is not a crucial property in the interpretation of the particle-like nature of light, as light also has energy in the classical model. Further, `momentum' and `mass' are irrelevant to the particle-like nature of light in the context of the photoelectric effect, even though they are important when considering other aspects of wave-particle duality, for example the de Broglie wavelength.

In summary, when we asked students about the logical link between the particle-like nature of light and the photoelectric effect within the framework in which the photon model of light and particle-like nature of light were distinguished, students demonstrated serious difficulty in making the links. Many students could not directly explain the particle model of light and could not provide appropriate explanations to the particle-like nature of light in the context of the photoelectric effect.

6) The gap between the typical properties of particles and the photoelectric effect

In the interview, students showed difficulty arguing about the particle-like nature of light in connection with the photoelectric effect. We explored the origin of the difficulty through a questionnaire. Sixteen of nineteen interviewed students responded to it. They were asked “What are important properties or features of the particle in the classical view?” Students listed important particle properties in their conception. In Table 3, we summarized the list of particle properties of which more than two students mentioned and the number of students who mentioned the properties. Eleven particle properties were mentioned by at least two students. Only five of the eleven particle properties had been mentioned by at least one student as particle-like properties in the photoelectric effect.

Table 3

Properties of a particle in students' conceptions. (Third column ‘O’ indicates particle-like properties of photon of which students had mentioned in the interview.

Property of a particle provided by students	Number of mentioning students	Particle-like property of photon, had been mentioned by students in the interview	Relevance of the particle property to the particle-like nature of light
Mass (weight)	14	O	Irrelevant
Momentum	11	O	Irrelevant
Position	6
Collision (momentum conservation)	5	O	Controversial
Energy	5	O	Controversial
Velocity or acceleration	5
To be numerable	3	O	Relevant
Volume (size)	2
Discontinuity	2
Electric charge	2
Gravitational interaction	2

In the Table 3, important particle properties in students' conceptions such as ‘mass’ and ‘momentum’ are not particle-like properties of light revealed in the photoelectric effect. As to ‘collision’ and ‘energy’, we already argued that it is unclear whether ‘collision’ and ‘energy’ are crucial particle-like properties of light in connection with the photoelectric effect. Only the property ‘to be numerable’ is clearly considered as a particle-like property of light in the photoelectric effect, however, only three students mentioned it as an important property of the particle.

Considering the results, it would be possible to conclude that students cannot adequately argue about the relationship between the photoelectric effect and the particle-like nature of light because the photon in that context is not clearly relevant to the crucial properties in their particle conception. A curriculum-reform program utilizing interactive computer simulation reported that linking the photoelectric effect to the particle-like nature of light is still a challenge contrary to its success on other aspects of the effect[5]. In this study we suggest that the photoelectric effect does not sufficiently support the particle-like nature of light in students' conception. It might be the reason why students fail to make a logical link between the two.

In this paper we investigated characteristic patterns of university students’ understanding about the physical process of the photoelectric effect and their reasoning in linking the effect and particle-like nature of light. It was found that students had a brief energy description for the interaction between light and an electron, but lacked further elaboration. Their further descriptions for the behavior of light were below what was expected at their level or even incorrect. Consideration of the multi-photon involvement caused serious difficulty when students attempted to explain the experimental results with the quantum model. They also thought that the photoelectric effect is evidence against the classical model, yet, had difficulty in directly linking the effect and the particle-like nature of light. They did not well describe the particle-like properties of light revealed in the effect. Their conceptions of the particle were not clearly related to the particle-like properties of the photon in connection with the effect, which could be a reason for the difficulty in making the link between the photon model and the particle-like nature of light in the context of photoelectric effect.

