What's a guy to do? (A paean for laziness)

Halfway through our winter break I paused in my course preparations, panicking that I'd spent nearly all of my time planning my HON 479 course and had done nearly nothing to prep for Linear Algebra II. Though I had a rough framework for the course's structure (semiregular homework assignments, a few take-home exams, student-led projects, presentations, and discussions), I had almost no idea what content I would include in the course.

After a few moments (okay, maybe a few hours), the panic passed. I realized the futility of overplanning, a futility reconfirmed by the survey of my Linear II students' background I performed on Monday. The 23 students in that class come to me having taken Linear Algebra I from no fewer than five different faculty members in my department, as long ago as two and a half years back. These faculty include me and one of my colleagues who shares my penchant for student-centered, application-based teaching, a couple folks who typically offer a blend of applications and theory (one with a much more student-centered approach than the other), and a fifth who focuses exclusively on abstraction and theory and whose teaching style can only be described as "traditional." Needless to say, my 23 students come to me with extremely diverse linear algebraic backgrounds. It's unlikely that, beyond a few basic principles (row reduction, linear (in)dependence, bases, determinants, eigenvalues and -vectors, etc.) they all will have studied, they'll have any content knowledge in common. In the end there's really very little I can do to accommodate them all: no matter what static plan for the course that I could come up with, it would no doubt lose some and bore most of the others.

This realization was liberating. Instead of putting forth a particular course of study, I could let the students take the lead, offering them the chance to investigate topics in which they are interested, sharing their investigations with each other in the form of in-class presentations, discussions, and problem sets. I'm going to ask every student to take a turn, working with one or two of her or his peers, leading the class in the study of a topic of her or his choosing. For those who might not know what direction they'd like to head in, I made a list of potential topics, many of which likely made an appearance in some students' first-semester Linear I courses:

• orthonormalization methods
• orthogonal systems of polynomials (e.g., Chebyshev polynomials, Hermite polynomials, and Legendre polynomials)
• Gröbner bases
• LU factorization
• abstract vector spaces and modules
• network flow analysis
• unitary and Hermitian matrices and their applications
• finite element methods (e.g., in atmospheric science)
• Google's PageRank algorithm
• the basics of functional analysis
• linear codes and linear cryptography
• applications to differential equations
• linear programming (e.g., the simplex method)

To help everyone get to the point where we can approach some of these topics, I'm spending the first week or two on review, wherein the students are taking turns, in groups of three or four, presenting on the various "basic principles" I listed above. It's going well so far. "Is this useful at all?" I asked after a couple of presentations this morning. "Should we keep doing this?" There was almost unanimous agreement that yes, we should. So we'll keep it up.

How'll it go? Who knows? Not me. I'm excited to find out, though.