The other day, Skanda and I headed to a TC Colloquium on Visualizing Mathematics
with Jonathan Rogness, a professor of Mathematics at the University of Minnesota, and the creator of the hugely popular YouTube video Moebius Transformations Revealed
. After the talk, we were able to ask Prof. Rogness a few questions regarding his experience with the use of visualization in math education.
One of our foremost curiosities centered around the use of visualization in lower-level mathematics. During the Colloquium, Rogness provided a number of examples illustrating how well-designed visualizations could make lofty mathematical concepts very reachable. However, can the same practices be applied to math lessons at even the primary school level? According to Rogness, academic literature on the use of visualization to teach math actually increases as you head down the educational hierarchy. Subjects like geometry and algebra are laden with opportunities for visualization, only the simplest of which are used by most educators today. More challenging, though, and even more widely ignored, is the use of visualization in higher level mathematics. Rogness attributes this to a simple cost-benefit analysis - developing something that can help teach 5th graders about math is more easily marketable than a product targeted towards only those students who have surpassed calculus-level math.
Of course, visualization, like any teaching tactic, is heavily dependent on class and concept. Rogness explained that he typically uses more visuals with those audiences that he expects to have less background in the subject matter (a somewhat humbling statement for us, given that he had just dedicated much of his presentation to showing off example visuals). In this way, however, Rogness explains that visualizations can be used as a sort of lingua franca in a classroom full of students with varied backgrounds. They have even been found to provide cultural bridges when designed properly. Still, assessment of these strategies is still somewhat lacking. Rogness specified one occasion in which he found a study with a sample size of 1. Concrete testing and data is more or less lacking, especially for higher level stuff.
Personally, after seeing the presentation, I felt as if the real crux of the visualization problem is figuring out how to get the visuals to help deepen the understanding of a concept, as opposed to just granting an intuitive or elementary knowledge of it. Recognizing this, Rogness explained that he typically uses images to complement, rather than convey knowledge. That is, visualizations function as a mode of conceptual, rather than numerical understanding. In my experience, visual java applets for physics teaching, sometimes called 'physlets', corroborate this claim. In high school, our physics instructor would introduce these after we'd gotten lost in all the numbers. They helped to bring things back to reality, letting us truly understand a concept rather than just being able to push numbers around correctly.
Ultimately, I think Rogness' work evidences the potential for a reorganization of mathematics education into a curriculum images, and interactive visualizations play a greater role. Further research is certainly required, but I think that carefully developed images tailored to the illustration of concepts could help to enrich a true understanding of mathematics, as opposed to just a proficiency in calculation.