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The word modeling is becoming more and more common in physics, chemistry, and general science instruction. In physics, students learn models of the solar system, light, and atom. In biology courses they encounter models of joints, the circulatory system, and metabolic processes. The benefits of engaging students in model building are described in the literature.1-5 ``Modeling instruction'' is an example of a whole curriculum based on the idea of modeling.6 However, in a traditional physics class students do not have a clear understanding of what the word model means, and thus do not appreciate the role of this notion in physics.7-9 Physics teachers also have difficulties defining this word.10,11 The purposes of this paper are (a) to reexamine the word model as it is used in science, and (b) to suggest several types of tasks that engage students in the construction of models in a regular-format introductory physics course.
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The Role of Models in
Physics Instruction
Eugenia Etkina, Aaron Warren, and Michael Gentile, Rutgers University, New Brunswick, NJ
T
he word modeling is becoming more and more
common in physics, chemistry, and general
science instruction. In physics, students learn
models of the solar system, light, and atom. In biology
courses they encounter models of joints, the circula-
tory system, and metabolic processes. The benefits of
engaging students in model building are described in
the literature.
1-5
“Modeling instruction” is an example
of a whole curriculum based on the idea of model-
ing.
6
However, in a traditional physics class students
do not have a clear understanding of what the word
model means, and thus do not appreciate the role of
this notion in physics.
7-9
Physics teachers also have
difficulties defining this word.
10,11
The purposes of
this paper are (a) to reexamine the word model as it is
used in science, and (b) to suggest several types of tasks
that engage students in the construction of models in a
regular-format introductory physics course.
What Is a Model?
The modeling approach in physics began with Rene
Descartes. He was the first to propose that the mental
constructs of a scientist about the world were not to be
considered “postulates representing his own beliefs but
as useful models from which one could deduce conse-
quences in agreement with observations.
12
In physics education research the word model is
associated with David Hestenes and his colleagues,
who advocated the use of models in physics instruc-
tion more than 20 years ago. He defined a model in
the following way: “A model is a surrogate object, a
conceptual representation of a real thing. The models
in physics are mathematical models, which is to say
that physical properties are represented by quantitative
variables in the models.
1
In general, physicists share several common ideas
about models:
a) a model is a simplified version of an object or pro-
cess under study; a scientist creating the model
decides what features to neglect;
b) a model can be descriptive or explanatory; explana-
tory models are based on analogies—relating the
object or process to a more familiar object or pro-
cess;
c) a model needs to have predictive power;
d) a model’s predictive power has limitations.
13
Mastering these ideas is difficult. How does one
know what to neglect while simplifying an object or
a process? Can the same object or process be modeled
differently in different situations? How does one make
a decision whether a particular model is appropriate?
How can one use models to make predictions?
In the following section we explore the meaning of
the word model at a deeper level, which will ultimately
allow us to devise tasks for students that address the
questions raised in the previous paragraph.
Classifying Models
As discussed above, scientists use models or simplifi-
cations to describe and explain observed physical phe-
nomena and to predict the outcomes of new phenom-
ena. We suggest that when simplifying a phenomenon
to make a model, we simplify 1) objects, 2) interac-
THE PHYSICS TEACHER Vol
tions between objects, 3) systems of objects together
with their interactions, and/or 4) processes (Fig. 1).
This classification gives us four types of models.
1. Models of objects: When we choose to investi-
gate a physical phenomenon, we first identify the
objects involved. We then decide how we will sim-
plify these objects. For example, we can model the
same car as a point particle, or as an extended rigid
body, or as multiple extended rigid bodies.
2. Models of interactions: When there are multiple
objects involved, we need to consider interac-
tions between those objects. We make decisions
to neglect some interactions and take others into
account. We can model interactions quantitatively
in terms of the strength and the direction of a force
or a field, or the magnitude and the sign of a po-
tential energy. When we quantify this picture we
get some mathematical expressions that we call
interaction equations. An example of an interac-
tion equation is Coulombs law.
