Having the end in mind

So, I'm dusting off Linear Algebra ahead of the coming semester. I expect I'll need at least some of it for my course Bayesian Statistics. I'll definitely need it for the Qualifying Exam next fall. I don't want to say I'm struck by how easy it is, because I recognize it as a meaty subject. And, while it's rusty, this is a subject that I have mastered previously. However, I am surprised at how much more sense it makes the second time through.

My experience in just about all math courses is that, while I can work the problems and pass the tests, I don't really grasp them until I'm well into the following course. It was a month into Differential Equations when I really understood how derivatives and integrals from Calculus worked together and why integration was the more difficult task. A few weeks into Measure Theory, continuity and countably infinite subsets, concepts that were taught in Real Analysis, went from mysterious to obvious. I have a vivid memory from my last semester at Cornell of looking up from a proof I was working on for Extreme Value Theory and realizing that, while I had barely a clue of what I was doing or why, I suddenly knew exactly what a Poisson Point Process was (the subject of the prerequisite course).

Some of this is just the fact that this stuff takes a while to sink in. Math beyond Calculus gets increasingly non-intuitive and by grad school it's downright weird. However, I think it also has to do with the fact that it helps to know the end at the beginning.

By this I mean that when new concepts are presented without knowing how they will be used, it's more difficult to fit them into the framework of one's existing knowledge. Math instruction generally builds on itself; starting with definitions and axioms and then working through theorems so that each piece is supported by the work before. While this is logically correct, I'm not sure it's the best way to learn.

I'm willing to take a few items on faith at the beginning of a course, knowing we'll get to the proof later on. I think it would be helpful to put the later results out there right at the start simply so the class had some sense of where this is all heading. I think I'll try to do this the next time I teach a math class, or any class, for that matter. Spend a lecture up front just taking the class on the express train all the way through the material. Then, proceed with the normal building-block approach. I wouldn't expect the students to learn anything from that opening lecture other than where we are headed (and I'd make that clear, so they didn't freak out). Having a destination is no small thing, however, and I think that many would appreciate knowing that sooner rather than later.