Visualizing Science Using
Learning Objects Created with
Mathematica

Andy Anderson
http://www.amherst.edu/~aanderson/

Amherst College

Amherst, Massachusetts, USA

Presentation Overview

Learning Objects for Scientific Concepts
Learning Objects Created with Mathematica
Advantages/Disadvantages of Mathematica
Learning Object Development at Amherst

The Complexity of Scientific Concepts

Many scientific concepts involve complex relationships in space and time, for example:

Astronomy: Planetary, Stellar, and Galactic Motion
Biology: Cellular Development, Neuronal Discharge
Chemistry: Molecular Structure, Chemical Reactions
Geology: Ground Water Flow, Plate Tectonics
Mathematics: Geometry, Vector Calculus
Physics: Just About Everything!

Hand-drawn classroom representations of scientific phenomena are typically sketchy and are neither reusable nor distributable.

Dynamic phenomena require multiple images to indicate time progression, and can be difficult to interpret.

Classroom representations are often difficult to reproduce by students.

Learning Objects for Scientific Concepts

Digital media provide a means to illustrate complex structures and motions, using:

Static Images
Animations
3D / Virtual Reality Objects
Interactive Tools

They can help bridge the gap in student understanding.

Digital images and animations are examples of learning objects that are commonly viewable by any computer.

Such learning objects are reusable and improveable.

The World-Wide Web can integrate learning objects together with text and other media, and distribute them to the lecture hall or to students in their dorm rooms.

The portability of these learning objects also means that they can be easily shared with other educational institutions.

Learning Objects Created with Mathematica

Mathematica can be used to create scientifically realistic learning objects.

It offers:

Stellar mathematical abilities
Full-fledged graphics capabilities
Web Interface

Example: 3D Image for Geography

This image helps to visualize the projection of the Earth's surface onto a plane.
It is used in an introduction to geographic information systems.
This image was created using Mathematica and a set of geographic coordinates for the boundaries of the continents (found on a USGS Internet site).

The coordinates were drawn directly on a sphere to produce the blue globe.
They were also transformed with the mathematical formula for an orthographic planar projection, obtained from MathWorld, and drawn on the adjacent plane.
A number of other projections were also generated.

Example: 3D Image for Astronomy

This image helps to visualize the three possible structures of the Universe due to the curving of space by its mass and energy, according to General Relativity: spherical, flat, and hyperboloidal.
It has been used in an introductory astronomy class, and is in the MERLOT catalog.

Textbooks commonly paint these structures a dull gray.
Using Mathematica, a Hubble Space Telescope image of the distant Universe was layered onto these surfaces, more like the real thing!

Example: VR Object for Astronomy

General relativity explains the force of gravity as the result of the curvature of space.
This virtual reality object illustrates the bending of the paths of the planets in our Solar System due to the Sun.
As in the previous example, the 3D solar system is represented as a 2D surface, with the Sun at the bottom of a gravity well.

Mathematica was used to generate a large number of different views of this 3D image, at regular angular intervals.
The images were stitched together into a VR object with the Apple QTVR Authoring Studio.

The planets have circular or elliptical orbits, depending on their "tilt".

The light beam has a hyperbolic path.

Example: 3D Animation for Astronomy

This animation helps to visualize a pulsar: a massive, compact neutron star that rotates rapidly and beams electromagnetic energy along its tilted magnetic axis.
It has been used in an introductory astronomy class, and is in Wesleyan University's Learning Objects, Learning Activities catalog (which is planned to be linked into MERLOT at some point).

Mathematica was used to define the various structures.
A viewpoint is chosen so that that when the beam rotates in our direction, we see a "pulse" of radiation. (Actually, the viewpoint rotates!)
The beam isn't infinitely long, so some artifacts appear around its edges, and Photoshop was used to clean it up.

The background image of this presentation is the Crab Nebula, which has a pulsar at its center.

Example: 2D Animation for Astronomy

This animation helps to visualize stellar motion in a spiral galaxy: stars move in correlated elliptical orbits, with roughly the same linear speed in the disk.
The result are spiral arms that travel more slowly than the inner stars, and faster than the outer stars (differential rotation).
It has been used in an introductory astronomy class, and is destined to for LOLA...

