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Asymmetric translation between multiple representations in
chemistry
Yulan I. Lin
a
,JiY.Son
b
and James A. Rudd II
a
a
Department of Chemistry & Biochemistry, California State University, Los Angeles, CA, USA;
b
Department of
Psychology, California State University, Los Angeles, CA, USA
ABSTRACT
Experts are more proficient in manipulating and translating between
multiple representations (MRs) of a given concept than novices.
Studies have shown that instruction using MR can increase student
understanding of MR, and one model for MR instruction in
chemistry is the chemistry triplet proposed by Johnstone.
Concreteness fading theory suggests that presenting concrete
representations before abstract representations can increase the
effectiveness of MR instruction; however, little work has been
conducted on varying the order of different representations during
instruction and the role of concreteness in assessment. In this
study, we investigated the application of concreteness fading to MR
instruction and assessment in teaching chemistry. In two
experiments, undergraduate students in either introductory
psychology courses or general chemistry courses were given MR
instruction on phase changes using different orders of presentation
and MR assessment questions based on the representations in the
chemistry triplet. Our findings indicate that the order of
presentation based on levels of concreteness in MR chemistry
instruction is less important than implementation of comprehensive
MR assessments. Even after MR instruction, students display an
asymmetric understanding of the chemical phenomenon on the
MR assessments. Greater emphasis on MR assessments may be an
important component in MR instruction that effectively moves
novices toward more expert MR understanding.
ARTICLE HISTORY
Received 5 May 2015
Accepted 18 January 2016
KEYWORDS
Science education; multiple
representations; chemistry
education; concreteness
fading; assessment
Introduction
Multiple representations
The ability to link information and ideas across multiple representations (MRs) for a given
concept is a more meaningful indicator of understanding than manipulation of symbolic
notation (Ainsworth, 1999; Prain, Tytler, & Peterson, 2009; Prain & Waldrip, 2006; Trea-
gust, Chittleborough, & Mamiala, 2003). For example, many chemistry students can learn
to balance reaction equations, but providing correct coefficients and chemical symbols for
equations is not the same as understanding the molecular behavior or macroscale
phenomena represented by the equations (Gabel, Samuel, & Hunn, 1987; Yarroch,
© 2016 Informa UK Limited, trading as Taylor & Francis Group
CONTACT James A. Rudd II jrudd@calstatela.edu Department of Chemistry & Biochemistry, California State Uni-
versity, 5151 State University Drive, Los Angeles, CA 90032, USA
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION, 2016
http://dx.doi.org/10.1080/09500693.2016.1144945
1985). The importance of MR understanding is also true of other fields, including physics,
math, and biology (Hmelo-Silver, Marathe, & Liu, 2007; Kohl & Finkelstein, 2008; Kohl,
Rosengrant, & Finkelstein, 2007; McNeil & Fyfe, 2012; Schnotz & Kulhavy, 1994; Tsui &
Treagust, 2013). Previous work in MR has investigated expert use of MR (Chi, Feltovich, &
Glaser, 1981; Kozma & Russell, 1997; Larkin, McDermott, Simon, & Simon, 1980), student
understanding of MR, and the use of MR in instruction (Bibby & Payne, 1993; Dienes,
1973; Hennessy et al., 1995; Kaput, 1989; Oliver, 1997; Schwartz, 1995; Tabachneck, Koe-
dinger, & Nathan, 1994; Thompson, 1992).
Experts are better at manipulating and translating between MRs than novices. They
tend to represent underlying function and behavior in their models, while novices base
their models on surface features of appearance and structure (Chi et al., 1981; Hmelo-
Silver et al., 2007; Kozma & Russell, 1997; Larkin et al., 1980). Chemistry experts
connect MR that are conceptually related more efficiently and accurately than novices
(Kozma, 2003). Physics graduate students translate between MR during problem-
solving more easily than physics undergraduate students (Kohl & Finkelstein, 2008).
Even in domains outside of the natural sciences (e.g. economics), experts reason by seam-
lessly transitioning between MR (Larkin & Simon, 1987).
Unlike experts, novices generally find working with MR difficult. Studies assessing MR
understanding have demonstrated that students are worse at problems that require them
to translate between different representations than single-representation problems (Ains-
worth, Wood, & Bibby, 1996,1998; Ramnarain & Joseph, 2012; Tabachneck, Leonardo, &
Simon, 1994; Yerushalmy, 1991). However, there has been little work to address what
types of MR problems may be more difficult than others. It is commonly assumed that
deficiencies in MR understanding may be traced back to instructional practices: not all
representations are given the same weight in instruction. For example, there may be an
overpresentation of one particular representation type in instruction or it may be more
common to ask students to translate from one representation to another than vice versa
(e.g. typically students are asked to provide a graph from equation than the other way
around, Dugdale, 1982). There have been many attempts to improve MR translations
through innovative pedagogy (e.g. Hennessy et al., 1995; Kozma, Chin, Russell, & Marx,
2000; Thompson, 1992).
Given the centrality of MR in expert-like thinking, each knowledge domain should con-
sider which MR to include in instruction. One model for MR in chemistry instruction is
the chemistry triplet, first proposed by Johnstone (1982) and consisting of three represen-
tations: the macroscale, the nanoscale (also referred to as the ‘micro’or the ‘sub-micro’),
and the symbolic (Figure 1) (Johnstone, 2000a,2000b,2009).
The macroscale representation is at the human scale in which natural phenomena can be
observed through the senses (sight, touch, etc.). The nanoscale representation is at the
molecular scale of molecules, atoms, and other particles that cannot be directly observed
by human senses. The symbolic representation is the abstract representation of natural
phenomena through the use of symbols, equations, and so on. The chemistry triplet is
often shown as the corners of an equilateral triangle to symbolize the equal importance of
each type of representation and the links between them in understanding chemistry. The
edges of the triangle represent possible translations among the three representations.
Although this interpretation of the triplet is common, other valid frameworks and per-
spectives exist and are in use in chemistry education research and instruction (Gilbert &
2Y. I. LIN ET AL.