Since student samples in this study were limited, but not a small class size for the course, additional empirical studies are required to confirm the generality of the results. However, there are reasons that we believe that our finding would reveal a widespread trend. First, much of the extracted students' difficulty seems to reflect the insufficient treatment of the current textbooks and other media of the photoelectric effect[12-14]. Second, many aspects of the photoelectric effect seem insufficiently covered in the traditional instruction based on the textbooks. Even previous studies, which investigated students' understanding of the photoelectric effect did not concern many issues of this study[1-6]. Third, all the participants in this study were of a high academic standard in the sense that they passed highly competitive entrance examinations to be on the course. Thus we believe that our findings should be sufficiently considered, even before we look at a larger and possibly more fine-grained study.

McKagen et al. reported that there seemed to be a widespread perception that the photoelectric effect is straightforward and can be understood by students with relatively little effort[5]. On the contrary, this study and previous studies on the effect show a lot of difficulty of which students might encounter in the learning of the effect. The following suggestions for teaching of the photoelectric effect and research on students’ understanding can be drawn from this study.

First, explicit distinction between constructing a photon model of light and interpreting the nature of light from the photon model is required in teaching for a robust understanding of the effect. In this distinction, understanding of the physical process must be completed before the interpretation of the nature of light. Further the link between the photon model and the particle-like nature of light in the context of the photoelectric effect should be carefully treated in the teaching of this topic, since the link might not be self-evident in a student’s own conception as this study shows.

Second, considering students' difficulties in understanding the physical processes shown in this study, instructors need to pay close attention to the basic process of interaction between light and an electron (such as absorption, emission, and electromagnetic forcing of the electron) in the context of the photoelectric effect and certainly take it beyond the simple energy-transfer reasoning. In particular, the photon's behavior during the interaction with the electron should be treated explicitly during instruction. For this, instructors should elaborate upon the meaning of terms such as ‘collision’ used to describe the interaction between photon and electron, as the interaction differs from a typical two-particle collision.

Third, the photoelectric effect might not be compelling evidence of the particle-like nature of light even to a critically-reflective learner. This is contrary to the implicitly shared belief that the photoelectric effect is evidently non-controversial, falsifying the classical wave model of light and supporting the particle-like nature of light. There are many and diverse discussions about possible interpretations of the nature of light inferred from the photoelectric effect[7,8,15,16]. For instance, the idea that the effect shows the particle-like nature of light seems to be widely accepted in the physics education community. Some previous studies seemed to accept the belief in their research[4,5]. On the other hand, several modern physics textbooks mostly mention the quantization of light-energy concerning the photoelectric effect, without any compelling argument to link `explicitly' the effect with the particle-like nature of light[12-14]. Moreover, some researchers pointed out that the photoelectric effect was not as a compelling piece of evidence for the particle-like nature of light in the history of physics[7,8]. Many in the physics community did not accept the quantum model at the outset though it successfully predicts the experimental results of the photoelectric effect. Physicists’ negative attitudes to the particle-like nature of light dramatically changed after the discovery of the Compton effect and other developments[8]. The interaction between a photon and an electron in the Compton effect is very similar to typical two-particle collision. Therefore, the photon model strongly supports the particle-like nature of light in the Compton effect. On the other hand, the photon model in the photoelectric effect is not clearly related to the crucial particle properties in students' conceptions and perhaps even in the conceptions of professional physicists. Recently it has been known that the photoelectric effect can be explained by way of a semi-classical approach in which light is treated as a classical electromagnetic wave[15,16]. As a result, strictly speaking, the photoelectric effect is not an experiment that required the concept of photon. Thus, implicit assumption that the photoelectric effect supports the particle nature of light should be reconsidered.

The discussions related to the interpretation of light seem to reflect that the appropriate interpretation on the nature of light in the photoelectric effect is unobvious to the critically reflective learner. Thus, in our opinion, it is more appropriate for instructors to say that the effect merely prompts the issue of the particle-like nature of light although the concept of the photon can be introduced from the effect. If this position is accepted, then students' difficulty in interpreting the particle-like nature of light in the context of photoelectric effect might not reflect on their lack of understanding. Therefore, a more careful reinterpretation of the results of some previous studies is required, since the studies largely assumed that the effect provides good evidence for the particle model of light[4,5].