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Integrating Mathematica in the Undergraduate
Science Curriculum:
Teaching Computer Literacy with Mathematica
Fred Olness
Southern Methodist University
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"By the year 2000, American colleges and universities will be
lean and mean,
service oriented and science minded, multicultural and increasingly
diverse---if they
intend to survive their fiscal agony" ("Campus of the
Future," TIME,
April 13, 1992).
If our educational institutions are to remain competitive, it is
essential that our
students be prepared to enter a society that is increasingly dependent
on advanced
technology. Therefore, students must be exposed to advanced computer
tools in our science
and mathematics courses--even in introductory courses--to prepare them
for a world where
computers play an important role.
In the 10 years since Mathematica was introduced, it has
already had a significant
impact on the way physics is taught and how research is performed.
This presentation will
review our experiences during the last five years to integrate
Mathematica into the
physics curriculum at SMU, starting from the introductory-level
courses up to the graduate
level. By examining canonical problems from the undergraduate
curriculum and contrasting
the standard solution "before Mathematica" with the
modern standard
solution "after Mathematica," we will also examine
how Mathematica
allows the student to better focus on the underlying concepts. To be
specific, I will
consider two examples.
The double pendulum is a topic that is generally treated as an
"advanced" topic,
but with Mathematica the solution is elementary. Additionally,
using the animation
capability of Mathematica, it is trivial to animate the final
result, allowing
students to discover results on their own by varying the amplitudes of
the separate normal
modes.
Boundary value problems in electrostatics is a topic that beginning
students find
overwhelming. With the power of Mathematica, it is easy to show
these solutions are
quite straightforward--especially with the help of the different
coordinate systems built
into Mathematica. When students finish the problem with pen and
paper, they have
only a set of formulas that may mean very little. With
Mathematica's 2D and 3D
graphics, we can plot the final solution to verify visually that the
boundary conditions
are satisfied. This technique encourages the student to think about
the solution and not
simply grind out the math.
For the lower-level courses, it was the user-friendly interface and
the intuitive graphics
capabilities of Mathematica that encouraged and tempted the
student to experiment
with different methods of solving problems. The tedium of the algebra
or calculus no
longer was an impediment. For example, in 1993 we added
Mathematica as a component
of our calculus-based introductory physics course. The students were
given an initial
tutorial at the beginning of the semester and then assigned problems
from their text that
were particularly well suited to the capabilities of
Mathematica.
With intermediate-level courses, the students were able to delve
deeper into the
capabilities of the program in solving more complex problems.
It is said that there are only two problems in physics that can be
solved exactly: the
Kepler problem of an orbiting mass and the simple harmonic
oscillator--everything else is
perturbation theory.
With Mathematica we were not limited to analytic perturbation
theory, but could
easily devise numerical solutions to real-life problems that were not
solvable using
traditional analytic methods. Examples include projectile motion with
air resistance,
simple harmonic motion with a damping force, and chaotic double-well
systems.
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