Teaching ODEs via the World Wide Web

The Original Vision (1995-1997)

"Computers in the classroom - teaching ODEs via the World Wide Web. When given the opportunity, I enjoy teaching students about computers and how they can be used to learn about differential equations and dynamical systems. A hope is to one day create an online classroom for the individual investigation of differential equations using some of the more sophisticated machinery available in the industry."

This vision, articulated when the web was only four years old, imagined a future where:

• Students could explore mathematics interactively from anywhere
• Sophisticated computational tools would be accessible via web browsers
• Individual investigation would complement traditional instruction
• The "machinery" of research mathematics would be democratized

MAE 305: Mathematics in Engineering

At Princeton, I developed one of the first fully online components for a differential equations course. This wasn't just posting lecture notes—it was interactive learning:

Students could explore concepts like stability, oscillations, and chaos through interactive demos—revolutionary for 1995 when most math was still taught exclusively on blackboards.

Technical Innovation in 1995

Creating online math education in the mid-90s meant solving problems that modern educators take for granted:

Every interactive element had to be coded from scratch. There were no frameworks, no Stack Overflow, no npm packages—just raw HTML, Perl scripts, and a vision of what online education could become.

Educational Philosophy

The approach to online ODE teaching was grounded in several key principles:

• Visualization First: Complex concepts become intuitive when you can see them evolve in real-time
• Interactive Exploration: Students learn by doing, not just reading
• Immediate Feedback: Mistakes become learning opportunities when caught quickly
• Accessible Anywhere: Education shouldn't be limited by geography or schedule
• Research-Grade Tools: Why shouldn't undergraduates use the same tools as researchers?

"The individual investigation of differential equations using some of the more sophisticated machinery available in the industry"—this wasn't dumbing down; it was opening up.

Topics and Interactive Modules

The online ODE curriculum covered traditional topics with non-traditional methods:

Each module included interactive elements where students could change parameters and immediately see how solutions evolved—making abstract mathematics tangible.

Legacy and Impact

While specific course materials from 1995 may seem primitive by today's standards, the vision was remarkably prescient:

• MOOCs (2010s): Massive Open Online Courses realized the vision of accessible education—15 years later
• Interactive Textbooks: Now standard, but revolutionary when equations could "come alive" in 1995
• Computational Thinking: Using computers not just as calculators but as exploration tools
• Asynchronous Learning: Students could learn at their own pace, on their own schedule

Connection to SCIGMA

The ODE teaching project and SCIGMA software development were deeply intertwined:

• Both democratized access to sophisticated mathematical tools
• Both emphasized visualization as key to understanding
• Both pushed the boundaries of what computers could do for mathematics
• Both believed in making research-grade tools accessible to students

SCIGMA's interactive manifold computations showed what was possible; the ODE teaching project brought similar capabilities to undergraduate education.

The Vision Realized

Today, online mathematics education is ubiquitous. Students routinely use Desmos, Wolfram Alpha, and countless other tools to explore differential equations. MOOCs teach millions. Interactive textbooks are the norm.

But in 1995, teaching ODEs via the World Wide Web required imagination, technical innovation, and belief in a future where anyone, anywhere could explore the beauty of mathematics. That future is now our present—and the journey continues.