Article pubs.acs.org/jchemeduc
Unpacking the Meaning of the Mole Concept for Secondary School Teachers and Students Su-Chi Fang,*,† Christina Hart, and David Clarke International Centre for Classroom Research, Melbourne Graduate School of Education, University of Melbourne, Melbourne, Victoria, 3053, Australia ABSTRACT: The “mole” is a fundamental concept in quantitative chemistry, yet research has shown that the mole is one of the most perplexing concepts in the teaching and learning of chemistry. This paper provides a survey of the relevant literature, identifies the necessary components of a sound understanding of the mole concept, and unpacks and presents these components in the form of a concept map. The concept map incorporates the atomic−molecular concept with the mole concept, and connects the two concepts by two linking ideas: the number aspect of the SI definition (linking idea 1) and the connection between relative atomic−molecular mass and molar mass (linking idea 2). This concept map not only provides a conceptual framework for making meaning in relation to the mole but also sheds some light on how the concept might be better taught and learned. KEYWORDS: High School/Introductory Chemistry, Misconceptions/Discrepant Events, Stoichiometry, Atomic Properties/Structure
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BACKGROUND This paper reports the findings from an in-depth content analysis relating to the concept of the mole. This content analysis formed a significant part of a larger project that investigated the process of teaching and learning the mole concept in secondary school classrooms. The content analysis was expressed in the form of a concept map and provided a framework for the analysis of classroom observations. The authors consider that the concept map itself may provide a useful tool for teachers, and that the underlying content analysis could make a useful contribution to the pedagogical content knowledge1 relating to this fundamental chemical concept.
that, while it is necessary that student conceptions of the mole should be consistent with the SI definition, this does not imply that the SI definition is the most effective or appropriate instructional representation of the mole concept. This distinction between the need for consistency with the SI definition and its effectiveness as an instructional tool is at the heart of the argument put forward in this paper. One significant finding to emerge from the research of the past 40 years is that the way the mole is conceptualized in educational settings is actually inconsistent with the meaning of the mole expressed in the SI definition. For example, Cervellati et al.4 designed a diagnostic test to investigate secondary Italian students’ conceptions about the mole. The results showed that the students did not understand the mole to be the SI unit of the physical quantity “amount of substance”. Also, while the students all knew the magnitude of Avogadro’s number, they did not, however, understand how this is determined experimentally. Gabel and Bunce5 reviewed the literature on students’ conceptions of the mole from 1970 to 1990.4,6−9 Their review concluded that students lacked a coherent understanding of the mole concept and articulated understandings that were conceptually inconsistent with the SI definition. Strömdahl et al.10 interviewed 28 chemistry educators (teachers, lecturers, and textbook authors) about how they conceptualized the mole. They identified several qualitative differences among the educators’ definitions of the mole, with only three of the educators articulating a conception that was consistent with the SI definition. Although each educator stated
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LITERATURE REVIEW The mole was defined in 1971 as the unit of a physical quantity, the “amount of substance”, and adopted as one of the basic units in the SI system.2 According to the IUPAC (International Union of Pure and Applied Chemistry) (14th CGPM: ref 2, p 70): A mole is the amount of substance of a system which contains as many elementary entities as there are carbon atoms in 0.012 Kg of 12C. The elementary entity must be specified. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. In the same year Johnstone et al.3 published surveys on Scottish secondary students’ difficulties with topics in school chemistry courses. They found that secondary students’ difficulty with the mole concept was widespread. Several studies have investigated and discussed the difficulties in teaching and learning the mole concept. It is important to note © 2014 American Chemical Society and Division of Chemical Education, Inc.
Published: February 11, 2014 351
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a preferred definition, they often called on an alternative definition when they encountered a different problem. When the same group of researchers11 interviewed upper-secondary students, they found that the students’ conceptions basically mirrored those of the educators. Most of the students defined the mole either as Avogadro’s number or as a mass, a finding confirmed by Staver and Lumpe.12 Strömdahl et al.10 pointed out that the educators were able to switch easily between the different descriptions in a “professional manner” as required by different problems. In contrast, a great proportion of the students were confused about when it is appropriate or beneficial to use which definition.11 Furió et al.13 also found that teachers themselves were confused about the meaning of “amount of substance”. In their research, few participating teachers correctly defined the mole as the unit of the quantity “amount of substance”. Most perceived “amount of substance” as either a mass or a number (of elementary entities). The literature reviewed highlights the complexity of the mole concept. Both students and educators ignore the relevance of the physical quantity “amount of substance” and of the SI definition of the mole and, instead, rely on multiple, frequently inconsistent, interpretations of the meaning of the mole.
