A study of undergraduate physics students’ understanding of heat conduction based on mental model theory and an ontology–process analysis

LEARNING
Maria Varelas and Michael Ford, Section Coeditors
A Study of Undergraduate Physics
Students’ Understanding of Heat
Conduction Based on Mental
Model Theory and an
Ontology–Process Analysis
GUO-LI CHIOU
Graduate School of Technological and Vocational Education, National Taiwan University
of Science and Technology, Taipei 106, Taiwan
O. ROGER ANDERSON
Mathematics, Science & Technology, Teachers College, Columbia University, New York,
NY 10027, USA
Received 25 February 2009; revised 1 November 2009; accepted 9 November 2009
DOI 10.1002/sce.20385
Published online 29 December 2009 in Wiley Online Library (wileyonlinelibrary.com).
ABSTRACT: This study first used a new approach, combining students’ ontological beliefs
and process explanations, to represent students’ mental models of heat conduction and then
examined the relationships between their mental models and their predictions. Clinical
interviews were conducted to probe 30 undergraduate physics students’ mental models
and their predictions about heat conduction. This study adopted a constant comparative
method to discover patterns of the participants’ responses across the various sources of
data, such as verbal utterances, writings, and drawings. The results indicate that, based
on the identified five process analogies for how heat is conducted and three ontological
beliefs about the material basis for heat conduction, the combinations of these two aspects
can better represent their mental models in terms of both the underlying mechanisms and
emergent processes of heat conduction than using either alone as has sometimes been done
in prior research. In addition, while a scientifically accepted mental model had a better
Correspondence to: Guo-Li Chiou; e-mail: gc2158@columbia.edu
C 2009 Wiley Periodicals, Inc.

826 CHIOU AND ANDERSON
chance to be accompanied by a correct prediction, a correct prediction might not result
from a scientifically accepted mental model. However, as suggested by some cognitive
psychologists, regardless of which mental models the participants possessed, they tended
to automatically retrieve their learned rules or past experience, instead of manipulating
their mental models, to generate predictions for the encountered problems. C 2009 Wiley
Periodicals, Inc. Sci Ed 94:825–854, 2010
INTRODUCTION
One of the challenges in teaching physics is to help students develop a scientifically
compatible understanding of heat phenomena based on their existing ideas and beliefs. To
successfully achieve this goal, this study argues for a complementary approach to represent
students’ understanding of heat and the dynamic processes that accompany heat transfer.
Past studies have identified that the mismatch between students’ and scientists’ ontological
beliefs of heat can be attributed to be the major barrier to learn a coherent and scientifically
accurate understanding of this concept (Chi, Slotta, & de Leeuw, 1994; Vosniadou, 1994);
while most students tend to treat heat as a material substance, scientists define heat as a
dynamic process of energy transmission. However, even though most researchers recognize
the essence of heat as a process, few studies have specifically focused on students’ detailed
conceptualization of this process. Furthermore, past studies usually investigated students’
understanding of heat at a concept level, instead of a system level that corresponds to
the actual physical system in which heat transmission occurs. Thus, this study argues
that to better understand students’ understanding of heat, it is necessary to employ a set
of cognitive representations that can adequately demonstrate the systematic and dynamic
aspects of heat transmission in addition to the presentation of an appropriate physical
system. There is a growing consensus from various research fields that the construct of
mental models can serve as a highly suitable set of cognitive representations to meet the
previously stated need (e.g., Brewer, 2005; Clement & Rea-Ramirez, 2008; Glynn & Duit,
1995; Nersessian, 1999, 2008; Norman, 1983). Moreover, since one of the major functions
of mental models, among other facilitatory roles, is to help people generate predictions
and explanations (Norman, 1983), this study highlights the relationships between students’
mental models relevant to thermal systems and their resultant predictions and explanations.
Promisingly, the investigation of students’ mental models of heat transfer in terms of its
dynamic processes, as well as how students use mental models to make predictions and
explanations, can greatly extend our understanding of students’ learning about heat.
Nature of Mental Models
In the present study, mental models refer to individuals’ internal, mental representations
of external, physical phenomena or systems (Gilbert, Boulter, & Elmer, 2000; Vosniadou
& Brewer, 1992, 1994). The major feature of this mental representation is its analogous
structure to what is represented. That is, a mental model can be thought of as an imaginary
structure that corresponds to the externally represented or perceived system in terms of
the spatial arrangement of elements involved in the system and the relationships between
or among these elements. From this perspective, a mental model of a specific domain is
not merely a collection of memorized facts or beliefs relevant to that domain (Clement,
2008), but a set of mentally perceivable elements which can be manipulated within specific
conceptual constraints that determine the relationships between or among those elements
under certain conditions. Moreover, if the relationships between elements are causal in
nature, these relationships can help not only to reveal the mechanisms underlying the
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MENTAL MODELS OF HEAT CONDUCTION 827
processes of the system but also to determine the sequence of the changes in the states of
each element based on the system’s initial condition. It is in this sense that mental models
can be run “in the mind’s eye” (de Kleer & Brown, 1983, p. 185), or through mental
simulation (Nersessian, 2002, 2008), to generate predictions and explanations, which are
among the crucial functions of mental models.
In addition to the analogous and imagery-based representations mentioned above, rule-
based representations may have close relationships to mental model reasoning, particularly
in the domain of physical systems. Rules can be represented through linguistic or numerical
symbols (Schwartz & Black, 1996) and in the form of a production system, which uses
the “if–then” combination to describe what action an individual will take under a specific
condition (e.g., Anderson, 2005; de Kleer & Brown, 1983). The relationships between rules
and mental models can be postulated on the following two bases: the causal and practical
ones. First, with respect to the first kind, rules can represent the causal relationships
between elements in a physical system and thus can determine how the state of an element
will change resulting from the changes of other elements (de Kleer & Brown, 1983). These
causal rules may be either explicit or implicit and serve as physical constraints that delimit
the imaginary behaviors of the elements. Second, with respect to the practical kind, rules can
be pieces of practical knowledge that are constructed from repeatedly manipulating related
mental models in similar contexts and can thus be further applied to replace the process of
manipulating mental models to generate predictions more efficiently (Schwartz & Black,
1996). According to Schwartz and Black, people engage in the processes of manipulating
mental models only when they need to generalize rules, face novel situations, or when their
present rule fails. Otherwise, they tend to immediately apply a practical rule to make direct
predictions and skip the effortful processes of mental simulation. On the basis of these two
relationships between rules and mental models, this study argues that it is the inclusion
of rule-based representations that makes mental models in physical domains distinct from
those in general reasoning domains (e.g., Johnson-Laird, 1983), and that extends mental
models to encompass multiple representations beyond analogous ones (e.g., Hegarty, 2004;
Nersessian, 2008).
The Development of Mental Models
The construction of mental models is a continuously developmental process throughout
an individual’s lifetime. There is now a broad consensus that mental models first originate
from an individual’s constant interactions with related physical phenomena and systems
(Norman, 1983) and then develop on the basis of a series of assimilations and accom-
modations, or conceptual changes, stimulated by continuous exposure to diverse events
in her/his cultural and social environments (Glynn & Duit, 1995; Rea-Ramirez, Clement,
& Nunez-Oviedo, 2008; Vosniadou & Brewer, 1994). However, beyond this consensus,
there is no overall agreement about the detailed mechanisms underlying the development
of mental models accompanied by the processes of conceptual changes. Nowadays, there
are two competing theoretical approaches to address the developmental issue of mental
models. On the one hand, conceptions (a person’s understanding of a specific subject,
event, concept, etc.), as well as mental models, are assumed to develop within a common
theoretical framework, and thus a fundamental change at the theory level is necessary for a
na¨ıve mental model to be transformed into a scientifically acceptable one (e.g., Vosniadou
& Brewer, 1992, 1994; Wiser & Carey, 1983). On the other hand, conceptions are supposed
to be constructed through one’s isolated phenomenological primitives (p-prims), which
are abstracted from one’s perceptual experience of related phenomena, and thus the pro-
cesses of conceptual changes, as well as the development of mental models, are gradual
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reorganizations of the available, existing p-prims (e.g., diSessa, 1993, 2002, 2008). Since
the present study does not aim to resolve the complex issue of the development of mental
models, we will focus only on how the ontological issue relates to our study in the following.
Ontological aspects of mental representations, including a student’s ontological belief,
are defined in this research as students’ presuppositions about the ontological nature of
things, i.e., the representational entities or elements that comprise an interpretation of
phenomena, particularly those that have a correspondence to scientific concepts. Pupils’
ontological belief of a specific conception or mental model has attracted many researchers’
attention since the 1990s (e.g., Chi et al., 1994; Vosniadou & Brewer, 1994). Their under-
lying assumption is that to achieve a radical change in a conception or mental model, the
correspondent ontological belief of that conception or mental model must be changed. For
example, Vosniadou and Brewer (1994) claimed that children’s mental models are built
on the constraints of their ontological and epistemological beliefs (theses two are called
presuppositions). Moreover, they claim that when children encounter new information from
cultural or social interactions, if their entrenched ontological beliefs have not been changed,
their original, na¨ıve, mental models may simply combine with the new information to form
synthesized models, which are still under the original ontological constraint. That is, the
resulting models are merged and unrefined products without a radical change into a sci-
entifically acceptable model. Another example is Chi et al.’s (1994) proposal that some
scientific concepts are difficult to learn just because they are assigned to an incorrect onto-
logical category, such as the concept “heat” assigned to a “substance” category instead of
a “process” category. Accordingly, radical conceptual changes can be achieved only when
these concepts are reassigned to the correct, scientifically accepted ontological category.
