This is a description of an interdisciplinary first year course at Olin College. The course uses a series of projects to introduce students to major concepts in modeling and simulation.

1. Good Morning. My name is John Geddes and I am
going to describe a year long course on modeling
and simulation that my colleague Mark Somerville
and I teach at Olin College of Engineering. It is an
interdisciplinary, project-oriented course that
satisfies graduation requirements in Single Variable
Calculus, Multivariable Calculus, Mechanics, and
Electricity and Magnetism.
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2. Olin College of Engineering is located just outside of
Boston, Massachusetts, and was founded in 1997
with a major gift from the F W Olin foundation. The
first employees were recruited shortly thereafter, and
the first students arrived in 2001. Olin College oers
three engineering degrees, all of which were recently
accredited by ABET. The college emphasizes hands-
on and project-based learning, and all students
receive full four-year tuition scholarships. The first
class graduated in 2006, and we currently admit
approximately 75 students every year.
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3. The course I am going to describe this morning takes
place in the first year of the curriculum. It is part of a
closely coordinated set of courses which all students
take. It is interdisciplinary, and draws together ideas
in mathematics, science, and computing. Students
have several shared experiences and all courses
frequently leverage concepts and skills developed in
the other courses.
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4. I want to tell you about this course by giving you a
sense of what the student experiences are. As such,
I’d like to tell you the story of the course. These are
some of the characters in my story, and it’s worth
telling you a bit about their backgrounds and
personalities because this course was designed by
our faculty, for our students. A typical Olin student is
talented academically, has lots of extra-curricular
interests, and has strong social skills. Virtually all
students have seen calculus before and many have
seen mechanics. In general, they are very
independent-minded. The same description could
apply to the faculty, with an added dose of ego!
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5. I’ve told you who the people are, now let me tell you
about the setting of this course.
This course is taught in two locations. An auditorium,
which is a pretty normal space.
Here we meet with all 75 first year students in order
to deliver lectures.
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6. And a set of studios where we meet with 25 students.
At the beginning of the semester, a studio look like a
typical classroom or lab.
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7. By the end of the semester, the rooms look quite
dierent. The tables are usually covered with paper
or equipment, and there are dividers to create a
sense of ownership. Students actually own this space
- multiple courses are taught to the same students in
this room. It can get incredibly messy during project
time.
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8. Now I want to give you a sense of the overall
structure of this course, so that as we talk about
student experiences you have a sense of where they
are in the course.
The theme for this course is modeling and simulation
and it is a year long course. We cover both discrete
and continuous systems, in both compartment and
distributed form.
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9. In the fall, we begin with discrete time compartment
systems which involves concepts from sequences and
dierence equations. In the second half of the
semester, we work with continuous time
compartment systems which involves fundamental
concepts from dierential and integral calculus,
some dierential equations, and rigid body
mechanics. Students receive credit for single-variable
calculus and mechanics on successfully completing
this course.
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10. In the spring we begin with continuous space
distributed systems which involves concepts from
vector calculus (multiple integrals, surface integrals
etc.) and time-independent electric and magnetic
fields. We end the year with distributed systems
continuous in space and time which involves
concepts in partial dierential equations and waves.
Students receive credit for multi-variable calculus
and electricity and magnetism on successfully
completing this course.
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11. In this presentation I’m going to describe in detail
the student experience in the first quarter of the
course - the portion that concerns discrete time
compartment systems. I will then give you a sense
for what the students DO in the other parts of the
course.
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12. When students first arrive in studio, this is what they
see on their tables. It’s a simple game board, based
on the beer game which was devised by John Sterman
at MIT - it is typically used in MBA courses to teach
students about supply chain eects. We have adapted
and simplified it a little and we call it the production
game.
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13. Before we do anything else, we play the production
game. Students are placed in teams of 4 and each
one plays a dierent role - there is a retailer, a
wholesaler, a distributor, and a factory. Each player
picks up their mail, reads the product order, and
tries to fill it. They move items into shipping and
then place an order for product which they pass
down the line. The next player picks up their mail
and so on. After the factory plays their turn, the
items in inventory etc are recorded and the players
are ready to play the next round. The team with the
least costs at the end of the game are declared the
winners.
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14. As the game proceeds, fights start to break out. The
combination of feedback and long delays between
production and retailer leads to large piles of pennies
and silly order strategies. After about thirty rounds,
large oscillations in inventory set in and despite their
best eorts students are unable to minimize the
damage.
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15. Students record data while playing the game and
their first assignment is to present their data
graphically with an hypothesis about what happened.
Each player gives a brief 5 minute presentation using
a visual aid in their studio space. At this stage, the
graphics are usually sloppy, the hypothesis are weak,
and the presentations are poor. This sort of informal
presentation using visual aids is repeated over and
over during the semester and is a central theme in
the course. As you can imagine, students tend to
improve very quickly!