Figure 1. Meanings embedded in the SI definition of the mole.
In fact, there are two layers of meaning subsumed in the standard pack. The first one is straightforward: it represents a number of atoms. The other is indirect; in the case of 12C, this standard pack weighs 12 g. Across this binary meaning, the mole serves as a bridge, connecting the number of particles at the atomic or molecular level and the mass at the macroscopic level. This binary meaning is fundamental to the mole concept and to its use in chemistry (stoichiometry): to use weights of substances to ensure the correct ratio of particles participating in a chemical reaction. However, the SI definition in itself tells us explicitly only what one mole of 12C should be. How one mole of other isotopes and elements in different states (solid, liquid, or gaseous) can be worked out, or are related, is not explained in the scientific definition. Neither is it clear how the definition helps to ensure that for a given chemical reaction we have the same (or the correct ratio of the) number of reacting particles. In other words, the definition does not say how the designation of one mole in terms of 12 g of 12C enables arithmetic calculations in stoichiometry in relation to substances other than 12C.
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RESEARCH QUESTIONS The studies cited above appear to have implicitly assumed that the SI definition of the mole serves as a benchmark of a good understanding of the meaning of the mole concept. This further implies that a sound understanding depends on seeing the mole as the unit of a quantity “amount of substance”. This assumption raises the fundamental question: Can the SI definition of the mole on its own lead students to form a meaningful conceptual understanding of the mole? This paper aims to address this pivotal question by first pursuing the following questions: 1. What information is contained in the SI definition of the mole? 2. What might be accepted as a sound understanding of the mole? 3. What subordinate concepts are essential for making the concept of the mole meaningful and scientifically coherent? In the following section, we reflect on and reconceptualize the mole concept and try to describe what might be accepted as a sound understanding of the mole. We then develop a concept map that identifies relevant concepts and clarifies their interrelationships. This concept map is intended to provide a conceptual framework for making the mole concept meaningfulparticularly to studentsin a scientifically coherent way.
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The Underlying Concept 12
C is taken by chemists and physicists as an agreed standard for measuring the masses of atoms and molecules. According to the IUPAC Gold Book, the unified atomic mass unit (symbol: u) is the unit used to express the masses of atoms and molecules. “1 u” (one unified atomic mass unit) is defined as “one-twelfth of the mass of a 12C atom in its ground state”.14 Therefore the atomic mass of 12C is exactly 12 u. Other specific atomic isotopes have an atomic mass expressed in u. For example, the atomic mass of 16O is 15.9949 u. Based on the 12C scale, together with the scientific definition of one mole, we can deduce the masses of one mole, named the “molar mass” of other elements. The key point here is that when the amounts of two elements have the same number of atoms, the mass ratio of these two amounts will be the same as the ratio of their atomic masses. For example, one atom of 20Ne has a mass of 19.99 u. Provided that we have the same numbers of 12C atoms as 20Ne atoms, say three of each, the ratio of their total masses will still be 12 to 19.99. Now one mole of 12C weighs 12 g (from the scientific definition), so one mole of 20 Ne weighs 19.99 g. In other words, 19.99 g of 20Ne will have the same number of atoms as 12 g of 12C. The thinking route is shown in Figure 2.
RECONCEPTUALIZING THE MOLE CONCEPT
What Does the SI Definition Tell Us?
According to the SI definition (see Literature Review), one mole is an “amount of substance”, and this amount is defined by what can be called “a standard pack”. The standard pack is the number of 12C atoms in 12 g of 12C, that is, Avogadro’s constant, NA (see Figure 1). Experimentally, this number is found to be 6.02214084(18) × 1023. Therefore, any sample of any substance that contains the same number of identical particles (atoms, molecules, or ions) as the standard pack is called one mole.