Treagust, 2009a; Taber, 2013; Talanquer, 2011). Thus, implementation of the triplet model,
whether for research or instruction, requires clear identification of the triplet and the
aspects that are being emphasized. To provide clarity before proceeding further, our
implementation attempts to hold closely to Johnstone’s views of the macroscale, nanoscale
(submicro), and symbolic. Our view of the macroscale emphasizes the actual physical
phenomena experienced tangibly through human senses, rather than a macroscale property
or conceptual framework such as density or pH. Our view of the nanoscale emphasizes ball-
and-stick and space-filling models as descriptive, explanatory, and predictive represen-
tations of the nanoscale, rather than as mere symbolic icons (Talanquer, 2011). Lastly,
our view of the symbolic emphasizes that the symbols, formulas, equations, and so on,
span the macroscale and nanoscale (Taber, 2013), for example, H
2
O(s) symbolizes both
the macroscale ice and the nanoscale collection of water molecules vibrating closely
together in fixed positions in an ordered structure.
Perhaps due in part to the potential for ambiguity in interpretations of the triplet,
chemistry novices (students) are far less skilled than chemistry experts at translating
between the corners of the triplet and understanding the underlying concepts that tie
the different representations together. Not only do chemistry students correctly balance
equations without understanding the meaning of the equations (Gabel et al., 1987),
they are most comfortable manipulating symbols and symbolic representations using
flawed algorithms, rather than considering underlying concepts (Gabel, 1993; Gabel
et al., 1987; Nurrenbern & Pickering, 1987; Nyachwaya, Warfa, Roehrig, & Schneider,
2014; Smith & Metz, 1996). Students also misunderstand the relationship between macro-
scale properties and nanoscale processes (Griffiths & Preston, 1992; Lee, Eichinger, Ander-
son, Berkheimer, & Blakeslee, 1993). For example, some students assume that an atom
isolated from a gas will embody the bulk properties the gas exhibits on the macroscale,
just on a smaller scale (Ben-Zvi, Eylon, & Silbemein, 1986). A study of student perform-
ance on a standardized chemistry exam revealed that South African 12th graders per-
formed worse on questions that require translation between MR than on questions that
do not require translation (Ramnarain & Joseph, 2012).
Student proficiency in manipulating symbols without understanding the underlying
meaning and their discomfort in translating across MR may be a result of chemistry
instruction that concentrates on symbolic representations. Unsurprisingly, instruction
that explicitly teaches MR has been shown to strengthen student understanding of MR
and increase their ability to translate between the different corners of the chemistry
triplet (Gabel, 1993). In one study, tenth-grade Lebanese students who were explicitly
Figure 1. The chemistry triplet
Source: Adapted from Johnstone (1982).
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 3
taught the different corners and the relationships between corners performed significantly
better on a concept map task than students who did not receive the MR instruction (Jaber
& BouJaoude, 2012). In a study of eighth-grade Greek students who were taught macro-
scale representations first, then symbolic, and then nanoscale performed better on postas-
sessments and retained more information (Georgiadou & Tsaparlis, 2000). Even in
domains outside of chemistry, explicitly teaching MR has been found to improve
student understanding of MR (Kohl et al., 2007; Tsui & Treagust, 2013).
Although MR instruction in science appears to be more effective than single-represen-
tation instruction, less work has been done identifying what types of MR translations are
most difficult for students and whether MR instruction can alleviate those difficulties.
Translations between different corners of the chemistry triplet may vary in difficulty
depending on the corners involved and the direction of translation. In other words,
there may be an asymmetry in students’ability to move between different representations.
As a theoretical framework for investigating these issues, we look to the research on the
benefits of concrete and abstract representations in the learning sciences.
Concreteness fading: combining learning benefits of concrete and abstract
representations
There have been many empirical investigations regarding the role of concrete and abstract
representations in advancing conceptual and generalizable understanding. Concrete rep-
resentations are connected to their referents through perceptual similarity and are often
linked to learners’prior experience. For instance, a concrete representation of melting
could be a video of ice melting in a glass. In contrast, abstract materials are more arbitrarily
associated with referents and are perceptually stripped down in form. A chemical equation
represents melting in an abstract way because symbols such as (s) and (l) reference phys-
ical states only by convention. Studies in cognition have demonstrated two principles: (1)
instruction with MR generally leads to more robust understanding than with a single rep-
resentation (e.g. Brenner et al., 1997; Gentner & Markman, 1997) and (2) presenting the
most concrete instantiation first then presenting abstract materials, known as concreteness
fading, leads to better generalization (see Fyfe, McNeil, Son, & Goldstone, 2014 for a recent
review). University students in the USA who received instruction about complex systems
with concrete-then-abstract representations performed better on a transfer test than stu-
dents who either received abstract-to-concrete instruction, abstract representations only,
or concrete representations only (Goldstone & Son, 2005). Another study examined con-
creteness fading instruction in the context of learning algebra and found that undergradu-
ates who received concreteness fading instruction outperformed those who either received
abstract-only instruction or concrete-only instruction on delayed tests administered three
weeks after initial learning (McNeil & Fyfe, 2012).
The mechanism behind the success of concreteness fading may have to do with max-
imizing the benefits of concrete and abstract materials (Fyfe et al., 2014). Concrete
materials ground a new concept in a familiar or graspable context, while abstract materials
limit unnecessary detail, facilitating generalization. Progressing from concrete to abstract
examples initially anchors new knowledge in already familiar territory then moves the
learner toward a more abstract and transferrable understanding of the concept.
A second benefit of progressing monotonically along a concreteness continuum is that
4Y. I. LIN ET AL.
it minimizes the cognitive leaps needed to move from one example to the next (Freu-
denthal, 1983; Kotovsky & Gentner, 1996). For instance, moving from a macroscale rep-
resentation to a nanoscale representation may be more accessible than moving all the way
from a macroscale representation to a symbolic representation.
Concreteness fading and MR assessment
Much of the research on concreteness fading has focused on the presentation of MR to
novice learners (Fyfe et al., 2014). Only a few studies have examined concreteness in
assessment, and these studies contrast assessments utilizing concrete materials with assess-
ments utilizing abstract materials (e.g. Petersen & McNeil, 2013). Very little research has
focused on the design and implementation of assessing students’ability to connect and
translate across concrete and abstract MR. One study on pattern perception in pre-
school-aged children found that the direction of assessment, in this case an abstract cue
with concrete answer choices versus a concrete and perceptually rich cue with abstract
answer choices, can affect generalization (Son, Smith, & Goldstone, 2011). Research on
the direction of assessment is critical because the way in which we query students
defines the scope of understanding that students can demonstrate. A student might
appear quite competent on one type of assessment but not be able to demonstrate their
understanding on a different type of assessment. Thus, in considering how to measure
student understanding of MR, we need assessments that examine connections between
MR. An open question is whether science students can translate from concrete to abstract
representations as well as they can translate from abstract to concrete.