These two proposals imply that conceptual changes can be achieved by a change in people’s
ontological belief about a specific conception or mental model.
The Inclusion of Dynamic Processes to Represent Mental Models
The present study, however, argues that changes in an individual’s ontological belief
are necessary but not necessarily sufficient for a mental model to become a scientifically
acceptable one. Many findings from research on mental models can be reinterpreted to
support this argument. For example, among Borges and Gilbert’s (1999) five constructed
patterns of the participants’ mental models of electricity, one of them was characterized as
“current as moving charges,” in which the participants thought that moving charges were the
building elements of a current and were powered by the chemical interactions that occurred
in a battery, but the participants failed to treat the current as an interactive system formed
by batteries, wires, and resistances. This particular finding can be reinterpreted as follows:
while the participants used ontologically correct elements, i.e., moving charges to represent
a current, this did not guarantee they possessed a comprehensive mental model that placed
the elements of their conception of an electrical current into a systemic perspective. The
acknowledgment of the insufficient role of ontological belief encourages us to search for a
more complementary approach to study individuals’ mental models.
Hence, it is suggested that in addition to examining the ontological belief, researchers
should place equal, if not more, emphasis on the interactions between identified elements
and particularly on their dynamics processes in a system, whether they are ontologically
correct or not. However, a refocusing of the mechanism governing the involved elements
and their dynamic processes in a mental model does not mean these have never been noticed
by past studies before. Rather, they could have been either taken as a distinct accompanying
property of a specific portion of an ontological belief, or treated in an overly simplified
manner. Regarding the former situation, for example, in D. Gentner and D. R. Gentner’s
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(1983) classical study, they found that students’ understanding about the features of two
series batteries (which double the voltage) and two series light bulbs (which double the
resistance) in currents can be, respectively, predicted by different analogical entities they
adopted, i.e., flowing water or moving crowd. Their assumption is that the analogical entity
applied in a mental model predetermines the accompanying mechanism simulated in a
mental model. However, it is worth noticing that D. Gentner and D. R. Gentner’s result
might be biased because before collecting data, they had filtered out potential participants
who had no comprehensive understanding or consistent responses about the behaviors of
the source of their analogy, i.e., water or moving crowd.
Regarding the latter situation, for example, although Hubber (2006) reported that students
used three different mental models of light based on three different ontological natures,
i.e., ray beam, particle beam, and wave-incorporated ray, he used nearly the same wording
to describe the processes of reflection in these three models: “one side of the ray (can be
replaced as particle or wave) hits the optically denser material and slows, swinging the ray
(can be replaced as particle or wave) into a different direction” (p. 426). The wording of
this sentence implies that different models of light have a common mechanism underlying
the propagation of light. However, modern physics suggests that waves and particles as
phenomena of light have sharply different properties, and it will be quite surprising if
the students could use exactly the same mechanism to mentally simulate the propagation
of light as both wave and particles even if they had no comprehensive and systemic
understanding of light and its propagation. These two examples illustrate the weakness of
past studies in dealing with the mechanisms underlying the identified elements in a mental
model and their phenomenal processes. Moreover, it highlights the importance of further
differentiating between the mechanisms and phenomenal processes expressed in students’
explanations using mental models.
Refocusing the mechanisms underlying the identified elements and the phenomenal
processes of a mental model is consistent with many researchers’ interests in emergent pro-
cesses of specific science domains (e.g., Chi, 2005, 2008; Rappoport & Ashkenazi, 2008).
For example, Chi (2005) proposes that some scientific concepts representing emergent pro-
cesses are difficult to learn because they are straightforwardly mistaken as direct processes,
which can be conceived as a series of sequential cause–effects resulting from identifiable
agents in a system. However, although Chi recognizes direct processes and emergent pro-
cesses as two exclusively different categories and proposes conceptual changes can be
achieved by a shift from a direct to an emergent process, she makes the distinction between
these two processes on the basis of their ontological difference in agents’ properties. That is,
Chi’s underlying assumption is that if an individual can understand the difference between
direct processes and emergent processes and the properties of their correspondent agents,
then she/he can construct a scientifically acceptable conception or mental model. However,
as discussed previously, there might be no simple one-to-one correspondence between a
specific kind of agent (or elements) and a phenomenal process. It is quite possible that
the same ontological elements or agents can result in various kinds of direct or emergent
processes, which are also the major causes of misconceptions or flawed mental models.
Past Studies on Students’ Conceptions of Heat
As most of the scientific concepts which represent emergent processes and which are
difficult to learn, students’ understanding of heat has been extensively studied since the
1970s. Four patterns of students’ interpretative frameworks of heat can be found based on
relevant studies on misconceptions or conceptual changes (e.g., Clough & Driver, 1985;
G. L. Erickson, 1979, 1980; G. Erickson & Tiberghien, 1985; Lewis & Linn, 1994; Linn
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& Songer, 1991; Reiner, Slotta, Chi, & Resnick, 2000; Wiser & Carey, 1983; Wiser, 1988;
Wiser & Amin, 2001). First, heat is treated as an intrinsic property of a substance. For
example, as hotness is a property of wood, coldness is a property of ice. Accordingly,
children who possess this framework lack a clear understanding about the movement or
transferability of heat. Second, heat is treated as a material substance, hotness, which can be
envisioned as a group of particles and can move from a hot object toward a cold object; the
opposite of hotness particle is coldness, which can move from a cold object to a hot object.
Within this framework, heat and temperature are not well distinguished, and temperature
refers to a measure of the hotness involved in an object. Third, heat is treated as a nonmaterial
entity, caloric flow, which propagates from objects at higher temperatures to objects at lower
temperatures. In this framework, the change in temperature of an object can be measured
by the net amount of caloric flow moving in and out of the object. It is worth noticing
that the second and the third framework might not be exclusively independent according
to some researchers, and these two frameworks might form a hybrid framework called the
calorie view (G. L. Erickson, 1979, 1980) or the source–recipient view (e.g., Wiser, 1988;
Wiser & Amin, 2001). The fourth framework represents a scientifically acceptable view,
from which heat refers to a transfer of thermal energy (the total kinetic energy of all the
particles, such as atoms and molecules, in a substance) due to a temperature difference,
and temperature refers to the measure of the average thermal energy in a substance. In
this framework, heat transfer is a process of thermal energy transition and its underlying
mechanism is a series of particle motions. The sequence of these four frameworks also
represents the developmental stages of peoples’ conceptions of heat, developing from a
na¨ıve view toward a more scientific one.
Although these four views represent a substantial gain in our understanding of students’
general conceptions of heat, much remains to be clarified about how students conceptualize
the dynamic process of heat transfer. First of all, these four frameworks provide no further
elaboration about the specific dynamic processes of heat transfer, which is supposed to be
the essence of the concept heat, given that heat is defined as transmitted energy due to tem-
perature differences (Pushkin, 1997; Slisko & Dykstra, 1997). Moreover, although these
four frameworks reveal students’ ontological beliefs of heat, they imply there is a specific
set of mechanisms underlying each ontological belief, and thus underestimate the com-
plexity of students’ conceptualizations of the dynamic processes involved in heat transfer.
Second, although students may possess any of these four frameworks, how they apply these
interpretative frameworks to make predictions and generate explanations needs to be further
clarified. That is, whether students solve their problems based on their own interpretative
framework and mental models, or just on their practical knowledge constructed from their
fragmented experiences, is still unclear. Third, while these four frameworks might represent
different developmental stages of students’ understanding of heat, they are generated from
studies whose participants were almost entirely K-12 students. Thus, it is imperative to
investigate how college-level advanced learners make sense of heat and whether they apply
any of these four frameworks.
Aims of This Study
Within the context of the preceding discussion, the present study argues that investigating
college physics students’ mental models of heat conduction, which is a specific way of
heat transfer, can make a major contribution to our understanding of students’ learning
of physical phenomena related to heat transfer. Based on a previous definition of mental
models, adopting the construct of mental model highlights the importance of investigating
both the fundamental elements and their underlying mechanisms involved in students’
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mental models, and doing this can help to shift the ontology-dominating approach toward
a complement-based approach, which simultaneously considers both students’ ontological
beliefs and their conceptualized processes of thermal phenomena. With the recognition
of this complement-based approach, heat conduction can serve as a nearly ideal research
domain based on both its various potential ontological elements and its complex dynamics.
Hence studying students’ mental models of heat conduction may extend our understanding
of students’ learning of thermal physics from a less-sophisticated conception level of heat
toward a model level of the dynamic processes of heat transfer. Regarding the participants,
since college physics students have learned the major theories in thermal dynamics, they
are supposed to be able to use a more flexible approach to solve problems related to heat
conduction. Moreover, given that the scope of K-12 students’ mental models is rather limited
(Borges & Gilbert, 1999), studying more advanced students’ (college physics students)
manipulation of mental models has at least a twofold advantage. It may help first to
discover more diverse mental models of heat conduction, given the greater depth of content
knowledge of the respondents, and then to explore the ways in which different mental models
are utilized to generate predictions and explanations (e.g., Clement, 1989, 2003). Hopefully,
the findings of this study can provide some theoretical and instructional suggestions for
improving the teaching and learning of physical phenomena when represented by mental
models.