Figure 2. The relationship between molar mass and atomic mass of atoms of selected isotopes. 352
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mass is the same as relative atomic mass. Also, Padilla et al.15 found that all the professors participating in their research considered relative atomic mass to be a fundamental component of an understanding of the mole concept. Padilla et al.15 point out that “the concept of relative (atomic) mass appears most important because it represents the key for counting elementary entities by weighing”. In fact, the atomic−molecular concept relates directly to the significance of the mole concept and the reasons why chemists needed to invent the mole.16 On one hand, the atomic− molecular concept enables chemists to theoretically think about discrete atom(s) and molecule(s), and their atomic and molecular masses at the microscopic level. On the other hand, by means of the mole concept, chemists can practically manipulate an extremely large number of atoms and molecules on the macroscopic level by weighing them.
However, most of the time, chemists do not deal with only one isotope of an element. On the contrary, they have to deal with huge numbers of atoms and molecules, consisting of a mixture of isotopes. Therefore, chemists use relative atomic mass to simplify the situation. The relative atomic mass (or atomic weight, Ar) is defined as “the ratio of the average mass of the atom to the unified atomic mass unit”.14 “Average mass” means the average of the atomic masses of all the atoms of an element found in a particular sample, weighted by isotopic abundance. It is important to note that relative atomic mass is a ratio without a unit. Sometimes these ratios are given in the periodic table, for example, the relative atomic mass of carbon (C) is 12.01 and the relative atomic mass of neon (Ne) is 20.18. Because atomic masses are based on the 12C scale, its derivativesrelative atomic masscan also be traced back to this origin. Thus, relative atomic mass is a ratio indirectly based on the 12C scale. Consequently, as Figure 3 shows, we can have proportional relationships between relative atomic mass and molar mass.
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THE CONCEPT MAP In order to integrate the understandings and reasoning outlined in the discussion above, we developed a concept map. On this map, concepts are represented as rectangular shapes. The large rectangle framing the entire schematic diagram represents the mole concept itself. Smaller rectangular shapes represent the subconcepts, such as molar mass. The solid lines represent “linking ideas” or mathematical relationships between the subconcepts. Connecting the Number Aspect and the Mass Aspect
From the preceding discussion, the meaning of the quantity amount of substancecan be seen from two aspects, the number aspect and the mass aspect. The number aspect represents “a number of entities”. For example, 2 mol of 20Ne has (2 × 6.02 × 1023) 20Ne atoms. The mass aspect comes from the SI definition, which states the relationship between the number of elementary entities in a sample and its mass. In the case of 2 mol of 20Ne atoms, their total mass is 39.98 g. This two-aspect view (number and mass) forms the vertical dimension of the concept map (see Figure 4).
Figure 3. The relationship between molar mass and relative atomic mass.
Because molecules are composed of atoms, the ideas used for atoms can be extended to molecules. For this reason, we have a connection between molar masses of compounds and their relative molecular masses. Thus, one mole of an element or compound has a numerical value for its molar mass in grams that is identical to its relative atomic or molecular mass. This provides great convenience in arithmetic calculations in stoichiometry. From the numerical identity between molar mass in grams and relative atomic or molecular mass, one can easily calculate the amount of a substance in moles from the mass of the sample and its molar mass. A Conceptual Understanding of the Mole Concept
From the above reflection, we conclude that a sound understanding of the mole concept consists of much more than simply being able to articulate the SI definition. The concepts of relative atomic or molecular mass, of molar mass, and of the connection between molar mass and relative atomic or molecular mass have to be included. Underlying all of these ideas is the atomic−molecular concept, including the concepts of atom, molecule, atomic mass, and relative atomic or molecular mass. In addition, the reason why 12 g and 12C were designated needs to be made explicit, namely, how this enables us to figure out the molar mass of the elements other than 12C. There is strong support in the literature for this conclusion. For example, Staver and Lumpe12 tentatively proposed that student difficulties, both in defining the mole according to the SI definition and in making a connection between molar mass and relative atomic mass, explain how students come to form a common alternative conception about molar mass: that molar
Figure 4. The number and the mass aspect of the mole concept in the case of 2 mol of 20Ne atoms.