Concreteness and the chemistry triplet
The concreteness and abstractness of the three corners of the chemistry triplet can be
interpreted through different perspectives. Gilbert and Treagust (2009b) raise the issue
of relating ‘types’of representations to ‘levels’of representations that define the cognitive
relationship between the types of representations (i.e. the separate corners of the triplet)
from the human learner perspective. Level could mean differences in physical scale
from macro to meso to nano (submicro), but level could also indicate differences in the
language that describes phenomena, such as concrete descriptions of macroscale to
abstract chemical symbols. Thus, two psychological dimensions of concreteness may be
operating in the chemistry triplet: scale, where the macroscale that is more perceptible
to humans could seem more concrete and the less perceptible nanoscale could seem
less concrete, and language, where familiar language is more concrete and more special-
ized language that requires training and education is more abstract. Justi, Gilbert, and Fer-
reira (2009) identify concrete as one mode of representation used to communicate a
person’s mental model and indicate that the model can be expressed as a mixture of rep-
resentation modes (concrete, verbal, etc.) when providing the external representation to
the learner (p. 286). This dimension of concreteness in modes of communication allows
for representations to be viewed as more or less concrete, for example, a broad qualitative
analogy describing a concept may feel more concrete to a learner than a quantitative,
mathematical expression of the same concept.
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 5
In the work presented in this paper, our goal was to map the corners of the triplet to a
single concreteness continuum in order to situate our research within the broader frame-
works developed in the cognitive and learning sciences, and we began with the basis that
all representations (verbal, pictorial, symbolic, video, gesture, etc.) can be placed on a con-
creteness continuum. Specifically, we define the concreteness of a representation as simi-
larity to the referent or the intended meaning of the representation. Thus, the macroscale
is considered the most concrete of the three corners because macroscale representations,
whether they are verbal descriptions or videos of macroscale phenomena, most closely
resemble their referents. For example, a video of ice melting more closely resembles the
actual observable phenomena of ice melting, and is therefore more concrete than the sym-
bolic representation of ice melting: H
2
O(s)→H
2
O(l).
Symbolic representations, such as chemical symbols, formulas, and equations, are less
concrete because they are more arbitrarily connected to the referent. Also, symbolic rep-
resentations are very reduced representations of the referent, and the relationship between
the symbolic representation and the meaning of the representation is more obscured.
Specialized training is typically needed in order for a person to succeed in connecting
the representation with its meaning. Using ‘H’to mean hydrogen simply because hydro-
gen starts with the letter ‘H’is a connection between the symbolic representation and
referent that is arbitrary and reliant on conventions in our culture and language. For
instance, ‘Cu’as the symbolic representation of the substance ‘copper’is more arbitrary
for English speakers than for French speakers (‘cuivre’). In French, the symbolic represen-
tation at least resembles the word more closely than in English, but in both languages, the
symbols are still more arbitrary than the relationship between a macroscale representation,
such as a picture showing a sample of each metal, of these referents.
We place nanoscale representations in between the more concrete macroscale and more
abstract symbolic representations. Nanoscale representations are somewhat concrete in
that ball-and-stick and space-filling models capture some of the perceptual qualities of
the referent of atoms and bonds than the arbitrariness of symbols. However, ball-and-
stick nanoscale representations are also less concrete than macroscale representations
because nanoscale representations also have elements of arbitrariness, for example,
colors are assigned to different elements by convention, such as oxygen atoms are often
colored red and the size of the balls distort the actual and relative size of atoms.
Essentially, the more concrete end of our concreteness continuum captures more of the
qualities of the referent and requires less specialized training to understand the meaning of
the representation, whereas the more abstract end of the continuum has fewer and weaker
connections between the representation and the referent and requires more specialized
training for learners to interpret the representation. Although this approach is not the
only way of describing the concreteness and abstractness of the corners of the chemistry
triplet, it allows us to connect to the learning literature research on manipulating concre-
teness to help students move between concrete and abstract representations.
Concreteness fading and the chemistry triplet
In this study, we investigated the application of concreteness fading to MR instruction and
assessment in teaching chemistry. To do this, the chemistry triplet was mapped to a con-
creteness continuum (Figure 2(a)).
6Y. I. LIN ET AL.
For the purposes of this study, the macroscale representation of objects, phenomena,
and manipulations is on the human scale (i.e. observed or experienced through sight,
hearing, touch, etc.) and is considered the most concrete (Johnstone, 2000a,2000b,
2009). The nanoscale representation of the molecular level is less concrete because the
nanoscale is less perceptually accessible to humans; however, molecules, atoms, and so
on can be represented by graphical icons (e.g. pictures of spheres to represent atoms)
or physical models (e.g. ball-and-stick models) that provide a perceptual anchor. Lastly,
the symbolic representation is the least concrete (most abstract) because chemical
symbols and equations reference substances and processes by convention in a highly effi-
cient and simplified manner (Johnstone, 2000a,2000b; Taber, 2013). Thus, in mapping the
chemistry triplet onto a concreteness continuum, we have placed the macroscale and sym-
bolic at the extremes and the nanoscale as intermediate between them.
Because the chemistry triplet is typically shown as an equilateral triangle, there may be
an underlying assumption that each representation (corner) and each direction of trans-
lation between representations are somehow equal (Figure 2(b)). However, considering a
linear concreteness continuum may help us explore the hypothesis that these represen-
tations and translations may not be cognitively equivalent for chemistry novices. Although
the ideal may be that chemistry students should understand each representation equally
well, some translations may initially be easier than others (Figure 2(b)).
If deep understanding of chemistry includes understanding all three corners, how they
relate, and how to translate between them, then using a concreteness continuum provides
a framework for asking questions about MR instruction and assessment. Should the more
or less concrete representation be presented first in MR instruction? Are students equally
adept at translating from concrete to more abstract representations or from abstract to
more concrete ones? That is, does the direction of translation matter in assessment?