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16. Meanwhile, in the auditorium, formal lectures and
tutorials are delivered by faculty on relevant topics.
For example, at this early stage I will lecture on
sequences and dierence equations, and provide
simple examples of compartment models. The
teaching method in the auditorium uses typical active
learning strategies - short lectures, conceptual
questions, paired problem solving, etc. We also use
undergraduate course assistants to circulate and
help.
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17. Back in studio, students then apply the concepts of
sequences and dierence equations to the
production game. Their goal here is to develop a
formal description of the production game using the
language of dierence equations.
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18. Once students have developed the governing
equations, they now develop a simulation of the
production game. Our simulation environment is
Matlab and we cover basic programming and
simulation in the auditorium using short exercises.
Students then apply the core ideas to simulating the
production game itself.
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19. Then, using this simulation tool, they investigate
what actually happened during the production game.
In this way, they validate their tool, better understand
the game, and are now ready to use the simulation to
do some work. Here you see a comparison between
the date recorded during game play and date
generated by the simulation - 90% of the errors turn
out to be due to bad handwriting and faulty
arithmetic during game play!
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20. At this stage of the semester, which is about three
weeks in, the course moves into a project-oriented
phase during which students develop discrete time
models of various phenomena, they pose a question,
and develop a simulation in order to explore the
answers. Students choose their own projects and we
have seen all sorts of things. Some examples include
a model of owl populations, a model of a SARS
epidemic, a model of beetle populations, and
extensions to the production game.
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21. Here is an example of a question which Pam and Alex
posed about owl populations. They found a
published discrete model for owl populations, but
wanted to adapt it for small populations of breeding
owl pairs.
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22. They developed a discrete stochastic model suitable
for small population numbers. There are three
classes of owl, and each one has a probability of
dying. Adult pairs have a probability of mating and
produce a juvenile which may die before it transitions
to become a sub-adult. Sub-adults then become
adults if they don’t die first.
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23. As with the production game, they develop governing
equations and write a numerical simulation in order
to explore their question. This is largely self-directed
and takes place in studio. One of the interesting
aspects of studio is the way it changes faculty-
student interactions. We enter their space, respond
to their questions, and ask them to defend their
assertions. This can be hard to adapt to - we focus
our attention on posing questions rather than
providing answers.
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24. During this entire process, students are constantly
creating visual aids and making brief informal
presentations. The final deliverable is a formal poster
which summarizes the results of their investigation.
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25. In this case, for example, Pam and Alex ran lots of
simulations using their small population model and
compared their results to the deterministic large
population model. They also investigated the impact
of varying the number of initial breeding pairs. At
this stage, graphics have improved considerably as
well as hypothesis posing and testing.
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26. At the end of this project, students are expected to
publicly present and defend their work in a
professional manner.This is roughly five weeks into
the semester and students have had repeated
practice at creating visual aids and giving
presentations. There is usually an order of magnitude
improvement over their initial attempts.
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27. This is an overview of the piece of the course which I
have just walked through. Students play the
production game, they attend lectures on relevant
topics, they develop a model using dierence
equations, they create a simulation in order to
validate their model, and they pose a question and
explore it. They make frequent informal
presentations using visual aids and this stage of the
course ends with a formal poster presentation. From
the student point of view there was a constrained
phase consisting of introductory material and the
case study, and a project phase in which they have
more autonomy.
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28. We repeat this pattern four times over the course of
the year. After working on discrete time
compartment systems, students then work in
continuous time compartment systems (e.g., rigid
body mechanics) We then move to dealing with time-
independent fields – systems that are static in time,
but continuous in space (e.g., electric fields due to
charge distributions) And finally students learn
about, and model, systems that are continuous in
time and in space (e.g. propagating waves).
I’m now going to show you some examples of the
KINDS of things students DO in each of these phases.
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29. First we will look at compartment systems that are
continuous in time. We focus on rigid body dynamics
described by a finite number of ordinary dierential
equations for the most part, but students have some
freedom to choose. Students first investigate a
seemingly simple case study - a block that is free to
slide on a pivoting ramp. They create a model and a
simulation and use it to explore some basic
questions. In the open-ended phase they choose a
system to investigate themselves.
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30. For example, Greg and Kira decided to model and
simulate an inverted pendulum on a cart. The
pendulum is free to rotate, and the cart is free to
move on the horizontal surface. Most projects begin
with a general question like “can I control an inverted
pendulum” but eventually become more specific like
“what is the best PID control to use in order to
control the inverted pendulum”. Other examples of
projects include a caber toss, a ferris wheel, a
trebuchet, a segway, an elephant on a drum ....
30

31. Students begin by developing a model and the
relevant governing equations. For the inverted
pendulum on a cart, this involves the motion of two
bodies which leads to two coupled dierential
equations. The focus here is on understanding the
description of the model in terms of dierential
equations, and not on analytical solution processes,
none of which would be useful anyway.