Example: 2D Animation for Astronomy

Mathematica's matrix capabilities were used to move individual stars around strategically positioned elliptical orbits.
The star orbits were then rotated at a constant speed.
The size and speed parameters were chosen to match those of the Milky Way; the large yellow star represents the Sun.
The star densities and colors, from blue to reddish orange, were also chosen to represent the observed distribution.

Example: Correlated 3D Animation

This animation demonstrates the occurrence of solar and lunar eclipses...

Example: Correlated 3D Animation

Solar and lunar eclipses occur on an 18-year Saros cycle.
Mathematica was used to define general functions for the Earth-Moon-Sun system, allowing the display of two different, correlated views at the same time.
The size and speed parameters were chosen to match those of the Earth-Moon system; the Sun's position is distorted.

Example: Interactive Logarithm Exercises

Mathematica can also be implemented as a backend to a Java servlet called webMathematica.
This technology was used to create logarithm exercises for use in introductory chemistry classes.

The exercise page was written to translate more common notations (e.g. "log()") into Mathematica notation ("Log[]").
Student responses are transferred to the Java servlet, which hands it to Mathematica, which simplifies it, tests it, and gives back a response.
An introductory logarithm tutorial was written for the students to review before they practice with the exercises.

The Advantages of Mathematica

Mathematica can easily define mathematical functions and graph them in both two and three dimensions.
Mathematica allows accurate representations of scientific phenomena.

Scientific understanding can therefore drive development, with little need for artistic ability.

Often Mathematica is already familiar to scientists, from their research activities.
Often Mathematica is readily available with an existing campus license.

The Disdvantages of Mathematica

Mathematica's programming language is not as familiar as the more common Fortran or C/C++/Java.
Mathematica's graphics functions are somewhat inconsistent, so there is a learning curve even if one has a basic familiarity with the program!
Mathematica's graphics are relatively crude, being based on a Postscript model that's almost twenty years old.
Mathematica has no tools for the freehand definition of graphics elements, which is occasionally useful.
If Mathematica is not already available to you, it's one more expensive program to purchase.

The Development of These Learning Objects

Many of these learning objects were created by the presenter (a physicist and academic technology specialist) for his own teaching.
The others were created by students mostly supervised by the presenter, working on faculty technology projects during Amherst College Information Technology's Summer Intern Program.

These students typically know little or nothing about Mathematica, but otherwise have a good knowledge of math or science.
They quickly pick up a great deal of expertise with Mathematica, which they can use in their classes.
They are encouraged to become Student Software Consultants during the school year, where they are available to help other students, and work on additional projects.

Summary

Learning objects can illustrate scientific concepts in a way not possible on the blackboard or in textbooks.
Mathematica can produce learning objects for the natural sciences and mathematics with relatively little difficulty, as long as the developer has a good scientific perspective.
Even without extensive knowledge of Mathematica, faculty can oversee student projects that use it.
An early introduction to students can produce a skillful team of users that can more efficiently complete such projects, teach other students, and use Mathematica in their classes and theses.

Acknowledgements

Pulsar Animation

Development Assistance: Brian Lyons and Tarja "Dasha" Martikainen
Funding: Mellon Foundation through the Amherst College Information Technology Summer Intern Program
Sponsorship: Prof. George Greenstein, Amherst College Astronomy Department

Spiral Galaxy Animation

Development Assistance: Ding'an Fei
Funding: National Science Foundation, through the Amherst College Astronomy Department
Sponsorship: Prof. George Greenstein, Amherst College Astronomy Department

More Acknowledgements

Eclipse Animation

Development Assistance: Jen Roberge and Alexey Sokolin
Funding: Amherst College Information Technology Summer Intern Program
Sponsorship: Prof. George Greenstein, Amherst College Astronomy Department

Logarithm Exercises

Development Assistance: Lucia Cucu and Christian Wilson
Funding: Amherst College Information Technology Summer Intern Program
Sponsorship: Deborah Lee, Amherst College Quantitative Skills Center; Mark Marshall, Amherst College Chemistry Department

This slide show uses S5, an open-source browser-based presentation system.

MERLOT International Conference | Ottawa, Ontario — August 9, 2006