Here, we report new findings on MR instruction and assessment, and our findings shed
light on students’ability to translate between representations. Specifically, our investi-
gation addressed the following research questions to examine the role of concreteness
in learning MR in the domain of phase changes in chemistry:
(1) Type of first presentation: do students perform better with concrete-first or abstract-
first instruction? That is, do students perform better with macroscale instruction first
or symbolic instruction first?
Figure 2. (a) The relationship between the chemistry triplet and concreteness fading. (b) Directionality
of transitions between the corners of the chemistry triplet.
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 7
(2) Inclusion of a progression: do students perform better with instruction that progresses
monotonically along a concreteness continuum vs. instruction that does not include a
progression? That is, would students benefit most from macroscale, then nanoscale,
then symbolic instruction?
(3) Directionality of assessment: do students perform better on concrete-to-abstract or
abstract-to-concrete questions? That is, are students more adept at translating from
more concrete to less concrete corners of the chemistry triplet (black arrows in
Figure 2(b)) or vice versa (white arrows in Figure 2(b))?
Experiment 1
Students were shown chemistry instructional videos on phase changes in small groups that
were randomly assigned to one of four instructional conditions. Each participant com-
pleted pen-and-paper pre- and postassessments that contained two types of questions
(concrete-to-abstract, abstract-to-concrete).
Methods
Participants
One hundred forty-seven undergraduate students (100 female, 44 male, 3 declined to
state) from the Psychology department subject pool (most of whom were enrolled in intro-
ductory Psychology courses) at a mid-sized comprehensive public university on the west
coast of the USA participated in Experiment 1. Participants were given course credit for
being in the study. Twenty additional participants were excluded from analysis because
they did not complete the study.
Instruments
Instructional videos
Participants viewed three videos during the instruction, and each video presented phase
changes from the perspective of one corner of the chemistry triplet.
The macroscale instructional video introduced the macroscale, showed videos of water
freezing and ice melting (obtained from vimeo.com by permission of the owners), and had
added voiceover narration that described the melting and freezing of water being shown.
The nanoscale instructional video first introduced the nanoscale, including a brief
nanoscale drawing tutorial. Then, the video showed a screencast recording of one of the
researchers interacting with the ‘States of Matter: Basics’simulation (PhET States of
Matter: Basic Simulation, n.d.), which again had added voiceover narration.
The symbolic instructional video introduced symbolic notation, and presented sym-
bolic representations for phase changes and three properties associated with phase
changes (velocity, kinetic energy, and temperature). Because velocity, kinetic energy,
and temperature were presented to the participants as symbols (i.e. V, KE, T), we con-
sidered these to be symbolic representations (although the actual underlying concepts
could be interpreted through other perspectives of the triplet, e.g. temperature as a macro-
scale construct and kinetic energy as a nanoscale concept).
8Y. I. LIN ET AL.
All videos discussed kinetic molecular theory; the relationship between velocity, kinetic
energy, and temperature; and the relationship between the energy of the atoms or mol-
ecules and the phase or state of matter.
Assessments
Participants completed two-part assessments before and after instruction, which were
designed to assess students’ability to translate between the different representations of
the chemistry triplet. Each question required students to connect different representations
by giving one representation of a concept and asking students to provide a different
representation.
In Experiment 1, questions were written to assess two directions of translation: con-
crete-to-abstract and abstract-to-concrete. Figure 3 shows the relationship between ques-
tion directionality and the chemistry triplet.
The questions were also designed to assess transfer beyond memorization of presented
materials. Though the instruction focused on melting and freezing of water, participants
were asked about other substances and states of matter. Two parallel versions of the assess-
ments were created with nearly identical questions. For example, if Assessment A Ques-
tion 1 asked about a substance melting, then Assessment B Question 1 asked about the
same substance freezing. Participants randomly received either Version A of the pretest
and posttest or Version B. Sample questions are shown in Table 1.
The pretest and posttest had concrete-to-abstract and abstract-to-concrete translation
questions. At the end of the posttest, participants ranked their confidence in their answers,
estimated their ability to learn science, and reported how much they liked science.
Experimental design and procedure
The experiment used a pretest, intervention, and posttest procedure and a 2 × 2 × 2 mixed
repeated-measures design corresponding to the three research questions. The two
between-subjects factors were first presentation (concrete-first, abstract-first) and pro-
gression (progression, no progression) (Table 2). The within-subjects factor was question
directionality (concrete-to-abstract, abstract-to-concrete).
For the concrete-first, progression condition, the videos were presented in the order
macroscale–nanoscale–symbolic (M–N–S). The M–N–S condition is a progression
Figure 3. Assessment questions classified by directionality. (a) Concrete-to-abstract; (b) abstract-to-
concrete; and (c) abstract-to-abstract.
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 9
because the examples progress monotonically along the concreteness continuum from
most concrete to most abstract. For the abstract-first, progression condition, the videos
were presented in the order symbolic–nanoscale–macroscale (S–N–M). The S–N–M con-
dition is a progression because the examples progress monotonically along the concrete-
ness continuum from most abstract to most concrete. For the concrete-first, no
progression condition, the videos were presented in the order macroscale–symbolic–
nanoscale (M–S–N). For the abstract-first, no progression condition, the videos were pre-
sented in the order symbolic–macroscale–nanoscale (S–M–N). Both M–S–N and S–M–N
conditions did not progress monotonically on the concreteness continuum.
Participants signed up for experimental sessions in groups of 10–16 students. Each
session was randomly assigned to one of the four presentation conditions. Participants
were allotted 11 minutes to complete the pretest and were then shown the instructional
videos for 23 minutes. Participants were then given 11 minutes to complete the postassess-
ment. At the end of the posttest, participants completed a brief demographic survey.
Data analysis
Coding
The responses were scored using a rubric for each question and yielded for each partici-
pant an overall score, a concrete-to-abstract score, and an abstract-to-concrete score for
the pre- and postassessments. For each participant, a gain score was calculated as posttest
score minus pretest score.
Responses involving drawn nanoscale images were independently scored by another
rater using the same scoring rubric, and discrepancies were resolved by an experienced
chemistry professor. This coding was blind to presentation condition.
Statistical analysis
The gain scores were analyzed using a 2 × 2 × 2 mixed repeated-measures ANOVA (question
direction × first presentation × progression) with participants’pretest scores as a covariate.