Thus, this study aims to investigate college physics students’ mental models of heat con-
duction in terms of both their correspondent ontological elements and dynamic processes.
In addition, special attention is paid to the relationship between the students’ mental models
of heat conduction and students’ predictions relevant to the given problems.
METHODS
To better understand the participants’ mental models of heat conduction, this study used
semistructured, clinical interviews to elicit their mental models. In-depth clinical interviews
have been widely used to induce participants to externalize expressions about their inner
representations of the target system (e.g., Clement, 1989, 2003; Coll & Treagust, 2003;
Vosniadou & Brewer, 1992), i.e., their expressed models (Gilbert et al., 2000), which serve
as a primary source from which researchers can construct their own understanding of the
participants’ mental models. That is, this study recognizes the limitation that we have no
direct access to “see” the participants’ mental models, nor for that matter any internal
cognitive representation, and can only attempt to comprehend their mental models through
their expressed models, which is the information shared through verbal or other expressive
means (Norman, 1983).
Participants
Participants in this study were 30 senior undergraduate physics majors from a national
university in Taiwan. All of the participants had completed the required course not only in
fundamental physics but also in thermodynamics before the interview. They were selected
from a larger sample based on their ability to respond fluently to interview questions in
a preliminary interview. The participants were told that the goal of the interview was
to understand college physics majors’ thinking and problem solving about some general
thermal phenomena and were encouraged to think aloud and use all other means to fully
express their understanding about the questions.
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Interviews and Materials
To efficiently investigate the participants’ mental models of heat conduction, this study
developed a three-phase semistructured interview protocol. The three-phase interview pro-
tocol was refined based on the evidence gathered in a pilot study that was conducted to
assess how well the interview questions and related procedure worked using a different set of
college students with similar academic backgrounds. The first phase of the interview aimed
to probe the participants’ conceptual understanding of related concepts in thermal physics,
such as heat, temperature, thermal transfer, and so on. For each concept, the interview started
with a definition of the concept and then continued with a series of follow-up questions to
probe the participants’ understanding of this concept. All participants were given the same
set of initial questions to ensure consistency in questioning, but clinical-type, follow-up
questions were also used to probe more deeply into their thinking. For example, regarding
the concept of heat, the questioning began with “Can you tell me about your understanding
of heat?,” and some follow-up questions included “Can you give some examples of your
everyday experience in which heat was involved?,” “Can heat be measured? . . . How? (If the
participants responded, for example, with “yes.”),” and for a follow-up, “Do you know any
other scientific concept, law, or theory related to heat?” Also, to probe the participants’ on-
tological beliefs, some specific follow-up questions were used, such as “What is the nature
of heat?” and “What is the difference or relationships between heat and temperature?”
The second phase of the interview was a series of generative questions (Vosniadou &
Brewer, 1992), which were intended to encourage the participants to “run” their mental
models. The interview-about-event technique (Carr, 1996; White & Gunstone, 1992) was
used in this phase to probe the participants’ generative predictions and explanations of
the given events based on their manipulations of related mental models. The following is a
sample of the interview questions continuing with a set of follow-up questions (see Figure 1):
An iron disk, which is connected to both an aluminum rod and a wood rod, has been heated
by an alcohol lamp for a very long time. Can you rank temperatures of the different sections
within both rods? That is, what is the ranking of the temperatures at points A, B, C, D, E,
and F?
Follow-up questions:
• How did you determine the ranking of the temperatures at A, B, C, D, E, and F?
• Is there any heat transmission in this system? How do you know?
• How do you image the process of heat transmission in this system?
• How do you determine the means and the direction of heat transmission in this
system?
Figure 1. An example of interview question used in this study.
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• Are temperatures at location A, B, and C (and D, E, and F) the same during different
periods of the heating process? How do you know?
• Are temperatures of the two rods the same during different periods of the heating
process (i.e., comparing A and D, B and E, and C and F)? How do you know?
The procedure for each generative question and its follow-up questions was developed on
the basis of the characteristics of mental models defined in this study. That is, the interview
questions aimed to probe the following aspects of the participants’ mental models:
1. Predictions about the changing states of the system: The participants were required to
make two kinds of predictions about each generative question. First, every generative
question started by asking the participants to offer an ultimate prediction for the final
state of the target system. Then, the participants were requested to make three
stepwise predictions in a temporal sequence to describe the gradual changes of the
state in the system. The three stepwise predictions represented the beginning, the
middle, and the final stage of heat transfer in each generative question.
2. Elements involved in the target system: In addition to a series of predictions men-
tioned above, the participants were asked to describe their thought processes from
which predictions were generated. For example, the following probing questions
were used: “Can you tell me how you imaged the process of thermal transfer in this
situation?” Then, follow-up questions, “Based on the imaginary process you just
described, can you tell me what were the major elements engaged in the thermal
system? What made the system change?”
3. Relationships between the identified elements: The participants were further re-
quested to explain the underlying mechanism that causes the changes in the system
states. For example, the probing question “Can you tell me why the temperatures at
points A, B, and C are rising? How do the changes in temperature have to do with
the elements you just mentioned?” Or, “Why do temperatures at different locations
change in this sequence?”
The third phase of the interview was a set of reflective questions. The participants were
first asked to review their responses to every interview question and were encouraged
to restate their answers if they were not satisfied with the previous ones. Then, they
were requested to think about whether they adopted any scientific laws or theories to
make predictions and explanations during the interview. In addition, the participants were
required to reflect on whether they had learned the kinetic-molecular theory of gas and
whether they ever tried to use the molecular perspective to represent the process of heat
conduction before and during the interview. Last but not least, the participants were asked
to reflect on whether their understanding of related thermal phenomena or concepts had
been changed during the interview.
It is worth noting that, among these three phases of the interview, the generative ques-
tions in the second part were the major probe to the participants’ mental models of heat
conduction. The first and third phases of the interview were used to collect data for the
triangulation of the participants’ mental models.
Data Collection
Each participant’s behavior throughout the interview was videotaped by a digital cam-
corder. The camcorder recorded the processes during which the participants were producing
their responses in the following three formats, which are supposed to reflect the participants’
inner representations of their mental models.
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Verbalization. Three types of verbalized reports were collected throughout the interview.
First, the participants were requested to think aloud while they were solving the generative
questions. Second, if the participants paused to think about the generative questions, they
were asked to reflect about their thinking processes immediately after they gave the answers.
According to K. Erickson and Simon (1993), thinking-aloud and retrospective reports are
valid sources to elicit the participants’ in-progress thinking processes and thus are supposed
to provide valuable information about their manipulations of mental models. The third kind
of verbalized reports was the participants’ introspections about their understanding of
related scientific concepts and theories, about how they obtained the answers, and about
their explanations of the content of their thoughts.
Drawings. For each generative question, the participants also drew how they imagined the
processes of heat conduction under the given situation. Moreover, they were requested to
draw the imaginary processes of heat conduction in three consequent steps correspondent to
their three stepwise predictions mentioned above. Their drawings were intended to provide
not only a complementary source of how they represented their mental models in addition to
their verbal reports but also a means of gaining rich information about their inner analogue
representations of heat conduction.
Writings. The participants were also asked to write down their answers to each gener-
ative question, including their three stepwise predictions. The written answers served as
verifications of the participants’ expressed thoughts.
Data Analysis
To search for patterns of the participants’ mental models of heat conduction, this study
adopted grounded theory (Glaser & Strauss, 1967) in analyzing and interpreting the data.
To comprehensively illustrate the participants’ mental models of heat conduction in a
specific context, this study only attempted to analyze and present the data collected from
the interview question shown in Figure 1. The participants’ verbal reports collected from the
interview question were first transcribed into text, and the text together with the participants’
drawings and writings formed the materials for data analysis.
The processes of data analysis followed Chi’s (1997) guidelines for verbal data analysis
and the constant comparative methods offered by grounded theory (Charmaz & Henwood,
2008; Glaser & Strauss, 1967). First, according to the characteristics of mental models
that were delineated, the text of each participant’s verbal reports was reduced and coded
into three main categories: (1) predictions for both the ultimate and the stepwise changes
in thermal states of the given system, (2) references to the fundamental elements in the
given system, and (3) explanations of the mechanisms underlying the changing states of the
given system. Second, for each category, the second-round coding focused on the content of
the participants’ responses. For example, regarding the category of fundamental elements,
the participants’ responses were coded into “using heat particles” or “using caloric flow”
to describe the process of heat conduction. Third, based on the result of second-round
coding, temporary patterns and features of the participants’ responses were identified
through constantly comparing their similarities and differences, and then these were used
to combine or differentiate the emergent patterns accordingly. Fourth, the emergent patterns
that resulted from the text were compared with the participants’ drawings and writing, and
the categories were refined if any incoherence existed between different formats of data.
Then, the characteristic features of each pattern were determined and were further used as
the criteria to repeatedly check the consistence of the categorization.
