Whether scaling Mount Everest, examining the Mona Lisa in the Louvre, or simulating surgery, Virtual Reality (VR) makes almost any experience possible. Head-mounted display devices plunge users into an array of immersive, interactive 3D environments, from operating rooms to outer space. Now, this same technology can help students learn vector calculus in a groundbreaking new way, thanks to Nomad Calculus, a VR experience developed by Isaac Berman ‘19.
“Vector calculus can be difficult for students to understand conceptually, mainly because unlike single-variable calculus, most of the concepts cannot be easily represented graphically in two dimensions,” explained Isaac in his project proposal. “2D chalkboards and graph papers don’t do the 3D space justice. In a VR experience—which is the single best way to place users into a 3D world, giving them a sense of personal presence, scale, and depth perception—users can freely experiment with 3-dimensional immersive graphs and demonstrations that clearly illustrate what the calculus means.”
Introducing Nomad Calculus
Nomad Calculus (so-named because “…like a nomad wandering through the desert, users proceed at their own pace…and the program offers calculus without frustrating the user: No Mad…”) revolves around the Oculus Rift, an experience Isaac describes as one that “transports you into another world, one you can inhabit almost as fully as the real world. You turn or move your head, and it shifts your viewpoint accordingly.” The Rift’s VR headset, and companion touch–a tool that enables developers to sync up multiple devices into one 3D world–controls and cameras, immerses students in narrated lessons featuring 3D examples and interactive demos in which they can draw, move, and rotate animated 3D graphs appearing right before them.
Isaac’s work comprised several distinct areas. First, he designed and built the entire VR environment from scratch—including full graph interaction and motion control—using Unity 3D, a cross-platform game engine widely used to develop video games and simulations. He learned how to program for Unity and for VR at Carnegie Mellon University’s (CMU) National High School Game Academy during summer 2018; that Fall, Isaac began applying his new-found knowledge to the Computer Science Innovations project that became Nomad Calculus, revising and refining design and programming issues right through a spring break overhaul. To write the code enabling his program to “do the same things that a graphing calculator does, but in 3D” Isaac applied Unity’s tools to concepts like graphing vectors, vector-valued functions, and functions of multiple variables on coordinate axes in 3D space.
The Immersive Side to Calculus
Finally, Isaac created lesson plans, wrote voiceover scripts, and programmed the content. “The two lessons that currently exist are the tutorial—teaching the controls of the program—and Riemann sums, a method for estimating the volume of a 3D solid.” He conversed briefly with CMU professor Dr. David Kosbie about the development process; for constructing lesson plans, advice on research methods, and ongoing consultation, Isaac worked closely with Computer Science Department Chair David Nassar and Director of Upper School Dr. Anne Fay, whose background includes cognitive psychology and digital learning.
Each element of Nomad Calculus evolved through several iterations during the year, based on user feedback from at least 20 volunteers. Isaac tested the final version on volunteers from various math classes, none with any calculus background.
“Half of the volunteers used Nomad Calculus to learn about Riemann sums, and the other half watched a video lesson on exactly the same content taught by Mr. Nassar. Both groups took a multiple choice test after their lesson, and I found that students who used VR achieved about 12% higher test scores. Many students sought me out days later to tell me how amazing they thought it was.”
Bringing His Big Idea to PRESEF
Judges for the Pittsburgh Regional Science & Engineering Fair agreed, awarding Isaac 4th place in the Computer Science Senior Division for Nomad Calculus. All of this, however, was nearly eclipsed by the reaction of a certain Science Department Chair: “Mr. Marx is generally unflappable, serious, and usually the first to offer criticism. But when he tried Nomad Calculus, he was jumping up and down, screaming about how amazing the whole experience was. I definitely attribute a lot of that to the Oculus technology and not to myself, but such vigorously positive feedback from Mr. Marx was pretty inspiring.”
Isaac, attends the University of Chicago, has enjoyed programming “for almost as long as I can remember,” and appreciates that “WT has let me embrace it during school hours and for credit. The process of programming is always engaging for me. I plan to follow both math and computer science as far as my intellectual curiosity will take me.”