Table 1. Sample questions classified by direction.
Sample concrete-to-abstract question Sample abstract-to-concrete question
(Given macro →provide symbolic)
Given a chunk of solid aluminum, represent it in symbols.
(Given macro →provide nano)
Draw a nanoscale picture to represent solid aluminum.
(Given nano →provide symbolic)
Given a set of nanoscale pictures below, pick out the
corresponding chemical equation.
(Given symbolic →provide nano)
Select the nanoscale picture that most accurately represents a
sample with the symbol H
2
O(s).
(Given symbolic →provide macro)
Name the physical process represented by the chemical
equation N
2
(g)→N
2
(l).
(Given nano →provide macro)
Name the physical process represented by the nanoscale
image below.
Table 2. Four instructional conditions based on first presentation and progression.
Progression No progression
Concrete-first M–N–SM–S–N
Abstract-first S–N–MS–M–N
10 Y. I. LIN ET AL.
Results and discussion
The ANOVA results revealed that first presentation and progression had no statistically
discernable effect (F< 0.20, p> .700) and no significant interactions (F< 1.5, p> .23).
However, there was a main effect of question directionality (F(1, 147) = 233.67, p< .001,
η
2
= 0.62). Students improved significantly more on concrete-to-abstract questions than
on abstract-to-concrete questions (Figure 4) for all presentation conditions.
Because participant credit was based only on attendance, rather than performance on
the postassessment, participants had little incentive to learn the material and perform well
on the assessment. To address this limitation, Experiment 2 was designed to include
general chemistry students who were motivated to learn the material.
Experiment 2
Two types of students were randomly assigned to one of four instructional conditions
(Table 2). Each participant completed an online assignment composed of instructional
videos and a postinstructional assessment with three types of questions (concrete-to-
abstract, abstract-to-concrete, abstract-only).
Methods
Participants
Two hundred forty-nine undergraduate students enrolled at a mid-sized comprehensive
public university on the west coast of the USA participated in the study for course
credit. Students from the Psychology department subject pool (n= 168; 124 females, 36
males, 8 declined to state) received credit for attendance. Students enrolled in a general
chemistry course (n= 81; 43 females, 38 males) received credit scaled to their assessment
performance. The inclusion of chemistry students enhances the validity of the study
results because these students had incentive to learn the material and perform well on
the assessment.
Figure 4. Mean gain scores for the two different types of questions (n= 146).
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 11
Datasets were excluded from analysis if participants did not complete the study, includ-
ing those who spent less than 20 minutes on the study because the instructional videos
were over 17 minutes in length and the assessment required a minimum of three
minutes for one of the researchers to complete.
Instruments
The instructional videos and assessment questions were the same as for Experiment 1,
except the questions were slightly modified for online delivery. Also, a third type of assess-
ment question (abstract-to-abstract) was added because most general chemistry tests focus
on symbolic understanding and manipulations.
Experimental design and procedure
The experiment used an intervention and posttest procedure and a 2 × 2 × 2 × 3 mixed
repeated-measures design. The pre-assessment was removed to reduce participant time
and pretest effects. The between-subjects factors were first presentation (concrete-first,
abstract-first), progression (progression, no progression), and student population (psy-
chology students, chemistry students). The within-subjects factor was question direction-
ality (concrete-to-abstract, abstract-to-concrete, abstract-to-abstract).
Participants were randomly assigned to one of the four presentation conditions using
Qualtrics software (Provo, UT). The postassessment and demographic survey were also
presented on Qualtrics immediately after the instructional videos.
Students from the Psychology department subject pool signed up in groups of 1–20 to
participate in a computer lab. Each participant had an individual workstation with head-
phones and was given 45 minutes to complete the study.
Students enrolled in the general chemistry course received the link to the Qualtrics
experiment from their course instructor after the first midterm exam. Students completed
the study independently as a homework assignment and were asked to provide their name
and instructor name to award them appropriate class credit. Before data analysis, all iden-
tifying information was decoupled from the performance data.
Data analysis
Coding
The responses were scored using a rubric for each question and yielded for each partici-
pant an overall score, a concrete-to-abstract score, an abstract-to-concrete score, and an
abstract-to-abstract score. These raw scores were converted to proportion correct postas-
sessment scores. All coding was blind to presentation condition.
Statistical analysis
The postassessment scores were analyzed using a 2 × 2 × 2 × 3 mixed repeated-measures
ANOVA (first presentation × progression × student population × question direction).
Results and discussion
The ANOVA results showed that first presentation and progression did not have a stat-
istically significant effect (Fs < 0.1, p> .2); however, there was a statistically significant
12 Y. I. LIN ET AL.
effect of student population, F(1, 241) = 53.01, p< .001, η
2
= 0.24, and question direction, F
(1, 241) = 74.56, p< .001, η
2
= 0.34. There were no reliable interactions.
Unsurprisingly, students enrolled in the chemistry course (M= 0.71, SD= 0.20) per-
formed significantly better than students enrolled in the psychology course (M= 0.54,
SD = 0.22). Post-hoc-corrected simple comparisons revealed that participants performed
the best on abstract-to-abstract questions, then concrete-to-abstract questions, and
worst on abstract-to-concrete questions (ps < .001). Mean scores on these three question
types are shown in Figure 5. The asymmetry in student performance based on question
direction that was found in Experiment 1 was confirmed here with a sample of students
that had more motivation to learn chemistry and do well on the assessment, and presum-
ably, more initial chemistry knowledge.
Discussion
Taken together, the experiments reveal unexpected findings regarding concreteness and
MR chemistry instruction and assessment. First, neither experiment yielded differences
in student performance between different instructional conditions. The order of MR
instruction, whether starting with most concrete vs. most abstract or including a pro-
gression, did not observably impact student performance, meaning that concreteness
fading did not appear to have an effect, at least for learning how to translate between
MR in chemistry. Although our experimental design did not yield any significant influence
of instructional order, it may be that the brief instructional period limited students’ability
to gain sufficient understanding of the triplet, and additional research may detect an effect
of MR instructional order over longer time periods (e.g. an entire class period or course)
and/or with a wider variety of chemistry topics.
Differences in the learning goal between this study and most concreteness fading
studies may also explain the absence of any effect. In most concreteness fading studies,
the goal is for students to learn to generalize to a new and completely different concrete
example of the abstract concept being examined (Goldstone & Son, 2005; McNeil &
Figure 5. Mean postassessment scores for the two different types of questions (n= 249).