3. Models of systems: By combining the models of
objects and interactions for a physical system, we
get a model of the system. For example, if we sim-
plify a gas as many point particles that interact with
the walls of their container via elastic collisions, we
have a model of a system known as an ideal gas.
4(a). Models of processes (qualitative): Due to the
interactions between the objects in a system or
with objects outside a system, the system may
change in some manner. We will refer to a model
that describes the changes in a system as a process
model. For example, we can explain qualitatively
a thermodynamic process involving a gas in a con-
tainer with a movable piston using the model of an
ideal gas and considering its interactions with the
piston.
4(b). Models of processes (quantitative): When we
quantify our models of systems and processes, we
get mathematical expressions that we call state
equations and causal equations. A state equation
describes how one or more properties of a system
vary in relation to each other, but the cause of the
change is unspecified. A causal equation, however,
describes how the properties of a system are af-
fected by its interactions with the environment.
A state equation is a mathematical expression in
which each quantity corresponds to various properties
of a single system. For instance, x = x
0
+ v
x
t is a state
equation, a model of a process involving a point par-
ticle. Another example is the ideal gas law, a model of
a process involving an ideal gas. Each quantity in the
equation corresponds to a property of a gas (which is a
system of point particles). On the other hand, a causal
equation is any mathematical expression that includes
quantities that correspond to physical interactions
between a system and its environment. For example,
in the first law of thermodynamics heating and work
cause changes in the internal energy of a system.
Other examples of causal equations are the impulse-
momentum equation, the work-energy equation, and
Schrödingers equation. The most fundamental causal
equations are based on symmetries.
14
Each of these models can be represented in many
ways, including words, mathematical functions,
graphs, pictures, and model-specific representations
such as motion diagrams, free-body diagrams, energy
bar charts, ray diagrams, and so forth. Students need
to learn how to use these representations to solve spe-
cific problems. Much has been written about the im-
portance of representations in physics instruction and
16 THE PHYSICS TEACHER Vol.
Fig. 1. We can model nature by focusing on an object, an
interaction, a system, or a process. Quantifiable models
include mathematical expressions such as interaction equa-
tions, state equations, and causal equations.
successful instructional strategies.
15-17
Students often engage in modeling during our
classes but are unaware of it. In the section below we
suggest several types of tasks that make this process
explicit and encourage students to consciously engage
in and reflect on modeling.
Engaging Students in Making and
Testing Models
In physics education, modeling of phenomena for
investigations and problem solving has been done
mostly by Hestenes and his colleagues.
1-3, 6
Their ap-
proach assumes consistent use of special vernacular,
representations, and problem-solving strategies during
instruction. We suggest that tasks engaging students
in deliberate modeling of real situations can be used
in any physics course, while students are engaged in
problem solving or laboratory exercises. We describe
examples of activities to help students practice model
construction, evaluation, and revision of models. The
tasks are grouped under three categories: the types of
models identified in the “Classifying models” section
(models of objects, interactions, systems, and pro-
cesses), the purposes of modeling as identified in the
section “What Is a Model” (describing, explaining,
and predicting), and the limitations of the models. The
wording of the tasks follows the recommendations of
Heller and colleagues.
18
Different Types of Models
1) Choosing a model of an object
(a) A 70-m long train leaves a station accelerating at
2 m/s
2
. You are at the platform entrance 30 m
from the tracks. To determine if you can catch the
train, would you model the train as a point particle
or as a rigid object with a definite length? Explain.
(b) The same train travels for 10 hours and covers
630 miles. To determine the trains average speed,
would you model the train as a point particle or as
a rigid object with a definite length? Explain
2) Choosing a model of an interaction
You have been hired as a consultant for NASA and
the following task is given to you: You are in charge
of a group whose job is to design a computer pro-
gram that can quickly calculate the energy of an
Earth-rocket system. The rocket will travel from
the Earths surface to an orbit high above the Earth.
