Thursday, January 1, 2015

The Benefits of Confusion

This past September, The Chronicle of Higher Education published an article with which I strongly resonated, “Want to help students learn? Try confusing them.” [1]. It described an experiment in which two groups of students were each presented with a video of a physics lecture. The first lecture was straightforward, using simple animations and clear explanations. The second involved a tutor and a student in which the student struggled to understand the concepts and the tutor provided leading questions but no answers. Coming out of the videos, students found the first to be clear and easy to understand, the second very confusing. Yet when later tested on this physics lesson, students who had seen the second video demonstrated far more learning than those who had seen the first.

This illustrates the problem with so much of standard instruction, especially in undergraduate mathematics. The ideas have become so polished over decades if not centuries, and we who teach this material understand its nuances so thoroughly, that what we present glides easily past our students without opportunity to grasp its true complexities. For learning to take place, students must engage and wrestle with the concepts we want them to understand.

I am not advocating confusion for confusion’s sake. As Courtney Gibbon’s cartoon illustrates, a polished lecture can also be very confusing, and not in a good way. Confusion is most productive when it provides a focus for personal investigation. An example of positive confusion is the cognitive dissonance produced when student expectations confront convincing evidence that they are wrong. My prime example of this is George Pólya’s Let Us Teach Guessing (see my Launchings column Pólya's Art of Guessing).

I like to think of this as “gritty” mathematics rather than confusing mathematics. One of my favorite examples from personal experience was a Topics in Real Analysis course that I taught in Spring 1997 using Thomas Hawkins’ doctoral dissertation, Lebesgue’s Theory of Integration: Its Origins and Development, as the text. My experience teaching that course laid the foundation for my textbook A Radical Approach to Lebesgue’s Theory of Integration. Back in 1998, I wrote a paper about this experience, “True Grit in Real Analysis.” I never published it, but I still like it, and as a New Year’s gift to readers, I offer a link to that paper.

[1] Kolowich, S. Confuse Students to Help Them Learn. The Chronicle of Higher Education. September 5, 2014. Available at