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 13
Fyfe, 2012). For example, in a typical study, US undergraduates were taught the commu-
tative rule using generic symbols, cups, and pies and were assessed by being asked to gen-
eralize to examples with ladybugs, vases, and rings (McNeil & Fyfe, 2012). In this study,
however, the goal was for students to understand and translate between MR of the
concept, and students were assessed by being asked to translate across MR for new
examples of the concept, rather than simply by identifying new instances of the
concept. Similarly, another study assessing MR use in chemistry evaluated participants
on their ability to translate between videos, graphs, animations, and equations of the
same concept (Kozma & Russell, 1997).
When translation is the goal, rather than just identifying examples in new contexts,
presentation order may matter less. In addition, the MRs in the chemistry triplet may
be less like a continuum of concrete to abstract examples of a given concept and more
like three very different aspects of the concept. For example, a realistic video of ice
melting and a cartoon depiction of ice melting can represent more and less concrete
examples of melting, respectively. In contrast, a realistic video of melting, a cartoon
depiction of molecular movement during melting, and the chemical symbols for
melting are less like different examples of melting, but instead, are more like three differ-
ent dimensions or perspectives on the same phenomena that highlight very different
information.
Second, the experiments reveal unbalanced or asymmetric student understanding of
chemistry. Students were better at translating from concrete to abstract representations
vs. abstract to concrete representations, regardless of instructional condition for both
experiments. In other words, students apparently possessed an asymmetric understanding
of phase changes because they exhibited a greater ability to translate from representation A
to representation B (i.e. concrete-to-abstract) as compared to translating from represen-
tation B to representation A (i.e. abstract-to-concrete).
Little published work has examined the idea of symmetric and asymmetric under-
standing of and translation between MR. In mathematics, symmetric understanding
has been suggested as a necessary requirement for a complete understanding (Rider,
2007), and one study found asymmetric understanding in math students who were
more proficient at translating from equations to graphs versus translating from graphs
to equations (Yerushalmy, 1991). The same study found that students were instructed
to generate graphs from equations more often than the reverse (Yerushalmy, 1991),
so asymmetric understanding may be a result of asymmetric instruction and/or
assessment.
Depending on the specific concept and even the scientific domain, different trans-
lations may be more or less challenging for students. The nature of the subject itself
could foster asymmetric understanding, which may be further compounded by or
result in asymmetric instruction. For example, in chemistry, the macroscale represen-
tation is usually more familiar and more concrete to students (e.g. melting of an ice
cube), but in astronomy, the macroscale may be so expansive (e.g. stars light years
away and appearing as points of light) as to be unfamiliar and less concrete to students.
Students who exhibit stronger ability to translate from concrete-to-abstract vs. abstract-
to-concrete in chemistry may exhibit the opposite asymmetry in astronomy. Even
within one domain, such as chemistry, different asymmetries may exist when translating
between MRs for different concepts.
14 Y. I. LIN ET AL.
Conclusions
This study shows that the order of MR chemistry instruction by levels of concreteness does
not impact student understanding and that comprehensive MR assessments reveal asym-
metric understanding of chemistry. First, the order of presenting macroscale, nanoscale,
and symbolic representations of chemistry in very brief instructional periods may not
matter as much as providing instruction on all three corners of the chemistry triplet in
any order to develop more expert understanding. Alternatively, the pedagogical approach
may be related to the ideal order of presentation. For more passive pedagogy, such as the
viewing of video lectures in this study, the order may not have an effect, but for more
active learning methods, such as problem-based and inquiry-based learning, the presen-
tation order may have a greater role (e.g. the problem or question being investigated
may benefit from initially being presented at the concrete macroscale). For any approach,
MR instruction should explicitly teach translation between representations in multiple
directions to develop more symmetric understanding and translation ability. Such an
approach may help students move beyond one mode of thinking to develop stronger con-
ceptual understanding of a subject.
Second, we find asymmetric understanding even after MR instruction on all three
corners of the chemistry triplet. Typical chemistry instruction does not cover all three
corners to the same extent but rather focuses on symbolic representations, and our
ongoing research confirms that additional asymmetries can exist when instruction is
limited to fewer corners. Further research may also ascertain whether asymmetries
occur across scientific domains as well as elucidate the role of asymmetries in pedagogy.
Importantly, the use of MR assessments may play a critical role in framing MR instruc-
tion and research on MR instruction. Assessments not only reveal the limits of student
understanding, but they also circumscribe the range of understanding that students can
demonstrate. Because the types of assessments signal to students what concepts and
skills they should learn, asymmetric assessments focus students on developing asymmetric
understanding and translation skills. Designing and implementing comprehensive MR
assessments that translate in multiple directions would be more likely to promote
student ability to translate in all directions. In other words, a greater focus on MR assess-
ment (i.e. MR testing as part of teaching) would enhance MR instruction for all students
(primary, secondary, tertiary). Such assessments would also be useful for informing the
instructional design cycle by indicating the types of translations that are most challenging
for students.
Similarly, the design of symmetric assessments is an area where further research is
needed for all levels (primary, secondary, and tertiary) of MR instruction. As found in
this study, assessment design revealed more about student performance than pedagogy
design. There is a tendency in the research to compare different types of instruction
more than different types of assessments, but for investigations of MR instruction, an
increased focus on the assessment design may yield useful insights about highly effective
MR instructional approaches. It is likely that different pedagogies bias student learning
toward asymmetries in MR translation but need symmetric assessments to identify the
asymmetric performance.
Finally, MR and concreteness fading are broad concepts encompassing multiple modes
of thinking. Although the approach in this study may not apply to all types of MR, it may
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 15
be relevant to disciplines that utilize models potentially analogous to the chemistry triplet
(Table 3).
The ability to translate between MR goes beyond understanding and learning science
and math in academic contexts. Phenomena such as traffic jams, global trade, poverty,
and the preservation of ecosystems all function at multiple levels, with different infor-
mation and concepts relevant at each level (Holland, 2006). Information about these
real-world problems are presented using MRs, and being able to translate effectively
between these representations is critical for decision-makers and stakeholders. Developing
effective MR pedagogy and assessments is a modest but crucial contribution that educators
and educational researcher can make.