The Relationships among Constituent Subconcepts of the Mole Concept
The relationships among amount of substance, number of elementary entities, and mass are labeled as “relationship 1”, “relationship 2”, and “relationship 3”, respectively. The number of atoms in one mole, namely, the number in 12 g of 12C, has been experimentally determined as 6.02 × 1023. 353
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This means that we can figure out the amount of substance in a given sample, provided we know the ratio between the total number of elementary entities in a sample and NA (6.02 × 1023). Therefore, the total number of elementary entities divided by 6.02 × 1023 is the amount of substance expressed in moles, that is, N/NA = n (Figure 5, relationship 1). Similarly, as
Figure 6. The incorporation of “the atomic−molecular concept” in the concept map.
Linking Idea 1
At first glance, linking idea 1 (see Figure 7), between a single atom or molecule and a large number of elementary entities Figure 5. The relationships among constituent subconcepts of the mole concept.
molar mass means the mass of one mole of a substance, the mass of a sample of a substance divided by its molar mass gives the amount of substance expressed in moles. The formula m/M = n, stands for this relationship (Figure 5, relationship 2). Relationship 3 is the combination of relationships 1 and 2 and connects the number aspect of the mole to its mass aspect. These three relationships amount to arithmetic conversions between amount of substance, mass, and number of elementary entities. If one thinks deeply about the conceptual meaning of relationship 3 (m/M = n = N/NA), it becomes apparent that the amount of substance is like a bridge, linking the number of elementary entities (N) to the mass of the sample (m). That actually presents the essence of the mole concept: how chemists make sure of the right number, or rather the right ratio, of the number of particles for chemical reactions by weighing the substances involved. Up to this point, we have included in the concept map the following relevant subconcepts: • Amount of substance • Molar mass and number of elementary entities • The mathematical relationships between them
Figure 7. Linking idea 1: The number aspect of the SI definition. Linking idea 2: the connection between relative atomic or molecular mass and molar mass.
(i.e., the number aspect of the SI definition of the mole), seems to be straightforward. This is the idea of aggregating, or accumulating atoms or molecules from a single entity to one mole, namely, 6.02 × 1023 entities. However, the idea of aggregating atoms and molecules to a standard pack cannot on its own explain why the number 6.02 × 1023 is chosen, or what this number has to do with the number of atoms in 12 g of 12C. The first question (why 6.02 × 1023?) has to be answered by drawing on the mass aspect of the SI definition: according to the SI definition, the standard pack is the number of atoms in 12 g of 12C. Then, one might ask the second question: how do we know that 6.02 × 1023 is the number of atoms in 12 g of 12 C? In fact, this number was determined experimentally. Historically, this was first determined by Johann Joseph Loschmidt in 1865 by measuring the rate of diffusion of gases.17,18 One of the modern methods makes use of X-ray crystallography.19 The essential point is that 6.02 × 1023 has been empirically determined and its magnitude is a consequence of the choice of “gram” as the macroscopic unit of mass. To sum up, linking idea 1 is the number aspect of the SI definition of the mole. Additionally, it relates to the mass aspect indirectly because “12 g” of 12C defines the “number” of atoms or molecules in one mole of a substance. For this idea to be
Two Approaches to “Amount of Substance”
However, as we have suggested above, a sound understanding of the mole concept necessarily incorporates the atomic− molecular concept (see Figure 6). The two aspects that apply when viewing the mole concept, the number and the mass aspects, also apply to the atomic−molecular concept. The key idea emphasized from the number aspect is that substances are composed of atoms or molecules. The mass aspect relates to the concept of relative atomic or molecular mass. To connect the atomic−molecular concept to the mole concept, two critical linking ideas are required. These are shown in Figure 6, and each is discussed below. With these linking ideas, each aspect of the atomic−molecular concept may be connected to the corresponding aspect of the mole concept. Together, these two ideas are crucial for making meaning of the mole concept. 354
