Mathematics as Literature

Three students sitting at a table; female student looks toward speaker at front of room; male student leans forward, holds pen, and looks toward speaker; another male student looks down at paper.

Students listen as their colleague presents a lecture on rational homotopy.

By Joe Pickett, OCW Publication Director

Normally when you think of a literature seminar, you think of a bunch of students sitting around a table talking about Kafka or George Eliot.

OCW has just published a literature seminar of a different stripe—in mathematics. The course is 18.915 Graduate Topology Seminar: Kan Seminar. It is named after its founder, Daniel Kan, an MIT Professor known for his unconventional, conversational style of teaching. Lectures simply did not suit him. He developed an alternative style of instruction more reminiscent of the humanities than the traditional chalkboard session in mathematics. Professor Kan passed away in 2013, but his legacy has been carried on at MIT, most recently by Professor Haynes Miller, who led the published version.

The course serves as a sequel to a first year graduate course in algebraic topology. It introduces graduate students and occasional undergraduates to a broad range of more advanced algebraic topology by requiring them to read some of the classic papers in the field. The idea is “to push students through the transition from someone who takes courses to someone who thinks more actively about mathematics,” as 18.915 alumnus (and Kan PhD student) Philip Hirschhorn puts it.

It’s a process: Students give practice talks to their classmates (no instructor present). These are typically “fractious,” full of interruptions, questions, and criticism. After that, it’s on to formal presentations, with the instructor attending and attentive. In his office after delivery, Professor Miller then “debriefs” students individually on the effectiveness of their presentations and on the mathematics involved.

According to Professor Miller, a second course objective is to help students learn “to scan an article quickly, to glean the essential points and relate them to the rest of their evolving intellectual infrastructures, and to express this understanding.” So seminar participants must write “reading responses” to all of the papers they are not presenting. Professor Miller responds to the responses. The course site has a full list of readings and some sample student reading responses, along with Professor Miller’s replies.

There must be something contagious about this approach to teaching mathematics, because the OCW site has links to Kan seminars taught at other institutions. As Professor Miller sees it, instructors shouldn’t be put off by the list of long and difficult papers that have been used in the course. “You’re not there to teach the material to the students; you’re there to help students learn the material.”

18.915 is the second project-based course by Professor Miller published as part of OCW’s Educator initiative. The first was 18.821 Project Laboratory in Mathematics, in which students acquire a taste of what it’s like to do mathematical research.

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