Acknowledgements
The authors would like to acknowledge Donna Chen for assisting in coding and Angela Guererro
for assisting with literature review. Krzystof Dwornik, Jonathan Shrader, and youtube user 33342
shooting channel provided permission to use their videos.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes on contributors
Yulan Lin graduated with honors with the Bachelor of Science degree in chemistry from California
State University, Los Angeles in 2014.
Ji Son is an assistant professor of psychology at California State University, Los Angeles.
James A. Rudd II is a professor of chemistry and biochemistry at California State University, Los
Angeles.
References
Ainsworth, S. (1999). The functions of multiple representations. Computers & Education,33(2–3),
131–152.
Ainsworth, S., Wood, D., & Bibby, P. (1996). Coordinating multiple representations in computer
based learning environments. In P. Brna, A. Paiva, & J. A. Self (Eds.), Proceedings of the
European conference on Artificial Intelligence in Education (pp. 336–342). Lisbon: Edicoes
Colibri.
Ainsworth, S., Wood, D., & Bibby, P. (1998). Analysing the costs and benefits of multi-represen-
tational learning environments. In S. Vosniadou, K. Matsagouras, K. Mardaki-Kassotaki, & S.
Kotsanis (Eds.), 7th European conference for research on learning and instruction (pp. 500–
501). Athens: Gutenberg University.
Table 3. Multiple levels of concreteness in representations from different disciplines.
Math Physics Chemistry Biology
Concrete example Macroscale Macroscale Macroscale
Faded example Invisible (forces) Nanoscale Microscale
Symbolic notation Symbolic Symbolic Biochemical
Source: Adapted from Johnstone (2000a) and McNeil and Fyfe (2012).
16 Y. I. LIN ET AL.
Ben-Zvi, R., Eylon, B., & Silbemein, J. (1986). Is an atom of copper malleable? Journal of Chemical
Education,63(1), 64–66.
Bibby, P. A., & Payne, S. J. (1993). Internalizing and the use specificity of device knowledge.
Human-Computer Interaction,8(1), 25–56.
Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Duran, R., Reed, B. S., & Webb, D. (1997).
Learning by understanding: The role of multiple representations in learning algebra. American
Educational Research Journal,34(4), 663–689.
Chi, M., Feltovich, P., & Glaser, R. (1981). Categorization and representation of physics problems
by experts and novices. Cognitive Science,5, 121–152.
Dienes, Z. P. (1973). The six stages in the process of learning mathematics. Windsor, Ontario: NFER
Publishing.
Dugdale, S. (1982). Green globs: A micro-computer application for graphing of equations.
Mathematics Teacher,75, 208–214.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structure. Dordrecht: Kluwer
Academic.
Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics
and science instruction: A systematic review. Educational Psychology Review,26(1), 9–25. doi:10.
1007/s10648-014-9249-3
Gabel, D. L. (1993). Use of the particle nature of matter in developing conceptual understanding.
Journal of Chemical Education,70(3), 193–194.
Gabel, D. L., Samuel, K. V., & Hunn, D. (1987). Understanding the particulate nature of matter.
Journal of Chemical Education,64(8), 695. doi:10.1021/ed064p695
Gentner, D., & Markman, A. B. (1997). Structure mapping in analogy and similarity. American
Psychologist,52(1), 45–56.
Georgiadou, A., & Tsaparlis, G. (2000). Chemistry teaching in lower secondary school with
methods based on: a) psychological theories; b) the macro, representational, and submicro
levels of chemistry. Chemistry Education Research and Practice,1(2), 217–226.
Gilbert, J. K., & Treagust, D. F. (Eds.). (2009a). Multiple representations in chemical education.
Dordrecht: Springer.
Gilbert, J. K., & Treagust, D. F. (2009b). Towards a coherent model for macro, submicro, and sym-
bolic representations in chemical education. In J. K. Gilbert & D. F. Treagust (Eds.), Multiple rep-
resentations in chemical education (pp. 333–350). Dordrecht: Springer.
Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and ideal-
ized simulations. The Journal of the Learning Sciences,14(1), 69–110.
Griffiths, A. K., & Preston, K. R. (1992). Grade-12 students’misconceptions relating to fundamental
characteristics of atoms and molecules. Journal of Research in Science Teaching,29(6), 611–628.
doi:10.1002/tea.3660290609
Hennessy, S., Twigger, D., Driver, R., O’Shea, T., O’Malley, C. E., Byard, M., …Scanlon, E. (1995).
Design of a computer-augmented curriculum for mechanics. International Journal of Science
Education,17(1), 75–92.
Hmelo-Silver, C. E., Marathe, S., & Liu, L. (2007). Fish swim, rocks sit, and lungs breathe: Expert-
novice understanding of complex systems. Journal of the Learning Sciences,16(3), 307–331.
doi:10.1080/10508400701413401
Holland, J. J. H. (2006). Studying complex adaptive systems. Journal of Systems Science and
Complexity,19(1), 1–8.
Jaber, L. Z., & BouJaoude, S. (2012). A macro–micro–symbolic teaching to promote relational
understanding of chemical reactions. International Journal of Science Education,34(7), 973–
998. doi:10.1080/09500693.2011.569959
Johnstone, A. H. (1982). Macro and microchemistry. School Science Review,64, 377–379.
Johnstone, A. H. (2000a). Chemical education research: Where from here. University Chemistry
Education,4,34–38.
Johnstone, A. H. (2000b). Teaching of chemistry –Logical or psychological? Chemistry Education
Research and Practice Europe,1(1), 9–15.
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 17
Johnstone, A. H. (2009). Multiple representations in chemical education. International Journal of
Science Education,31(16), 2271–2273. doi:10.1080/09500690903211393
Justi, R., Gilbert, J. K., & Ferreira, P. F. M. (2009). The application of a ‘model of modeling’to illus-
trate the importance of metavisualisation in respect of the three types of representation. In J. K.
Gilbert & D. F. Treagust (Eds.), Multiple representations in chemical education (pp. 285–307).
Dordrecht: Springer.
Kaput, J. J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner & C.
Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 167–194). Reston,
VA: NCTM.