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chemical reactions appeared to “force us to focus attention more on the relationship between amount of particles that intervene than on that of combining weights” (p 277).16 Nevertheless, in practice, whatever their beliefs about the nature of matter, chemists’ quantitative calculations were based on experimentally determined equivalent weights. The attempt to make two apparently incompatible paradigms, the equivalentist and the atomistic paradigm, converge helped the scientific community to reach agreement on what the word “mole” actually means. However, the consequences of the convergence were that multiple meanings and conceptual contradictions resulting from the historical evolution of the mole have become sedimented in the SI definition of the mole. These multiple meanings continue to circulate and underpin ongoing conceptual ambiguities in chemists’ understandings of the quantity, amount of substance, and its unit, the mole. In scientific settings, for practicing chemists who are comfortable with using the mole in stoichiometric calculations, these ambiguities seem to pose no problems. However, in educational settings, the complex ideas embedded in the SI definition and the exceptional epistemology of the mole actually contribute to students’ learning difficulties.21 Padilla and Furió21 argued specifically in the case of the mole that the absence of historical knowledge in science teaching leads to distorted views of science and impedes meaningful learning. Certainly, we would argue that an understanding of the historical development of the mole concept is one useful approach to assisting students learning to construct a conceptually complete and coherent understanding of the mole. We further suggest, however, that coherent understanding can be promoted by the encryption of this history in suitable instructional media, such as the concept map proposed in this paper.
meaningful, the experimental methods for determining this number have to be described and explained. Linking Idea 2
Linking idea 2 in Figure 7 represents the connection between relative atomic mass and molar mass. We have already pointed out that, as long as the numbers of particles of two different elements are the same, the mass ratio would be the same as the ratio of their relative atomic masses (proportional reasoning). By definition, one mole of any substance always has the same number of atoms or molecules as 12 g of 12C. Consequently, based on the 12C scale (the concept of relative atomic or molecular mass) and proportional reasoning, the relative atomic or molecular mass of a substance in unified atomic mass units (u) always has the same numerical value as its molar mass in grams (g). Linking idea 2 actually accounts for this numerical identity. Thus, the reasoning behind linking idea 2 incorporates three components: the mass aspect of the SI definition, according to which 12 g defines the standard pack; the concept of relative atomic mass; and proportional reasoning in mathematics.20 In other words, linking idea 2 uses proportional reasoning and the standard pack to connect relative atomic mass and molar mass. Summarizing the Concept Map
To summarize, the concept map (see Figure 7) introduced above represents the components of, and concepts relevant to, the mole concept, together with their interrelationships. Undoubtedly, neither the SI definition of the mole nor the mathematical relationships, such as n = N/NA or n = m/M, alone constitute an adequate base for a sound understanding of the mole concept. The atomic−molecular theory and the two linking ideas indicated in the concept map are essential for making meaning of the concept of the mole. Indeed, the mole is a complex concept; this might be one of the reasons that the expression “the mole concept” is often used. This contrasts with other SI units such as “the gram” or “the second”one seldom sees reference to “the gram concept”.