You know that the gravitational potential energy of
a system consisting of two objects of masses M and
m (when the smaller object is outside the more mas-
sive object) is U = –GMm/r. To make your computer
program as fast as possible, though, you want to
know when it is OK to treat the gravitational poten-
tial energy of the system as U = mgh.
a) Where do we set U = 0 when we use U = mgh?
b) Using your answer to part (a), show that we can
use U = –GMm/r to derive U = mgh. [Hint: 1/(R +
h)
(1/R) – (h/R
2
) if h << R].
c) Based on the approximation used in part (b),
when do you think it is reasonable to use U = mgh?
3) Choosing a model of a system
You are an assistant for the physics labs. You just
found a resistor in your desk drawer and are curi-
ous about its resistance. You have a battery, some
connecting wires, an ammeter, and a voltmeter. You
decide to measure the voltage across the resistor and
divide it by the current through the resistor. You
build a circuit as shown below (Fig. 2a) and then
realize that there is another way to do it (Fig. 2b).
What modeling assumptions about objects and
processes in the circuit do you need to make to go
with the first circuit or with the second? What infor-
mation about the elements of the circuit do you need
to have in order to decide which method (a or b) is
applicable to the circuit? (Hint: Remember that both
measuring devices have internal resistance.)
THE PHYSICS TEACHER Vol 17
Fig. 2. Using the reading of the ammeter and voltmeter
to calculate the resistance of the resistor leads to differ-
ent results for different circuit arrangements.
4) Choosing a model of a process
State equations
Given a rigid 0.50 m
3
container with 4.46 mol of
air inside, and an initial temperature of 500 K, you
measure a pressure of 37.06 kPa. You then cool it to
a temperature of 133 K while compressing the gas to
a volume of 0.10 m
3
. The pressure of the gas is now
measured to be 49.10 kPa. Two possible models of
the gas are the ideal gas model and Van der Waal’s
model. Determine how consistent each of these
models are with the reported measurements (a =
0.1358 J·m
3
/mol
2
, b = 3.64 x 10
-5
m
3
/mol for air).
If one model is more accurate for a certain measure-
ment, propose an explanation for why this is the
case.
Causal equations
You are analyzing a video of a falling beach ball (m =
500g, R = 20 cm) by viewing it frame by frame. You
find that the acceleration of the ball is constant and
equal to 8.8 m/s
2
. You decide to analyze the situa-
tion by modeling the interactions of the ball with the
Earth and air. What modeling assumptions about the
interactions and processes do you need to make to
explain the acceleration of the ball?
Purposes/Uses of Models
5) Using models to describe phenomena
You bought a motorized toy car for your little sister.
How can you find out which model describes the
motion of the car best: the model of motion with
constant speed, constant acceleration, or changing
acceleration?
6) Using models to explain phenomena
You have a cart on an air track attached by a string
that passes over a pulley down to a hanging object
[see Fig. 3(a)].
You push it abruptly toward the left. The cart
moves to the left, slows down, stops, and starts mov-
ing to the right with increasing speed. The graph for
the cart’s acceleration versus time is shown in Fig 3(b).
When you repeat the same experiment with a cart on
a regular track, the acceleration-versus-time graph
looks different [see Fig. 3(c)].
Identify models of objects, interactions, systems,
and processes that can help you to explain each graph
and discrepancies between them.
7) Using models to predict new phenomena
A helium-filled balloon is attached to a light string
and placed inside a box made of transparent plastic.
The box has wheels on the bottom that allow it to
roll. Explain why the balloon and string are verti-
cal.
19
What models of objects and interactions did
you use? Predict what will happen to the thread
and the balloon if you abruptly push the box to the
left. To make the prediction, explain what models
of objects and interactions you will include in your
system and how you will model any processes that
occur. Then observe the experiment. If your predic-
tion does not match the result, revise your model in
order to get a new prediction that does match the
result.