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Moreover, as mentioned before, to better capture the participants’ mental models in
terms of both of their ontological beliefs and conceptualized processes, previous results
were further integrated into several ontology–process combinations that represented the
participants’ construction of the emergent processes of heat conduction. First, patterns
grounded in the participants’ stepwise predictions together with their corresponding draw-
ings and descriptions were used to construct the participants’ process analogies of heat
conduction. For example, P1’s three stepwise predictions for the changes in the tempera-
tures of different locations in the aluminum rod were (1) A > B = C, (2) A = B > C, and
(3) A = B = C in sequence, and his process analogy was coded as a marching analogy,
because he thought heat was moving in a step-by-step fashion and there was a clear bound-
ary between the heated and unheated area in the aluminum rod. Next, the participants’
ontological beliefs of heat and their explanations of the underlying mechanism of heat con-
duction were integrated with their process analogies and formed specific ontology–process
combinations. For example, given that heat conduction proceeded in a marching analogy,
S1 thought that this phenomenon was emergent from a caloric flow running from a warmer
object to a cooler object because of a temperature difference; thus, S1’s mental models was
coded as calorie–marching combination. Then, the constructed ontology–process combi-
nations were used to examine the participants’ different formats of responses to ensure the
consistence of these combinations.
This study used a series of triangulation strategies to improve its reliability and validity.
On the one hand, for example, to validate that the participants’ mental models were indeed
the object of study, several researchers’ definitions of mental models were examined (e.g.,
diSessa, 2002; Mayer, 1992; Norman, 1983), and from these the crucial characteristics of
mental models were elicited, such as the elements involved in a system, the relationships
between the elements, and the predictive power based on the mechanisms of these elements.
In addition, when coding the participants’ ontological beliefs about heat, results from past
research (e.g., G. L. Erickson, 1980; Lewis & Linn, 1994; Wiser & Amin, 2001) were used
to confirm the feasibility of the categorization scheme. Moreover, categorizations of the
participants’ mental models werefurther validatedthroughthetriangulationamongdifferent
formats of data. That is, the patterns that were found can be identified from concurrent
inspection of the participants’ verbal responses, drawings, and writings, respectively, and
there is a logical coherence between the participants’ predictions and their explanations
about the elements and their underlying mechanisms in the given system. On the other
hand, for example, to improve reliability, the consistency of the results of the categorization
across different formats of data and across the participants’ answers to different generative
questions was constantly checked. In addition, another science educator was invited to
examine the identified patterns and recoded the data based on the coding scheme. Regarding
the intercoder agreement, the kappa coefficients (Cohen, 1960) of the categorizations of the
participants’ ontological beliefs and process analogies were K = .92 and 1.00, respectively.
Any discrepancy, however, between the two coders was resolved through a concurrent
resolution method by which the coders discussed the data throughout the case until an
agreement was reached.
RESULTS
This section first presents the participants’ conceptualizations of the process of heat con-
duction and their ontological beliefs and then combines these two aspects as an integrated
framework to illustrate the participants’ mental models. Next, it indicates the relationships
between the participants’ predictions and their mental models.
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TABLE 1
Five Process Analogies of Heat Conduction
Progress of
Analogy Description State Change
Marching Heat conduction proceeds as a step-by-step
march.
(i) A = B = C
(1) A > B = C
(2) A = B > CAfter entering an object, heat steps ahead unitarily
and regularly toward the other end.
While a region swept by the marching heat
reaches thermal equilibrium immediately with
the anterior region, the region beyond the
marching heat remains thermally unaffected.
Flooding Heat conduction proceeds as a gradual rising
flood.
(i) A = B = C
(1) A = B = C
(2) A = B = CAfter entering an object, heat is instantaneously
and evenly distributed throughout the object.
The temperature of the whole object increases at a
constant rate till thermal equilibrium is reached.
Gradient Heat conduction proceeds in a gradient fashion. (i) A = B = C
(1) A > B > C
(2) A > B > C
After entering an object, heat rushes forward
dispersedly, and the amount of transmitted heat
gradually declines along the object.
Temperatures at different regions of the object
increase in inverse proportion to the distance
between the region and the heat source.
Gradient–marching Heat conduction proceeds in different analogies at
different stage.
(i) A = B = C
(1) A > B > C
(2) A = B > CHeat first proceeds in gradient analogy for a
transient phase and then continues in marching
analogy.
Gradient–flooding Heat conduction proceeds in different analogies at
different stage.
(i) A = B = C
(1) A > B > C
(2) A = B = CHeat first proceeds in gradient analogy for a
transient phase and then continues in flooding
analogy.
Note. The symbols (i), (1), and (2) refer to the participants’ predictions to the initial state,
the first and the second stage of heating, respectively. Process analogies contained no
information about the final state of the system. For interpretation of the notations, A, B, and
C, see Figure 1.
The Participants’ Conceptualizations of the Process
of Heat Conduction
Based on the participants’ stepwise predictions (as shown in the last column in Table 1),
drawings (Figure 2), and explanations in response to the interview questions, five process
analogies of heat conduction were identified (Table 1). Among these five process analogies,
three of them are fundamental analogies, which can independently but completely represent
a distinct conceptualized process of heat conduction, and the other two are dual analogies,
which involve a mixture of two fundamental analogies. The first fundamental analogy is
the marching analogy in which heat conduction proceeds as a step-by-step forward-moving
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Figure 2. Examples of the participants’ drawings of the five process analogies.
march. The front line of the march sets a clear-cut boundary of the heat transmission; when
heat is marching forward in an object, the area swept behind the front line reaches thermal
equilibrium immediately, whereas the area beyond the front line remains unaffected. The
whole object will reach thermal equilibrium when the front line reaches the end of the
object. The second fundamental analogy is the flooding analogy in which heat conduction
proceeds as a constantly rising flood. After entering an object, the transmitted heat will
be simultaneously and evenly spread throughout the object. Thus, the object as a whole
will have the same temperature at any moment during the process of heat conduction. As
heat transfer continues, the transmitted heat will keep spreading evenly throughout the
object and, thus, its temperature will keep rising as a whole at a constant rate. The third
fundamental analogy is the gradient analogy in which heat conduction proceeds in a grad-
ually declining fashion. That is, the amount of transmitted heat in an object declines in
proportion to the distance between a specific region within the object and the heat source;
the longer the distance, the less amount of heat will be transmitted into the correspondent
region. Accordingly, during the process of heat conduction, the temperatures at different
regions of an object vary in inverse proportion to their distances from the heat resource.
The fourth is a dual analogy, gradient–marching analogy, which combines the gradient
analogy and the marching analogy. In this conception, at the beginning stage of heat con-
duction, heat conduction proceeds as the progression of a gradient analogy. That is, the
quantity of heat in an object gradually declines in proportion to the distance between a
specific location and the heat source. Afterward, heat conduction continues as a marching
analogy in which the object reaches thermal equilibrium in a step-by-step fashion. The
fifth is another dual analogy, gradient–flooding analogy, i.e., a combination of the gradient
analogy and the flooding analogy. In this analogy, heat conduction also begins as the gra-
dient analogy. However, in contrast to a gradient–marching analogy, heat conduction then
continues as the flooding analogy in which the transmitted heat is immediately and evenly
spread throughout the object. Based on the features of the five analogies, it is apparent
that although all respondents attempted to describe a common phenomenon that, given
an object is being heated at one of its ends, the temperature at the other end will gradu-
ally increase, they adopted different perspectives to portray the potential progress of this
phenomenon.
To compare the distribution of the five analogies for rods of different composition,
the numbers of participants who used each process analogy in both the aluminum and
the wooden rods were combined into a cross-tabulation table (Table 2). As can be seen,
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TABLE 2
Numbers of Participants Who Used Each of the Five Process Analogies in
Describing Heat Conduction in the Aluminum and Wooden Rods
Aluminum
Gradient– Gradient–
Marching Flooding Gradient Marching Flooding Total
Wood Marching 1 0 0 0 0 1
Flooding 0 0 0 0 0 0
Gradient 0 0 20 1 5 26
Gradient–marching 0 0 0 2 0 2
Gradient–flooding 0 0 0 0 0 0
Other 0 1 0 0 0 1
Total 1 1 20 3 5 30
Note. The category “other” refers to a prediction that the temperature of the wooden rod
would not change throughout the heating process.
the gradient analogy, which represents the scientifically accepted analogy, was used by
20 and 26 participants in describing conduction in the aluminum rod and wooden rods,
respectively, and made up the largest proportion of analogies expressed by the 30 partic-
ipants all-totaled. In contrast, the other two fundamental analogies were adopted by only
one participant in describing what occurred in both rods. As shown in Table 2, while the
two dual-analogy categories, gradient–marching analogy and gradient–flooding analogy,
account for more participants than both the marching analogy and the flooding analogy,
the proportion of participants using these two dual analogies is still far lower than the
proportion of those who used the gradient analogy.
As also indicated in Table 2, it is apparent that not all participants’ process analogies of
heat conduction are consistent between the aluminum rod and the wooden rod, as would be
indicated if all tallies fell exclusively along the diagonal. Thus, any entries that fall outside of
the diagonal indicate a lack of consistency. Moreover, the degree of consistency varies with
different analogies; while the three fundamental analogies appear more consistent between
the two rods, the two dual analogies seem less consistent. For example, the participants
who expressed a marching analogy and gradient analogy in describing what was happening
with the aluminum rod (cell entries 1 and 20, respectively) tended to use the same analogies
in the wooden rod. In contrast, all five participants who expressed a gradient–flooding
analogy for heat conduction in the aluminum rod switched to a gradient analogy for the
wooden rod. Thus, it appears that fundamental models are more stable than dual models.
