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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 Coulomb’s 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

Schrödinger’s 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 train’s 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 Earth’s 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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1041 (1996).

3. I. Halloun, “Schematic concepts for schematic models

of the real world: The Newtonian concept of force,” Sci.

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4. I.M. Greca and M.A. Moreira, “Mental, physical, and

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physics,” Sci. Educ. 86, 106 (Jan. 2002).

5. A.G. Harrison and D.F. Treagust, “Learning about

atoms, molecules, and chemical bonds: A case study of

multiple-model use in grade 11 chemistry,” Sci. Educ.

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ing method for high school physics instruction,” Am. J.

Phys. 63, 606-619 (July 1995).

7. L. Grosslight, C. Unger, and E. Jay, “Understanding

models and their use in science: Conceptions of middle

and high school students and experts,” J. Res. Sci. Teach.

28, 799–822 (1991).

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“Students’ understanding of the role of scientific mod-

els in learning science,” Int. J. Sci. Educ. 24, 357 (April

2002).

9. J. Ryder and J. Leach, “Interpreting experimental data:

The views of upper secondary school and university sci-

ence students,” Int. J. Sci. Educ. 22, 1069 (Oct. 2000).

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education of modellers,” Int. J. Sci. Educ. 24, 369 (April

2002).

11. Private communication with Jane Jackson.

12. G. Holton and S.G. Brush, Physics, the Human Adven-

ture: From Copernicus to Einstein and Beyond, 3rd ed.

(Rutgers Univ. Press, New Brunswick, NJ, 2001), p.

526.

13. E. Etkina, T. Matilsky, and M. Lawrence, “What can we

learn from pushing to the edge? Rutgers Astrophysics

Institute motivates talented high school students,” J.

Res. Sci. Teach. 40, 958–985 (2003).

14. C.T. Hill and L.M. Lederman, “Teaching symmetry in

the introductory physics curriculum,” Phys. Teach. 38,

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review of research-based instructional strategies,” Am. J.

of Phys. 59, 891–897 (Oct.1991).

THE PHYSICS TEACHER ◆ Vol

16. J.R. Frederiksen, B.Y. White, and J. Gutwill, “Dynamic

mental models in learning science: The importance of

constructing derivational linkages among models,” J.

Res. Sci. Teach. 36, 809–836 (1999).

17. J.L. Lemke, “Teaching all the languages of science:

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