No regrets

I've said a lot lately about the way Calc I has been going this term, and I've said relatively little about Linear, perhaps because I feel that course has felt fewer obstacles along the way so far. I honestly feel that Linear has been going more smoothly than just about any course I've ever taught. (Fall 2006 Calc II and Fall 2009 Foundations are possible exceptions.) And I'm having a blast in it.

What's made it work so well? The high quality of the students, their outgoing nature, their friendliness, their willingness (nay, eagerness) to work together both in and outside of class...and, I'll own up to it, the course plan I've laid out is working very well.

I'm never planning too far ahead in that course. Rather, I'm responding to the way the students handle each new activity I give for them. If they need more time, we slow down; if they're bored, we speed up. More importantly, perhaps, no activity follows another without a reason for doing so. We introduced inverses because we needed them to solve a particular problem, and we introduced the determinant of a 2 x 2 matrix for the same reason. We defined matrix multiplication the way we did because it made sense to do so, not because the textbook told us to.

Moreover, I've avoided technicalities where I feel those technicalities tend to swamp out understanding and intuition. For instance, without knowing it, per se, the students have now worked with bases, matrix linearity and singularity, and Markov processes, generally without explicit mention of those terms. They don't yet know what a vector space is, nor a linear transformation, yet they do know how to apply the techniques of linear algebra to solve nontrivial problems in graph theory and geometry, and they have robust intuitive understanding of those problems, as well as the nature of linear equations and their solutions. I remain convinced that now, as we're finally getting around to proving conditions for singularity of a matrix (still without using that term), the students' understanding of those conditions is so much deeper than would be the understanding of a typical student by this point in the semester.

I do not regret the emphases I've chosen to give in this class. I hate to brag, but I've got to say that though we've not "covered" a number of the terms and techniques (for everyone's sake, do not focus your attention on the mechanics of row-reduction and matrix inversion for two or three weeks, people!), I'd bet that the students have a much richer understanding of linear algebra than would students in most Linear courses by this point in the term, and I'd also be willing to bet that that understanding will last, too, and not disappear immediately after this semester's over.

Any takers?