In addition, one of the three participants who held a gradient–marching perspective for
the aluminum rod changed into a gradient analogy perspective for the wooden rod. This
inconsistency of the dual analogies can be attributed to the contrasting difference in the
physical property, i.e., conductivity, between the two rods. For instance, one participant,
P20, who switched from using a gradient–flooding analogy for the aluminum rod into a
gradient analogy in the wooden rod, believed that while heat conduction occurred almost
instantaneously in metal, it proceeded extremely slowly in wood. In this case, one of
the central features of the gradient–flooding analogy, i.e., heat can be immediately and
evenly spread throughout an object, seemed unfeasible in wood for participant P20. Hence,
P20 needed to either abandon or adjust this analogy while representing the process of
heat conduction in the wooden rod. On the contrary, since the common characteristic
of the marching analogy and gradient–marching analogy, namely, heat proceeded in a
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TABLE 3
Participants’ Ontological Beliefs of Heat Conduction
Fundamental
Ontological Entity Components Underlying
Category of Heat of Mental Models Mechanism Number
Substance Heat as
physical
substance
Heat particles Heat conduction proceeds
as diffusion of heat
particles and is driven
by the difference in the
density of heat particles;
heat particles move
from a region with
denser heat particles
toward the region with
looser heat particles.
1 (3%)
Energy Heat as
massless
liquid
(calorie)
Caloric flows and
the lamp, the
iron disk, the
aluminum and
the wooden
rod
Heat conduction proceeds
as a flow of liquid and is
driven from the
difference in
temperature; heat flows
from the hotter toward
the colder object.
20 (67%)
Interaction Heat as
thermal
phenomena
resulting
from
molecular
collisions
Molecules Heat conduction proceeds
as sequential molecular
collisions and is driven
by the difference in
kinetic energy between
molecules; heat is
transferred from
molecules with higher
kinetic energy toward
the particles with lower
kinetic energy.
9 (30%)
step-by-step fashion, works adequately for both the aluminum and wooden rods, no major
adjustment, apparently, was needed when mentally simulating these two analogies in both
rods. Accordingly, the consistency of the five process analogies of heat conduction appears
to heavily depend on the material or context where they are applied. Since the process
analogies for the aluminum rod are more diverse than those used for the wooden rod,
henceforth the discussion of process analogies will focus only on the aluminum rod for
simplicity and adequate breadth of representation.
The Participants’ Ontological Beliefs of Heat Conduction
Three ontological categories of heat were identified as shown in Table 3. The first category
is substance in which heat is treated as a kind of physical substance, i.e. heat particles.
These heat particles, which serve as the fundamental components of a mental model of heat
conduction, are denser in a region whose temperature is higher and are sparser in a region
whose temperature is lower, and vice versa. If there is a temperature difference between
two regions, heat particles will start to move from the warmer region toward the cooler
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region as if by a process of diffusion. For example, P6 mentioned these imaginary particles
when being asked to elaborate on his imaginary simulation of heat conduction:
R: How did you imagine the process of heat conduction in this question?
P6: I imagined that there were many tiny particles running around. I just imagined heat
as many moving particles. Because heat is a very abstract concept, it is easier to
make it concrete to think it through. . . . The motions of heat particles follow the
scientific law, and they just move from the warmer place to a colder place.
Moreover, in this category, the changes of thermal states in temperature are represented as
changes in heat particles’ density. Temperature increases in proportion to the amount of
accumulating heat particles in a specific region. Accordingly, the correspondent production
rules for running the mental model are as follows:
1. If a region contains more heat particles, it has a higher temperature, and vice versa;
2. If there is a difference in temperature between two regions, heat particles move from
the warmer to the colder region; and
3. If a region within an object receives heat particles from another place or other object,
its temperature will increase.
However, it is worth noticing that, for P6, heat particles were just a heuristic imaginary
entity. P6 did explicitly mention that he only used “heat particles” for helping him run
his mental model, and definitely knew that heat is not a kind of physical substance but a
specific form of energy, in the accurate realm of modern physics.
The second ontological category is energy in which heat is treated as a massless and
invisible calorie flow. Since calorie flow is not an intrinsic property of any substance, it is
improper to claim that an object contains a specific amount of calorie flow (heat). Instead,
a calorie flow is generated on the spot only to represent a transitional agent that helps
to reallocate the energy under the condition of a thermal disequilibrium. Based on this
ontological belief, heat conduction proceeds as a massless and invisible flow, running from
a warmer region toward a colder region either within an object or between different objects.
For example, when responding to how his prediction was obtained, P3 provided a succinct
explanation:
R: How did you get your answer? What is the underlying mechanism?
P3: Heat flows from the warmer to the colder till thermal equilibrium is achieved.
In this sense, calorie flow seems able to provide a suitable and efficient way to explain
the mechanism of heat transfer from a macroscopic perspective. Moreover, from this view,
the components involved in the participants’ mental models are the items presented in the
interview protocols; that is, the lamp, the iron disk, the aluminum rod, and the wooden rod.
The changes in the thermal states of these components result from the net gain of calorie
flow in a specific region of an object. The larger amount of net heat flow an object (or a
region) acquires the larger extent of change in its temperature, and vice versa. Accordingly,
the correspondent production rules in this category for running the mental models of heat
conduction are as follows:
1. If there is a temperature difference between any two objects (or regions), heat
conduction starts to occur;
2. If heat conduction starts, heat (calorie) flows from a warmer object (or region) to a
colder object (or region); and
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3. If an object (or region) obtains larger amount of calorie flow, its temperature becomes
higher; if an object (or region) releases a larger amount of calorie flow, its temperature
becomes lower, and vice versa.
However, it is worth noting that the massless fluid is also a heuristic imagery entity.
Participants who held this category claimed that they understood that heat was not any
form of substance, but it was easier to run their mental models of heat conduction with the
aid of this imagery entity. For example, when P3 was asked to elaborate his imagination of
heat conduction, he consciously used “water” as an analogy to envision his mental model
regardless the true status of this analogical entity:
R: You just used the verb “flow” to describe the process of heat conduction. Can you
tell me how you envision heat transfer in this question?
P3: Just like water. Just like water flows from a higher place to a lower place. It is
because heat flows from a hotter area to a colder area, I think it acts like water.
R: But it sounds like you treated heat as a substance. You just mentioned that heat is a
form of energy instead of a substance. . .
P3: Exactly. As long as the answer is correct, since I am not creating a theory or
something, I don’t have to worry about its reality (whether heat is a kind of energy
or a substance).
The third category is interaction, in which heat is treated as a phenomenon resulting
from molecular collisions. A molecule itself does not contain heat. Instead, it carries
kinetic energy and serves as an agent to allocate the kinetic energy held by countless
other molecules through constant mutual collisions. It is these molecular collisions that
generate heat conduction. According to this ontological belief, heat conduction proceeds
as sequential molecular collisions from molecules with a larger amount of kinetic energy
toward those with a less amount of kinetic energy. For example, when responding to the
mechanism of heat conduction, P7 said,
It is just like the principle of billiard-ball collisions. This surface (of the aluminum rod)
contacts with the heat source, and from here (the interface between the heat source and
the aluminum rod), a ball (molecule) hits another ball (molecule), one-by-one all the way
down to the end in a sequence. And this process causes energy to transmit from here to the
other end. I think that is the mechanism of heat transmission.
In this category, the unit of the components involved in the participants’ mental models is
every single molecule in interaction with other countless ones. The change in the thermal
states of temperature in an object (or region) is determined by the net amount of kinetic
energy that the molecules in the object acquire. The larger the amount of net kinetic energy
a group of molecules obtains, the larger extent of increase in temperature in the object
(or region) that carries these molecules, and vice versa. Moreover, the previous quotation
helps to indicate a crucial implicit assumption underlying this molecular perspective of heat
conduction: molecules move faster in a region whose temperature is higher, and move slower
in a region whose temperature is lower, and vice versa. Accordingly, the correspondent
production rules of this ontological category to run the mental models are as follows:
1. If the temperature of an object (or a region) is higher, the molecules it contains
moved faster; if the temperature of an object (or a region) is lower, the molecules it
contains moved slower, and vice versa;
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2. If two molecules with different amount of kinetic energy collide, the molecules with
a larger amount of kinetic energy will pass part of its energy to the other one;
3. If there is a temperature difference between two objects (or regions), the faster
moving molecules within the warmer object (or region) will pass part of their kinetic
energy to their neighbors in a sequence toward the colder object (or region); and
4. If the molecules within an object (or a region) gain a larger amount of kinetic energy
and thus moved faster than they used to, the temperature of the object (or region)
increases to a larger extent.