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A Proposal To Meaningfully Teach the Mole Concept
As we have pointed out in the literature review, since 1971 a large number of studies have looked into the difficulties of teaching and learning the mole. In fact, there are approximately 60 articles related to the mole published in the Journal of Chemical Education in the past four decades. We reviewed these articles and identified two main research themes: (i) to discuss and suggest instructional strategies; and (ii) to look into learning difficulties.16,22 Although instructional strategies such as analogies and practical work are useful in teaching and learning, they are of limited effectiveness in assisting students to understand the mole concept integratively and coherently. In our opinion, for better teaching of the mole, teachers need a conceptual framework that helps to transform their understanding of the concept into a teachable form. We suggest that the concept map presented in this paper could be of significant value for educational instruction. It provides a conceptual framework for teaching and points out explicitly that the key to making meaning of the mole is to relate the mole concept to the atomic−molecular concept by two linking ideas. Specifically, in order to meaningfully teaching the concept of the mole, teachers need to guide students to appreciate the relationship between the mole and the atomic− molecular concepts from both the number and the mass aspect. The number aspect of this relation (i.e., linking idea 1) provides a theoretical (atomic) view of chemical reactions by explaining and predicting what are found (experimentally) to be the right amounts of chemicals involved in a chemical reaction. The mass aspect in terms of the connection between molar mass and
DISCUSSION
The Historical Roots of the Mole Explain the Interweaving Concepts Embedded
Quantitative chemistry developed while two apparently incommensurable views of the nature of matter were in operation: the equivalentist (continuous) and atomistic paradigms.21 Chemists who viewed matter as continuous preferred to use equivalent masses to explain chemical reactions, whereas atomists thought in terms of atoms and molecules. The terms “mole” and “amount of substance” were introduced in the late 1880s by Wilhelm Ostwald.13 Ostwald was an advocate of the equivalentist paradigm, so he defined these concepts in terms of a mass quantity. From the 20th century, the worldwide acceptance of the atomic−molecular theory of matter to interpret chemical changes eventually led the scientific community to adopt amount of substance as a fundamental quantity and to define the mole as its unit.16 At first glance, the SI definition of a mole accords with the atomistic paradigm because it refers to a certain number of elementary particles. However this number is actually defined by a specific mass of a specific substance (12 g of 12C), an idea originally attached to the equivalentist paradigm. Adoption of the atomic−molecular framework to interpret how substances are restructured and formed in 355
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(15) Padilla, K.; Ponce-de-Leon, A.; Rembado, F.; Garritz, A. Undergraduate Professors’ Pedagogical Content Knowledge: The Case of Amount of Substance. Int. J. Sci. Educ. 2008, 30 (10), 1389−1404. (16) Furió, C.; Azcona, R.; Guisasola, J. The Learning and Teaching of the Concepts “Amount of Substance” and “Mole”: A Review of the Literature. Rev. Res. Pract. 2002, 3 (3), 277−292. (17) Porterfield, W. W.; Kruse, W. Loschmidt and the Discovery of the Small. J. Chem. Educ. 1995, 72 (10), 870−875. (18) Loschmidt, J. On the Size of the Air Molecules. Sitzb. Akad. Wiss. Wien 1865, 52, 395−413. (19) Fujii, K.; Waseda, A.; Kuramoto, N.; Mizushima, S.; Becker, P.; Bettin, H.; Nicolaus, A.; Kuetgens, U. Present State of the Avogadro Constant Determination from Silicon Crystals with Natural Isotopic Compositions. IEEE Trans. Instrum. Meas. 2005, 54 (2), 854−859. (20) DeSanabia, J. A. Relative Atomic Mass and the Mole: A Concrete Analogy To Help Students Understand These Abstract Concepts. J. Chem. Educ. 1993, 70 (3), 233−234. (21) Padilla, K.; Furió, C. The Importance of History and Philosophy of Science in Correcting Distorted Views of Amount of Substance and Mole Concepts in Chemistry Teaching. Sci. Educ. 2008, 17 (4), 403− 424. (22) Fang, S. Teaching and Learning the Mole Concept: An Investigation of Science Secondary Classrooms in Australia and Taiwan. Ph.D. Thesis, The University of Melbourne, November 2011. (23) Guisasola, J.; Furió, C.; Ceberio, M. Science Education Based on Developing Guided Research. In Science Education in Focus; Thomase, M. V., Ed.; Nova Science Publishers: New York, 2008; pp 173−202.
relative atomic mass (i.e., linking idea 2) explains why the molar mass of a substance has the same numerical value as its relative atomic or molecular mass, and which rationalizes practical laboratory work. Some promising instructional strategies for linking idea 2 have been shown.23 The proposal that the teaching of the mole should be structured around the concept map and linking ideas 1 and 2 resonates with previous research findings suggesting that to meaningfully understand the concept of amount of substance and the mole, “one has to be aware that it is a macroscopic quantity related directly to the microscopic world of substance (it serves to count atoms and molecules)” (p 287).16
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Present Address †
Graduate Institute of Science Education, National Taiwan Normal University, No. 88, Sec. 4, Ting-Chou Rd., Wunshan District, Taipei, Taiwan 11677, R.O.C. Notes
The authors declare no competing financial interest.
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REFERENCES
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