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32. Using the simulation tools developed earlier,
students go on to simulate the motion of the
pendulum on the cart by numerically solving the
dierential equations. Typically, they will begin by
validating their model and simulation. Here, for
example, Greg and Kira examined the various
contributions to the energy of the uncontrolled
pendulum on a cart. In addition to confirming that
total energy is conserved, they gain valuable insight
into the dynamics of the system.
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33. After validation, students now DO some work with
their simulation tool. Here Greg and Kira
demonstrates that the inverted pendulum on a cart is
controllable using some sort of PID control. In this
particular case they went onto explore the eect of
varying PID control on the state of the pendulum and
cart. As with the discrete time phase of the course,
this stage ends with a formal poster presentation in
which they present and defend their work.
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34. During the second half of the year we move onto
systems that are distributed. First, we begin with
systems that are distributed in space. Here we
develop concepts and skills in fields and multiple
integrals in the context of electricity and magnetism.
Again, I want to give you a sense for the sorts of
projects students DO.
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35. In this example, Bill and Ilari decided to model and
simulate an electrostatic precipitator for use a coal
burning power plant. This involves defining a plate
geometry, computing the electrostatic field and then
tracking the motion of ions as they travel through the
system. During the spring semester, we switch from
oral and visual communication to written
communication. Students write short paper
fragments over and over again before the project
ends with a formal written deliverable.
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36. Bill and Ilari investigated the eficiency of several
precipitator designs. In the first, a wire is held at
-10V while two plates are held at ground. In the
second, two wires are held at -10V while the two
plates are held at ground. In the final design, one
plate is held at ground while the other is held at
-10V. Bill and Ilari wrote a simulation to compute the
electrostatic field due to these configurations and
then tracked the motion of ions as they pass through
the system for dierent initial velocities.
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37. Here they present their simulation results as the rate
of particle capture versus particle velocity. They
demonstrate that the simple two plate design is best
and that there is an optimal velocity in order to
maximize the capture rate.
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38. We end the course with systems that are continuous
and distributed in space and time. This involves the
major vector theorems from multivariable calculus,
as well as some partial dierential equations. Again,
students complete a case study in electricity and
magnetism, but we then open it up to waves in
general.
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39. In this example, Rachel and Ilari decided to
investigate the tones produced by a real piano and
compare them to the tones produced by a
mathematical model of a piano. They again work in
studio in a self-directed manner, and produce a
written paper as the final deliverable.
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40. In this project, Rachel and Ilari did quite a bit of
background reading and decided to implement a
model that they found in the literature. A piano
hammer impacts the string which sets up a
propagating wave. It then resonates with the bridge,
sending vibrations through the piano. The model
takes the form of a partial dierential equation which
captures the motion of a damped, propagating wave
subject to forcing from the piano hammer. While this
may look out of reach for these students, it follows
naturally from the earlier case studies and projects
and they have little dificulty in adapting their
simulation tools to handle this.
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41. Here they present their simulation results and
compare them against a real piano. They made a
recording of an actual piano and then decomposed
the signal into its Fourier components. Here they
show the relative strength of each fundamental
frequency for each of the dierent notes. Then using
relevant parameter values and an accepted model for
the hammer impact, they ran a numerical simulation
for each note and also decomposed the signal into its
Fourier components. They concluded their paper by
comparing and contrasting the results and by
discussing the various ways in which the model could
be improved.
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42. That ends my description of the course. I’d like to
spend the remaining time reflecting on this
experience. I’m not going to pretend that we are
conducting a careful, pedagogical experiment here -
rather we have developed this course based on a
deep-seated belief in the power of modeling and
simulation. We also believe that oral and written
communication are fundamental competencies that
leads to a deeper knowledge base.
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43. I mentioned earlier that this course was developed by
our faculty for our students. I believe that, like
politics, all education is local. What works for our
students won’t necessarily work for your students
and vice versa. We developed this course using an
user-oriented curriculum design process - the
details are finely tuned for our students.
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44. Having said that, I’d like to make a couple of
observations. Our focus on student self-direction at
this early stage leads to a group of fearless students.
They are simply not intimidated, and will take on
anything. They are not boxed into dealing with
models that are simple enough to admit simple
analytical solutions - real problems with realistic
models just aren’t like that.
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45. It is also incredibly motivational. The limits of
simulation become quite apparent to them and they
are very ready for alternative approaches and
models. In future classes, they come in hungry for
knowledge. In a nonlinear dynamics and chaos
course which I teach I usually have trouble containing
them - I think we all know the benefits of dealing
with a motivated group of students.
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46. And, in conclusion, I have to say that I believe that
they understand the fundamental concepts better
and can better apply them - but again this is based
on my own experience over the years. I’d be
delighted to take some questions.
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