However, although the participants in this category could view the process of heat
conduction from a microscopic perspective, most of them adopted this perspective only
when they were asked to explain the detailed mechanism of heat conduction. When making
predictions of the system’s states, or plainly describing the process of heat conduction,
they tended to use the macroscopic, calorific perspective. For example, when P26, who was
coded in the molecule category, was asked about how she figured out that the temperature
of the region A in the aluminum rod is higher than that of B and C, she directly adopted the
calorie flow view to elaborate her answer:
P26: The lamp keeps on providing heat energy, so heat keeps on entering the aluminum
rod through this end. And then heat gradually transmits from this end to the
other end. So, this region (which is closer to the heat source) will have a higher
temperature.
However, when P26 was required to provide a further explanation of the process of heat
conduction, she switched to the molecular perspective:
P26: I remembered I mentioned that, heat conduction is just molecular interactions. The
process is that, one molecule collides with another, and on and on. Then, when
the first molecule has passed it energy to its neighbor, it is hit (by the molecules
with a larger amount of kinetic energy from the iron disk) again. So, molecules in
Region A will move faster and have larger amount of energy, and that’s why its
temperature is higher than that of Region B and C.
Accordingly, P26, as well as many other participants who were assigned to the molecule
category, seemed to hold a dual ontological belief of heat conduction. Moreover, a dual belief
was also expressed by the participant (P6) who possessed the substance view. For example,
P6 mentioned his tendency to apply the substance view only when further explanation was
needed:
R: When making the prediction, did you use the little [heat] particles to help you?
P6: Not this time. I only used them (heat particles) when I tried to explain the mechanism
of heat transmission.
This dual ontological belief of the underlying mechanism of heat conduction is crucial to
interpret the distribution of the participants assigned in each ontology category. As shown
in Table 3, while there were 20 participants who held the calorie view among the 30, 1 held
the substance view, and 9 held the molecule view. However, as mentioned above, most of
the 10 noncalorie participants held a dual ontological belief. That is, they preferred adopting
a calorie flow to manipulate their mental models to answer the initial interview question but
switched to using either molecular collisions or heat particles to make detailed explanations.
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TABLE 4
Numbers of Participants in Different Ontology–Process Combinations of
Heat Conduction
Process Analogies in the Aluminum Rod
Gradient– Gradient–
Marching Flooding Gradient Marching Flooding Total
Ontological beliefs
Substance 0 0 1 0 0 1
Calorie flow 1 0 14 0 5 20
Molecule 0 1 5 3 0 9
Total 1 1 20 3 5 30
Ontology–Process Combinations
As mentioned before, jointly considering both the process analogy and the ontological
category can better represent the participants’ mental models of heat conduction. Different
ontology–process combinations help not only to describe the conceptualized progression
of heat conduction but also to elaborate its underlying mechanisms. A good example
comes from the gradient analogy and its three correspondent ontological beliefs. As shown
in Table 4, the 20 participants who possessed a gradient analogy can be divided into
three different ontology–process combinations: substance–gradient, calorie–gradient,
and molecule–gradient. The first combination, substance–gradient, was expressed by only
one participant. From this view, the central feature of the gradient analogy, the temperatures
of different regions in an object decline in proportion to the distance between the region and
the heat source, can be attributed to the decreasing density of heat particles along the rod.
Given that heating continues, more and more heat particles will keep on entering one end
of the rod from the heat source, and then diffuse toward the other end. Consequently, at any
time before thermal equilibrium is achieved, the end that is closer to the heat source carries
the largest amount of heat particles, and the amount of heat particles gradually declines
toward the other end of the rod. Accordingly, it is the difference in the heat particles’ density
that forms the macroscopic phenomenon of the gradient analogy of heat conduction.
The second combination, calorie–gradient, was possessed by 14 participants. Based on
the view of this combination, a calorie flow runs from the lamp into one end of the rod and
then moves toward the other end. The calorie flow runs as water sluiced from a reservoir
and then moving toward an open channel; the further the water flowed, the shallower the
depth of the water. It is the varying depth of the running calorie flow that represents the
macroscopic phenomenon of a gradient analogy.
The third combination, molecular–gradient, was expressed by five participants. From
this perspective, a series of molecular collisions begins at the conjunction of the aluminum
rod and the iron disk and then continues along the aluminum rod. Although molecules at
the region closer to the iron disk move faster because they directly gain kinetic energy from
their anterior fastest moving neighbors in the iron disk, they pass only part of their gained
kinetic energy to their succeeding neighbors. Consequently, molecules that are closer to the
iron disk obtain a larger amount of kinetic energy. In contrast, molecules that are farther
from the iron disk gain the less amount of kinetic energy that they gain from the sequential
collisions originating from the iron disk. It is the decreasing amount of gained kinetic energy
that represents the macroscopic phenomenon of the gradient analogy of heat conduction.
Apparently, the gradient analogy of heat conduction can be elaborately explained by these
three different ontological categories.
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On the other hand, based on the identified ontology–process combination, a common
ontological belief could generate different process analogy. The calorie view and its three
related process analogies provide a good example. The first combination stemming from
the calorie view is calorie–marching, which was expressed by only one participant. Based
on the view of this combination, heat conduction can be modeled as water (calorie flow)
running along a narrow, empty pipe. As the front line of the running water reaches a specific
region of the pipe, the area behind the front line has been filled with the water, while the
area beyond the front line remains empty. Thus, the increasing region that is filled with
water in the pipe can properly represent the step-by-step thermal equilibrium predicted by
the marching analogy.
The second combination is the calorie–gradient view expressed by 14 participants. As
mentioned in the preceding paragraph, in this combination, heat conduction can be modeled
as water sluiced out of a reservoir flowing toward an open channel. The declining depth of
the running water along the channel adequately represents the macroscopic phenomenon
of the gradient analogy.
The third combination is the calorie–(gradient–flooding) view that was expressed by
five participants. From this view, the process of heat conduction can be modeled as water
pouring down into a container. At the first stage, the poured water flows toward the edge of
the container, and this process can account for the gradient phenomenon at the beginning
of a gradient–flooding analogy; the further the water flows, the shallower the depth of the
water. At the following stage, after the front line of the running water reaches the edge of
the container, the surface of the water rises up at a constant rate, and this process properly
represents the flooding phenomenon at the latter stage of the gradient–flooding analogy. As
these examples indicated, a common ontological belief can provide alternative accounts for
different process analogies of heat conduction, and vice versa. More importantly, different
ontology–process combinations indeed provide a better way to represent the participants’
mental models of heat conduction.
The Participants’ Predictions of the Final State
The participants’ responses to the first interview question, “What is the ranking of the
temperatures at different regions of the two rods?,” could be grouped into four categories:
an overall equilibrium, differentiated equilibrium, metal-only equilibrium, and a continu-
ous increase in temperature. The correspondent states of the categories and the numbers in
each category are presented in Table 5. As can be seen in Table 5, 17 of the 30 participants
predicted that the two rods would achieve an overall thermal equilibrium regardless of the
region and material of the two rods. Among the other 13 participants, while they did not
predict that an overall thermal equilibrium would occur, six of them proposed two kinds of
TABLE 5
Four Categories of the Participants’ Predictions to the Final State of the
System
Categories of Prediction Number
Overall equilibrium A = B = C = D = E = F 17
Differentiated equilibrium A = B = C > D = E = F 4
Metal-only equilibrium A = B = C > D > E > F 2
Continuous increase in temperature A > B > C > D > E > F 7
Total 30
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partial-equilibrium solutions. The first partial-equilibrium solution is differentiated equi-
librium, which is a claim that although the aluminum and the wooden rods will separately
reach thermal equilibrium, their final states of thermal equilibrium will be achieved at
different temperatures.
The second partial-equilibrium solution is a metal-only equilibrium, which predicts
that the aluminum rod will solely reach thermal equilibrium, while the temperature of the
wooden rod will keep rising. It appears that these six participants believed that the feasibility
of reaching thermal equilibrium depended on the material of the two rods. The aluminum
rod, a typical kind of metal, would easily achieve thermal equilibrium after being heated
for a long time. In contrast, the wooden rod, a typical kind of heat insulator, would either
never reach thermal equilibrium or reach thermal equilibrium at a lower temperature than
that of the aluminum rod. Moreover, the remaining seven participants predicted that the
temperatures at different regions of the two rods would keep rising at different rates as long
as the heating continued. The different rates of increase in temperature at different regions
corresponded to the prediction of the gradient analogy. That is, the temperature declined in
proportion to the distance between a specific region and the heat source. In brief, the four
categories of the participants’ predictions seem to be based on a critical criterion—whether
the system can reach thermal equilibrium or not.
Although different ontology–process combinations as discussed above provide a fruitful
way to represent the participants’ mental models of heat conduction, it seems the partici-
pants did not use these combinations to predict the final state of the system. Instead, the
participants tended to retrieve their learned rules or interpretations of robust experiences
to make the predictions. For example, P30 claimed that he was reluctant to manipulate his
mental model to obtain the final state of the system when he heard the salient cue to the
answer of the interview question:
R: Did you use any simulation to obtain your answer?
P30: No!
R: Then how did you get your answer?
P30: As you started to read the question, I attempted to simulate the process a little bit.
But while I was starting to do that, I heard you said “after a very long time” in
your question and I just spontaneously stopped doing that. Because every time I
hear “after a very long time,” thermal equilibrium just pops up into my head. And
equilibrium just means that all the temperatures will be the same.
It appeared that P30 had made an automatic connection between the phrase, “after being
heated for a long time,” and the more learned answer, “thermal equilibrium.”