Kohl, P., & Finkelstein, N. (2008). Patterns of multiple representation use by experts and novices
during physics problem solving. Physical Review Special Topics –Physics Education Research,
4(1), 010111. doi:10.1103/PhysRevSTPER.4.010111
Kohl, P. B., Rosengrant, D., & Finkelstein, N. D. (2007). Strongly and weakly directed approaches to
teaching multiple representation use in physics. Physical Review Special Topics –Physics
Education Research,3(1), 010108. doi:10.1103/PhysRevSTPER.3.010108
Kotovsky, L., & Gentner, D. (1996). Comparison and categorization in the development of rela-
tional similarity. Child Development,67, 2797–2822.
Kozma, R. (2003). The material features of multiple representations and their cognitive and social
affordances for science understanding. Learning and Instruction,13(2), 205–226.
Kozma, R., Chin, E., Russell, J., & Marx, N. (2000). The roles of representations and tools in the
chemistry laboratory and their implications for chemistry learning. The Journal of the
Learning Sciences,9(2), 105–143.
Kozma, R. B., & Russell, J. (1997). Multimedia and understanding: Expert and novice responses to
different representations of chemical phenomena. Journal of Research in Science Teaching,34,
949–968.
Larkin, J., & Simon, H. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive
Science,99,65–99.
Larkin, J. H., McDermott, J., Simon, D. P., & Simon, H. A. (1980). Models of competence in solving
physics problems. Cognitive Science,4(4), 317–345.
Lee, O., Eichinger, D. C., Anderson, C. W., Berkheimer, G. D., & Blakeslee, T. D. (1993). Changing
middle school students’conceptions of matter and molecules. Journal of Research in Science
Teaching,30(3), 249–270.
McNeil, N. M., & Fyfe, E. R. (2012). ‘Concreteness fading’promotes transfer of mathematical
knowledge. Learning and Instruction,22(6), 440–448. doi:10.1016/j.learninstruc.2012.05.001
Nurrenbern, S. C., & Pickering, M. (1987). Concept learning versus problem solving: Is there a
difference? Journal of Chemical Education,64, 508.
Nyachwaya, J. M., Warfa, A. R. M., Roehrig, G. H., & Schneider, J. L. (2014). College chemistry stu-
dents’use of memorized algorithms in chemical reactions. Chemistry Education Research and
Practice,15(1), 81. doi:10.1039/C3RP00114H
Oliver, M. J. (1997). Visualisation and manipulation tools for modal logic (Unpublished doctoral
dissertation). Open University, UK.
Petersen, L. a, & McNeil, N. M. (2013). Effects of perceptually rich manipulatives on preschoolers’
counting performance: Established knowledge counts. Child Development,84(3), 1020–1033.
doi:10.1111/cdev.12028
PhET States of Matter: Basic Simulation. (n.d.). University of Colorado, Boulder.
Prain, V., Tytler, R., & Peterson, S. (2009). Multiple representation in learning about evaporation.
International Journal of Science Education,31(6), 787–808.
Prain, V., & Waldrip, B. (2006). An exploratory study of teachers’and students’use of multi-modal
representations of concepts in primary science. International Journal of Science Education,28
(15), 1843–1866. doi:10.1080/09500690600718294
Ramnarain, U., & Joseph, A. (2012). Learning difficulties experienced by grade 12 South African
students in the chemical representation of phenomena. Chemistry Education Research and
Practice,13(4), 462–470.
18 Y. I. LIN ET AL.
Rider, R. (2007). Shifting from traditional to nontraditional teaching practices using multiple rep-
resentations. Mathematics Teacher,100(7), 494–500.
Schnotz, W., & Kulhavy, R. W. (Eds.). (1994). Comprehension of graphics (Vol. 108). North
Holland: Elsevier.
Schwartz, D. L. (1995). The emergence of abstract representations in dyad problem solving. The
Journal of the Learning Sciences,4(3), 321–354. doi:10.1207/s15327809jls0403_3
Smith, K., & Metz, P. (1996). Evaluating student understanding of solution chemistry through
microscopic representations. Journal of Chemical Education,73(3), 233–235. doi:10.1021/
ed073p233
Son, J. Y., Smith, L. B., & Goldstone, R. L. (2011). Connecting instances to promote children’s rela-
tional reasoning. Journal of Experimental Child Psychology,108(2), 260–277. doi:10.1016/j.jecp.
2010.08.011
Tabachneck, H., Koedinger, K., & Nathan, M. (1994). Toward a theoretical account of strategy use
and sense-making in mathematics problem solving. In A. Ram & K. Eiselt (Eds.), Proceedings of
the 16th annual conference of the cognitive science society (pp. 836–841). Hillsdale, NJ: LEA.
Tabachneck, H. J. M., Leonardo, A. M., & Simon, H. A. (1994). How does an expert use a graph? A
model of visual and verbal inferencing in economics. In A. Ram & K. Eiselt (Eds.), Proceedings of
the 16th annual conference of the cognitive science society (pp. 842–847). Hillsdale, NJ: LEA.
Taber, K. S. (2013). Revisiting the chemistry triplet: Drawing upon the nature of chemical knowl-
edge and the psychology of learning to inform chemistry education. Chemistry Education
Research and Practice,14(2), 156–168. doi:10.1039/C3RP00012E
Talanquer, V. (2011). Macro, submicro, and symbolic: The many faces of the chemistry ‘triplet’.
International Journal of Science Education,33(2), 179–195. doi:10.1080/09500690903386435
Thompson, P. W. (1992). Notations, conventions, and constraints: Contributions to effective uses
of concrete materials in elementary mathematics. Journal for Research in Mathematics
Education,23, 123. doi:10.2307/749497
Treagust, D., Chittleborough, G., & Mamiala, T. (2003). The role of submicroscopic and symbolic
representations in chemical explanations. International Journal of Science Education,25(11),
1353–1368. doi:10.1080/0950069032000070306
Tsui, C., & Treagust, D. F. (2013). Multiple representations in biological education (Vol. 7). Springer
Science & Business Media. doi:10.1007/978-94-007-4192-8
Yarroch, W. L. (1985). Student understanding of chemical equation balancing. Journal of Research
in Science Teaching,22(5), 449–459. doi:10.1002/tea.3660220507
Yerushalmy, M. (1991). Student perceptions of aspects of algebraic function using multiple rep-
resentation software. Journal of Computer Assisted Learning,7,42–57.
INTERNATIONAL JOURNAL OF SCIENCE EDUCATION 19
