Visualizing and learning mathematics in virtual reality

[This first-person story from San Jose State University provides another example of how even simple, abstract immersive virtual environments can evoke spatial presence and increase learning. The original version of the story includes a second image, and for much more information on the NSF-funded project (including the VR modules), follow the link at the end below. –Matthew]

[Image: An image from the virtual reality classroom featuring three vectors. Credit: Photo courtesy of Ferdie Rivera]

Math Adventures in Virtual Reality

by Cassie Myers
June 6, 2024

I haven’t taken a math class since 2009 and I’m about as frightened of virtual reality as anyone living in Silicon Valley can be. So naturally I jumped at the chance to test out a virtual reality headset that helps students learn linear algebra (no sarcasm — I genuinely did).

It was all part of a demonstration led by Ferdie Rivera, professor of math education and director of the Connie L. Lurie College of Education’s EdD leadership program, who received a grant to use VR to help students understand the basic concepts of linear algebra, a lower-division math class SJSU students typically take in their first two years.

A virtual world

It was my first time putting on the Quest VR headset and I expected it to be unnerving. But it was surprisingly comfortable and fairly easy to use, like learning how to operate a friend’s Nintendo Wii. I found myself at first in a comfortable blank, lobby-looking space filled with pleasing wordless music. Entering one door off the lobby (entitled “vector modification”) took me into a white room with a three-dimensional graph (complete with x, y and z axes).

It felt like falling into the world of a graphing calculator — in a good way. There were my vectors: three lines with arrows that extended into space in three bold colors: blue, green, and a purplish pink. Rivera proved to be an excellent, patient guide and teacher. He walked me through the basic steps of the controls, teaching me how to adjust and rotate the three-dimensional space so that I could see the vectors from various angles.

He also taught me how to adjust the magnitude of the vectors so they were short or long, how to hide one to better visualize the other, and how to connect two vectors to make a plane, a flat surface that extends through all three axes. Then he helped me build a sphere, a colorful, transparent globe built out of the planes and vectors and how they intersected.

Visualizing math

If it’s difficult to visualize what I saw based on my descriptions, that illustrates a key point of Rivera’s entire idea (and what made me put the VR headset on in the first place). Three-dimensional mathematical concepts — like the sums of vectors, the way a plane created by two vectors operates, and more — are hard to draw in two dimensions and even harder for students to extrapolate into three dimensions without being able to physically see them in some way. And yet it’s very difficult for a student to learn (or pass, or understand) linear algebra without them.

Even for a clueless mathematician like myself, the VR walkthrough with Rivera was helpful. Seeing the two vectors extending into space and how they became a colorful plane when connected helped me understand (even in 20 minutes) two basic linear algebra concepts: the fact that a vector is essentially a line in one dimension that continues infinitely, and that a two-dimensional object is a plane where two vectors intersect.

And now that I’ve experienced these concepts visually, I’ll have a much better base-level understanding of vectors (like any other student in the theoretical course) that will help me with all the other linear algebra concepts.

It also helps that the VR environment is pretty, stark and soothing. It doesn’t feel like a classroom; if anything, it feels like a game. Even my English-major fingers were itching to pull the VR trigger and play with those vectors.

As it turns out, what I was doing (and feeling) was what’s called “embodied cognition,” which sounds appropriately sci-fi for the VR setting, but also has roots in educational research. It means that when you see and feel something with your own body, you learn it more quickly and more deeply.

“It’s like ‘my actions determine the knowledge that I generate from the space itself,’” explains Rivera. “So the idea is that if students are immersed in VR, they’ll be able to understand the geometric relationships and then convey what they see more precisely in mathematical form.”

The beginning of the grant

This VR project is a result of Rivera’s collaboration with his co-PIs Plamen Koev, professor of mathematics, Yingjie Liu, lead instructional developer for eCampus, and Patrick Stafford, a computer programmer who helped create the VR software. Together they’ve developed several teaching modules to help students understand linear algebra in the three-dimensional VR space — something they quickly discovered was necessary when Rivera’s assessment of end-of-semester grades from SJSU from 2015-2022 revealed that at least 20% of students found the course difficult and challenging to learn.

“As math professors, we’re never taught how to teach,” Koev explains. “All our education on how to teach was exactly half an hour.” So he teamed up with Rivera, who works in both education and mathematics, to tap into the wealth of educational scholarship that explores how to teach mathematical concepts to students.

Eventually, this collaboration led to their grant: $400,000 from the National Science Foundation (NSF) to create, test and pilot this VR math education program. All their deliverables are free: they have a 2-D desktop version of the course for those who can’t afford VR headsets, and they hope to make headsets readily available for those who need them.

The work began in earnest in fall 2023, so it’s in its early days. So far only twelve students have tested the VR modules, and it may be a year or two until the course is refined enough to bring into full classrooms, with teachers and students existing together in a virtual space. But early results are promising.

“The tools can help people visualize the concepts of linear algebra in a 3D space to the point where it makes it easier to understand,” said one student tester. “Being able to grab the graph and manipulate it around helps me see more than just the frontal view of the vectors and planes to better understand where everything is really located.”

“These are concepts that I will not forget in the future,” said another.

As the project and the class expands, it could lend itself to other concepts in other mathematical fields. But for now, Rivera, Koev and their collaborators are all focused on what’s right in front of them: a chance to use the technology of the future to help students understand and visualize linear algebra.

They’ve made one convert in this math-phobic writer. It’s hard not to share their enthusiasm and interest when their descriptions of their work sound so poetic. “In linear algebra, you have to imagine things,” Koev explains. “Because the nature of it is that after three dimensions, nobody knows what’s really happening there. You have to sort of open your mind, believe that it works and it does.”

In a way, then, it’s all about bringing education and technology together: The Matrix meets Reading Rainbow.

And if all else fails, a cool VR headset doesn’t hurt.

Check out the VR team’s website here.


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