In addition to the learned scientific rule, some participants mobilized direct recall from
their past experiences to make the prediction. For instance, P12’s response is a typical
illustration of this trend of inference:
R: How did you figure out your answer (A > B > C > D > E > F)?
P12: Because wood is less capable to conduct heat. . . .Aluminum is more capable to
conduct heat. So, comparing to F, D’s temperature will be higher.
R: Why is D’s temperature higher?
P12: Because it is closer to the heat source.
R: Why being closer to the heat source makes its temperature higher?
P12: Because it (Region D) directly absorbs heat from the heat source. F isn’t directly
influenced by the heat source. D’s temperature is higher because it is closer to the
heat source. Just like the spatula. The spatula is made of wood, because wood is less
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TABLE 6
Cross Comparison Between Prediction Categories and Process Analogies,
Ontological Beliefs, and Ontology-Process Combinations
Prediction Categories
A > B > C > A = B = C > A = B = C > A = B =
D > E > F D > E > F D = E = F C = E = F Total
Process analogies
Marching 0 0 1 0 1
Flooding 0 0 1 0 1
Gradient 7 0 1 12 20
Gradient–marching 0 0 0 3 3
Gradient–flooding 0 2 1 2 5
Total 7 2 4 17 30
Ontological beliefs
Substance 1 0 0 0 1
Calorie flow 5 2 2 11 20
Molecule 1 0 2 6 9
Total 7 2 4 17 30
Ontology-process combinations
Calorie–marching 0 0 1 0 1
Molecule–flooding 0 0 1 0 1
Substance–gradient 1 0 0 0 1
Calorie–gradient 5 0 0 9 14
Molecule–gradient 1 0 1 3 5
Molecule–(gradient– 0 0 0 3 3
marching)
Calorie–(gradient– 0 2 1 2 5
flooding)
Total 7 2 4 17 30
able to conduct heat. But if you use the spatula to touch the fire, the temperature of
the front end will be higher. So, the temperatures at points A, B, and C will be very
high after a long time. But because A directly absorbs heat from the heat source,
and then gradually passes the heat to B and C, A’s temperature will be higher (after
a long time).
Since prediction is a major function of mental models that was examined in this re-
search, it is crucial to explore the relationship between the participants’ predictions and
their mental models. To better understand the possible relationships, a cross tabulation of
the participants’ predictions in relation to their process analogies, ontological beliefs, and
ontology–process combinations was made (Table 6). According to the data in Table 6,
there seems to be no strong relationship between the participants’ predictions and their
process analogies, ontological beliefs, and ontology–process combinations. However, two
interesting associations can be found among them. First, there is a one-directional relation-
ship between gradient analogy and the prediction that the temperature of the system will
keep rising. All of the seven participants who predicted that the temperature of the system
would keep rising possessed the gradient analogy interpretation. In contrast, not all of the
20 participants who possessed a gradient analogy predicted a continuous increase in the
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temperatures; while 7 of them did, the other 13 participants predicted a partial or overall
thermal equilibrium of the system. Apparently, within the limitations of the sample size,
the relationship between the prediction for a continuous temperature and gradient analogy
is nonreciprocal; it is possible to use the result of that prediction to infer what process
analogy the participants possessed, though it is not possible to use the process analogies
to anticipate their predictions. Next, all participants who possessed the gradient–marching
analogy also possessed the molecule view and made a correct prediction. This association
is also nonreciprocal because only the gradient–marching analogy can be used to infer
its correspondent ontological belief (molecule view) and resultant prediction (an overall
equilibrium), but not the ontological belief or the resultant prediction from it to predict the
process analogy. Beyond these two possible relationships, no other salient connections can
be found among the ontological beliefs, process analogies, and predictions.
DISCUSSION
The Participants’ Ontological Beliefs of Heat Conduction
The three patterns of the participants’ ontological beliefs identified in this study, the
substance, energy, and interaction views, are quite similar to those found in past studies
(e.g., Driver, Squires, Rushworth, & Wood-Robinson, 1994; G. L. Erickson, 1979, 1980;
G. Erickson & Tiberghien, 1985; Kesidou, Duit, & Glynn, 1995). However, special attention
is required to properly interpret the distribution of these three patterns. A quick glance might
suggest that the distribution can be reasonably inferred from findings of past studies on the
basis of the participants’ developmental stage and academic background. That is, given that
the participants had received substantial exposure to formal instruction on thermal physics,
one may predict that only a few, if not any, participants would possess the substance view
and the largest proportion would possess the calorie view, instead of the interaction view
(Keisdou & Duit, 1993; Lewis & Linn, 1994). Nonetheless, according to the analysis of
their verbal reports, while the macroscopic, calorie view was utilized by all participants
as a common base to make direct predictions, the substance and interaction views were
recruited only to generate detailed explanations about the mechanisms underlying these
two microscopic perspectives. Thus, those participants who used these two microscopic
views to make explanations indeed possessed a dual ontological belief of heat, and they
would flexibly use either belief based on the encountered contexts and requirements, as the
conceptual profile view as was suggested by Mortimer (1995). Moreover, this duality of
ontological beliefs supports the argument presented here that students’ ontological beliefs
on heat should not be used as the sole indicator for their understanding of heat and their
progress in conceptual changes.
Although many past studies had reported a low proportion of participants who used the
particle view in explaining thermal phenomena (e.g., Kesidou & Duit, 1993; Lewis & Linn,
1994), the cause of this low proportion in the present study remains an unsolved mystery,
especially under the condition that all participants had studied advanced thermodynamics.
Some assumptions could be made to account for this phenomenon. First, although the
participants had considerable exposure to formal instruction on thermal physics, they failed
to make conscious connections between thermal phenomena and kinetic theory. The failed
connections could be partly attributed to the fragmented nature of the content of both the
textbook and instructional materials. For example, in their textbook (Halliday, Resnick,
& Walker, 2005) for the fundamental physics course, the section on thermal physics was
introduced before the section on kinetic theory, and the authors did not use any microscopic,
particle model to explain relevant thermal phenomena, including heat conduction. Hence,
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some participants expressed that they never attempted to connect these two fields of study
in the reflective parts of the interview.
Second, given that the participants might have established the connections between ther-
mal phenomena and the particle model, they either might be reluctant to mobilize them or
may have forgotten to apply the particle model in the interview. In the former situation,
mentally manipulating a particle model to simulate the processes of heat conduction re-
quires more working memory load. Thus, according to cognitive load theory (Sweller, Van
Merriënboer, & Pass, 1998), participants might try to avoid running the particle model,
especially when the calorie model could successfully achieve the same goal. In the latter
situation, the participants might just frame an inadequate representation of the problem
(Chi & Glaser, 1985). Although the participants might understand the particle model, they
may just be stuck with the question context, which was quite familiar to them and had been
successfully and repeatedly solved by the calorie view before, and failed to frame a proper
representation of the problem. Consequently, they failed to retrieve the particle model to
solve the problem.
The Participants’ Process Analogies of Heat Conduction
While the elements of each ontological belief have their correspondent mechanisms un-
derlying heat conduction, the participants’ conceptualizations of the processes emerging
from the mechanisms could be sharply different. This study identified five process analogies
of heat conduction, which can serve as the representations of the participants’ conceptu-
alizations of the emergent processes of heat conduction. Among these five analogies, the
two fundamental analogies, marching and flooding analogy, were used by an extremely
low number of participants. However, it would be unwise to delete these two categories or
to treat them as outliers based on prior research and a concordant analysis of the current
findings. According to Vosniadou and Brewer (1992, 1994), the two fundamental analogies
might serve as the original models, which then could evolve into the two dual analogies
(or synthesized models in Vosniadou and Brewer’s term), i.e., the gradient–marching and
gradient–flooding analogy, under the impact of the gradient analogy.
The distribution of these five analogies may also reflect the participants’ developmental
stage and academic background. That is, the two fundamental analogies are supposed to
be the original and na¨ıve ones and thus be used by the least number of participants; the
gradient analogy, which is supposed to be compatible with modern physics, was used by
the largest number of participants; the number who adopted the two dual analogies is just
slightly higher than that of the fundamental analogies but much lower than that of the
gradient analogy. Moreover, the result that the two dual analogies were unstable between
different materials also illustrates the developing nature of the dual analogies. Although
the gradient analogy is a scientifically accepted one for describing the process of heat
conduction in most conditions, it has an intrinsic difficulty in accounting for how a system
achieves thermal equilibrium at the final stage of heat conduction. In contrast, both march-
ing analogy and flooding analogy can straightforwardly represent the process of reaching
thermal equilibrium. Thus, given that the participants were aware of this difficulty, they
could solve it by directly combining the gradient analogy with either the marching anal-
ogy or flooding analogy. From this perspective, the gradient–marching analogy (ideally
explained via a kinetic molecular interpretation) may be the best one, or the most scientif-
ically accepted one, to represent the whole process of heat conduction from its beginning
to equilibrium. Moreover, the strategy of combining two process analogies might be drawn
from a means-ends analysis (Newell & Simon, 1972), given that the participants predicted
the system would eventually reach thermal equilibrium as the end of heat conduction.
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Hence, the distinction between the fundamental analogies and dual analogies appear to
be crucial in terms of either Vosniadou and Brewer’s theory or the means-ends analysis
approach.