Model Limitations
8) Limitations of models objects,
interactions and processes
Your friend’s lab group has to figure out the specific
heat of a 0.50-kg rock. They plan to heat the rock by
letting it sit in a 200°C oven for five minutes. Then
they will put the rock in a thermos filled with 200 g
of ice (measured with a dietary scale), close the ther-
mos, and wait another five minutes. After doing all
18 THE PHYSICS TEACHER Vol.
t
a
a
t
v
0
x
(a)
(b)
(c)
Fig. 3. (a) A cart on the air track connected by a string
that passes over a pulley to a hanging object was
abruptly pushed in the negative direction and let go.
(b) The acceleration-versus-time graph for the cart.
(c) The acceleration-versus-time graph for the same cart
when the experiment was repeated on a regular track.
this, they open the thermos and find that the ice has
completely melted. They measure the final tempera-
ture of the rock by measuring the final temperature
of the melted ice water, which is 20°C. They calcu-
late the specific heat of the rock as follows:
m
ice
L
f
+ m
ice
C
water
∆T
ice water
+ m
rock
C
rock
∆T
rock
= 0
C
rock
= –[(0.2 kg)*(33.5 x 10
4
J/kg) + (0.2 kg) *
4186 J/(kg*K) * 20ºC)] / [0.50 kg * (–180°C)] =
930 J/(kg K) .
Identify all the modeling assumptions your friend’s
group has made about the objects, interactions, sys-
tems, and processes, and evaluate whether or not each
assumption should be accepted.
Conclusion
Choosing a productive model to describe or explain
a phenomenon under study is a routine part of the
work of scientists but a rare exercise for our students.
Students have difficulties understanding the mean-
ing of the word model and using it to analyze physical
phenomena and solve problems. We hope that by cre-
ating and using tasks similar to the ones shown here
students can become more proficient at modeling. To
help students you can engage them in “meta-model-
ing”—reflection on the purposes and outcomes of the
modeling process. We encourage instructors to de-
velop their own tasks like those shown above, and to
incorporate them into their curricula. Solutions to the
problems presented in the paper and more tasks are
available at http://paer.rutgers.edu/Scientific
Abilities/ModelingTasks/default.aspx.
Acknowledgments
We thank Michael Lawrence and Suzanne Brahmia
for providing suggestions for the tasks, and David
Brookes and Alan Van Heuvelen for helping with
the preparation of the manuscript. We also thank an
anonymous reviewer for helpful comments and sug-
gestions. The project was supported in part by NSF
grant DUE 024178.
References
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THE PHYSICS TEACHER Vol
16. J.R. Frederiksen, B.Y. White, and J. Gutwill, “Dynamic
mental models in learning science: The importance of
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Res. Sci. Teach. 36, 809–836 (1999).
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Words, symbols, images, and actions,” http://www-
personal.umich.edu/~jaylemke/papers/barcelon.htm.
18. Students engage in similar activities while solving con-
text-rich problems as in P. Heller, R. Keith, and S. An-
derson, “Teaching problem solving through coopera-
tive grouping Part 1,” Am. J. Phys. 60, 627–636 (July
1992).
19. Videos for modeling tasks and other videotaped experi-
ments can be found at http://paer.rutgers.edu/pt3. The
video of this experiment can be found at http://paer.
rutgers.edu/PT3/experiment.php?topicid=13&exptid=
121.
PACS codes: 01.40Gb, 01.55
Eugenia Etkina is an associate professor of science
education at Rutgers University. She works with pre- and
in-service physics teachers and with colleagues in physics
who are reforming undergraduate physics courses.
Graduate School of Education, Rutgers University,
10 Seminary Place, New Brunswick, NJ 08901-1183;
etkina@rci.rutgers.edu
Aaron Warren is a graduate student in the Department
of Physics and Astronomy working on his Ph.D in physics
education research.
Department of Physics and Astronomy, Rutgers
University, 136 Frelinghuysen Road, Piscataway, NJ
08854; aawarren@physics.rutgers.edu
Michael Gentile is an instructor in the Department of
Physics and Astronomy.
Department of Physics and Astronomy, Rutgers
University, 136 Frelinghuysen Road, Piscataway, NJ
08854; mgnetile@physics.rutgers.edu
20 THE PHYSICS TEACHER Vol. 43, 2005
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