Making the Most of a Mathematical Moment: “Möbius Transformations Revealed”
“It was a perfect storm,” says Doug Arnold, professor in the School of Mathematics and director of the Institute for Mathematics and its Applications, about the phenomenon of the video “Möbius Transformations Revealed,” which he and colleague Jonathan Rogness created and posted on YouTube just a couple of months ago. His collaboration with Rogness, who is an assistant professor of mathematics and associate director of the Institute of Technology Center for Educational Programs, began after he attended a Rogness lecture on graphics and teaching. After the lecture, the two talked and decided to create a video for the 2007 National Science Foundation’s Science and Engineering Visualization Challenge.
Motivated by the deadline and the challenge of creating an aesthetically pleasing scientific visualization, the two posted the video on YouTube as a convenient way of making it accessible to a few friends and colleagues. The video won honorable mention in the competition, but neither anticipated just how popular it would become. Once posted on YouTube, the video went viral. As of this writing, the video on YouTube has received more than 1.3 million hits; approximately 4,350 comments; and has been favorited about 10,750 times. National Geographic, Boing Boing, countless blogs, and many newspapers picked up on it as well. And as Arnold and Rogness observe, this extraordinarily popular video can come to mean different things for different audiences.
Many YouTube viewers have given rave reviews of this video, saying it’s “superb,” “brilliant,” “elegant.” While two busy mathematics professors don’t have the time to read each and every one of the comments, Rogness and Arnold noted a range of responses. Some say they wish they could have seen the video in math classes past, and instructors report they have used it in class because of its appeal and help in clarifying concepts. Others have become involved in discussions about the implications of the concept and have considered ways to make this visual explanation even more compelling and clear. And as Rogness and Arnold note with amusement, the video has attracted its share of colorful commentary as well. In the end, Arnold says that for him it is perfectly fine if people consider the video an “interesting and pleasant way to spend a short amount of time.” As he points out, “the media tend to shy away from anything with intellectual content. The response to this video suggests that people do like a little intellectual stimulation.” Both he and Rogness believe the video has the potential to create for the viewer a “mathematical moment,” or an appreciation for the sheer beauty of mathematics. At that level, the video works nicely as a motivational tool.
In the classroom, the video takes on a different function and both assert that the key to its educational value is careful contextualization. While the video is compelling because of the visual images and the soundtrack, as Arnold points out “the point is not the pretty picture. A good mathematician moves back and forth between the geometric picture and the algebraic.” Arnold further explains that the video—or any visualization—should not be at the center of the lecture, but should play more of a supporting role. In other words, the presentation of any visualization should lead to further exploration and discussion and contribute to greater understanding of the more rich and nuanced language of mathematics.
While Arnold has been interested in visualizations in mathematics since the 70s, Rogness notes that the use of visual representations—and indeed, technology generally—is not without controversy in mathematical circles. He notes there is a “constant tension in math about the use of technology.” Many feared that the introduction of calculators, in particular graphing calculators, meant that students would not develop some necessary skills. Similarly, Rogness explains, some mathematicians fear that students would use visualizations as a crutch. He agrees with Arnold that visualizations should play a small but significant role in the mathematics classroom. Moreover, both believe that mathematical visualizations have a particular role in the curriculum. For example, the “Möbius Transformations Revealed” video may pique the interest of the YouTube viewer. For students in the beginning of their education, visualizations provide a little extra help grasping new concepts. Students who progress from novice to expert rely on them less as they develop their skills and knowledge. And perhaps a smaller number of those students who become mathematicians will someday create “superb,” “brilliant,” and “elegant” visualizations of their own.