The Use of Ontology–Process Combinations to Represent
the Participants’ Mental Models
The ontology–process combinations adopted in this study provide a complementary
approach to represent the participants’ mental models of heat conduction. These combi-
nations not only highlight the insufficient role of ontological beliefs in representing the
participants’ understanding of thermal phenomena but also reinforce the analogue nature
of mental models. The result of this study supports the previously stated argument that
there may be no one-to-one correspondence between a specific kind of element and its
underlying mechanisms or emergent processes. For example, while some participants ap-
plied the calorie view to run their mental models, they might differently conceptualize
heat conduction as either a marching, gradient, or gradient–flooding analogy. Therefore,
it is not productive to use the approach that singles out an ontological element as a single
critical factor to characterize the participants’ conceptions of heat conduction (e.g., Chi
et al., 1994). In addition, the combinations identified in this study suggest the plausible
ontology–process associations as well as exclude the implausible ones.
For example, while heat particles in the substance view can successfully serve as the
agents to accomplish a gradient analogy through diffusion, they by no means can be applied
persuasively to the flooding analogy through diffusion. Similarly, while molecules in the
interaction view can straightforwardly lead to a gradient analogy interpretation by a series
of mutual collisions, this mechanism can hardly support the emergent process of a march-
ing analogy. Accordingly, the only feasible combinations found in this study support the
assumptions that the manipulation of mental models involves mentally simulating pertinent
analogue representations (e.g., Hegarty, 2004; Nersessian, 2008; Schwartz & Black, 1996).
Moreover, the combinations can explicitly highlight the relationships among the elements,
mechanisms, and emergent processes and provide a promising way to represent the process
of heat conduction as a dynamically changing system.
Nonetheless, while the combinations reveal the analogous nature of mental models,
they also indicate the negative effect of analog reasoning. For example, given that the
caloric flow is an analogy to running water, the participants’ construction of the progress
of running water might delimit the possible analogies of caloric flow. More specifically,
if the participants thought water is running in the fashion of marching, then they might
just characterize caloric flow as a marching analogy in the process of heat conduction. A
similar limitation could be imposed on the gradient analogy and gradient-flooding analogy
under the calorie view. Hence, this result suggests the need for a close examination for
the generalizability of D. Gentner and D. R. Genter’s (1983) conclusion that there exists
an intimate relationship between the analogous element (e.g., flowing liquids or moving
crowds as an electric current) and its correspondent process and effect (e.g., double voltage
or double resistance). As mentioned earlier, the subjects in D. Gentner and D. R. Gentner’s
study were purposely selected on the basis that “they could correctly answer questions
about the behavior of water system” (p. 117). Accordingly, the results from this current
study suggest that if the participants had been randomly selected, the one-to-one paired
relationships between the analogous elements and their correspondent processes or effects
likely would have become weakened, and instead, one-to-many relationships probably
would have emerged.
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The Relationship Between Mental Models and Predictions
The relationship between the participants’ mental models and their predictions about
the final state of the thermal system deserves a careful interpretation. On the one hand,
there seems to be no overall significant relationship between the mental models and the
predictions. According to Schwartz and Black (1996), this independence can be attributed
to the rule-based reasoning adopted in generating predictions. That is, the participants
might not have actively run their mental models to make the predictions, but instead
automatically retrieved either a practical rule, every thermal system will eventually reach
thermal equilibrium, or their intuitive beliefs constructed from their tactile perception, the
temperature of an object being heated will keep rising, to solve the problem. Thus, the
participants’ predictions could greatly be determined by whether they believed the thermal
system would achieve equilibrium on the basis of their preference to using their practical
rule or intuitive belief. Also, the participants’ conceptions about the thermal conductor and
insulators could partially account for their predictions.
For example, if they believed conductors and insulators have sharply different thermal
properties, such as while heat can transmit quickly in conductors, insulators can hardly
transmit heat, they might predict that only the aluminum rod would reach thermal equi-
librium. Furthermore, the distribution of different prediction categories found in this study
concurs with Arnold and Millar’s (1994) finding that not all adults applied a thermal equi-
librium framework in responding to interview questions. According to Arnold and Millar,
this was because most of the participants relied on their unchallenged na¨ıve principle,
on = hot/off = cold (which is similar to the above intuitive belief found in this study),
to solve the problems. Thus, both Arnold and Millar’s and the findings of the current
study confirm Schwartz and Black’s suggestion that participants might directly apply their
practical rule or intuitive belief, instead of running mental models, to obtain an efficient
answer.
Although no overall significant relationship between the participants’ mental models and
their predictions was found, some interesting associations between them deserve further ex-
ploration. For example, all of the participants who predicted a continuation of disequilibria
increase in temperatures possessed a gradient analogy of interpretation, and not vice versa.
This connection appears to be straightforward, because only a gradient analogy can result
in a disequilibrium system. In addition, all of the participants who possessed a molecule–
(gradient–marching) analogy predicted the system would reach equilibrium, and also not
vice versa. This association among the ontological belief, emergent process, and resultant
prediction as a whole may represent the scientifically accepted one. If this association did
not occur by chance, then it must be built on the basis of either the manipulation of running
mental models or the direct retrieval of the practical rule constructed from past experience
of running relevant mental models. In either case, the use of mental models is assumed
to play a direct or indirect role in generating the prediction. Hence, this study argues that
a scientifically accepted model has enormous potential to produce a correct prediction,
whereas a correct (false) prediction may or may not result from a scientifically accepted
(unaccepted) mental model.
IMPLICATIONS AND CONCLUSION
Several implications arise from the major findings of this study. Among other topics ad-
dressed here, although our research was not focused on examining conceptual change per
se, we present some comments on the relationship of our findings to prior research related
to conceptual change and offer some suggestions for possible further research in this area.
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In this study, while our findings are consistent with Vosniadou and Brewer’s (1992) theory
about the development of mental models in terms of their characteristic finding of synthe-
sized models (or dual analogies in this study), the results reported here challenges their
and Chi et al.’s (1994) proposal that changes in an individual’s ontological belief may be a
sufficient indicator for the development of mental models or conceptual changes. As shown
in this study, one single ontological belief can result in more than one conceptualization of
the emergent process. Also, the process analogies identified in this study appear to serve as
a better indicator for accounting for the participants’ predictions than ontological beliefs.
In the recognition that the essence of heat is a dynamic process occurring in a thermal
system (Pushkin, 1997; Slisko & Dykstra, 1997), it is necessary to put more emphasis on
investigating students’ various constructions of both the underlying mechanisms and the
emergent processes involved in the system in addition to their ontological beliefs. Accord-
ingly, this study argues that the achievement of a radical change in a mental model also
requires a compatible change in individuals’ understanding of its correspondent underlying
mechanisms and processes.
Results of this study also illustrate that while generating predictions and explanations are
two major functions of mental models, they may rely on different thinking processes. On the
one hand, when facing familiar situations, students tend to directly retrieve practical rules or
intuitive beliefs to make predictions (Schwartz & Black, 1996). On the other hand, to make
a satisfactory explanation, students need to manipulate their analogous, explanatory models
(Clement, 2008; Gilbert, Boulter, & Rutherford, 1998). This difference may not only result
from two different approaches of learning—rote learning and meaningful learning—but
also result in a performance gap between students’ surface answers and their in-depth un-
derstandings. To reduce this gap, this study suggests that model-based instruction can serve
as a bridge to generate meaningful connections between predictions and explanations. On
the one hand, as shown in Table 6, the participants who possessed scientifically acceptable
mental models had a better chance to make a scientifically acceptable prediction. On the
other hand, as suggested by Clement (2008), a scientifically acceptable mental model af-
fords students the capability to make a proper explanation of the underlying mechanism of
a corresponding physical phenomenon. Thus, given that model-based instruction provides
enormous potential for students to construct scientifically acceptable mental models, these
mental models can serve as a solid common platform to connect students’ predictions
with their explanations by understanding the underlying mechanisms that can result in the
predictions.
However, which model of heat transfer should be adopted for instruction is a contro-
versial issue. On the one hand, some researchers (Lewis & Linn, 1994) propose that the
macroscopic, calorie view can serve as a practical model for formal education because it
not only requires less cognitive demand but also is capable of generating correct predic-
tions and powerful explanations. On the other hand, Wiser and Amin (2001) suggest that
students should, and are able to, learn the scientifically compatible model, which is built on
the microscopic, particle view, as long as they are encouraged to engage in metacognitive
thinking to assess their own understanding of heat with a reference to the developmental
nature of science. Since this study found no significant difference in the participants’ per-
formance between these two models, it appears that the most prudent position is to remain
neutral about this issue, but at the same time argue that the decision should be made on the
basis of the intended instructional goals. For example, if the goal is to afford students with
a viable model for efficiently handling everyday problems, then the calorie model may be
adequate; if the goal is to prepare future scientists for successfully solving laboratory-based
problems, then the particle model should be the inevitable one to use. In either case, the
underlying mechanisms and emergent processes of heat transfer should also be the foci
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A study of undergraduate physics students’ understanding of heat conduction based on mental model theory and an ontology–process analysis

A study of undergraduate physics students’ understanding of heat conduction based on mental model theory and an ontology–process analysis

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