Why?

(My partner in conversation below is a fictionalized amalgam of real people and of inner discourse. He is a cipher, into whom I've placed words as freely as he's spit them out.)

Ethelred: Why?

Me: Why not?

Ethelred: Why not? Because you can't possibly cover as much. I mean look: how many of your students are going to be able to state the Cauchy-Schwarz Inequality, let alone prove it? Now it's come and gone, and you've never said a word about it in class.

Me: How many of my students in a traditional Linear class would be able to state the Cauchy-Schwarz Inequality a week after it's "covered"? And how many of them will ever use it?

Ethelred: Okay, bad example. But what about the Triangle Inequality? How are they supposed to understand more general inner product spaces later on if you're not even making sure they get the tools they need to understand the most important one?

Me: They've understood that most important one all their lives, just have them try to draw a triangle where one side's longer than the other two put together. So what if they don't know "why" that must be?

Ethelred: Now you're being pedantic.

Me: Probably.

Ethelred: You have to admit that Friday's class didn't get as far as you'd hoped it would.

Me: Maybe. Maybe at first. Maybe at first I thought that we'd fallen a little bit behind.

Ethelred: Ah. Ha.

Me: At first.

Ethelred: And then?

Me: And then I thought, "you know, if we'd finished up that chemistry exercise, we'd have worked our way through the solution of a linear system by row-reduction of matrices. We'd have finished up Section 1.4. We'd have been ahead..."

Ethelred: So you're still thinking in terms of "coverage"?

Me: "Coverage" is a four-letter word.

Ethelred: But you don't deny that it's on your mind?

Me: I don't deny it. A decade of teaching more or less one way, even if that one way's got some bells and whistles on it every now and then, doesn't disappear overnight. But you interrupted me.

Ethelred: Sorry.

Me: "...and they would have pushed their way ahead on their own. By themselves. They would have worked out the application for themselves. I wouldn't have done a single example for them."

Ethelred: So what?

Me: So how would you have had me teach Sections 1.1 and 1.2? "Definition, theorem, proof, theorem, proof, incomprehensibly inapplicable theorem, unnecessarily dense proof, useless abstract example, definition, long and boring (however impressive-looking) list of properties...half of which everyone'll forget a week after the exam, all of which half of them will forget a week after the exam..."

Ethelred: You're being pedantic again. There's a lot to say for the traditional Linear Algebra classroom. Are you really doing your math majors a favor? They, at least, deserve to see worked-out proofs of the major theorems from linear algebra. They, at least, deserve to gain access to the more technical aspects of the material.

Me: They'll get that in the reading. Besides, what would you have me do, sacrifice the rest of the class for the sake of the math majors?

Ethelred: There are a good many of them in there.

Me: There are just as many, if not more, atmospheric scientists, and you can bet most of them could give a monkey's red rear end for whether or not the Cauchy-Schwarz Inequality holds in any given inner product space.

Ethelred: Whether they care about it or not, exposure to that fact is good for them. Damn it, Patrick, you've said it yourself all of these years: above all else, the best thing that a student can get out of a math class is a little exercise in critical thought.

Me: And who's to say I've changed my mind? Besides, the person who said that was a younger version of my self-as-teacher, an alpha model. An inchoate form, Patrick-of-five-years-ago. I've rested safely and soundly in my old teaching method for years now, and though it's worked for me well all those years, there was something missing.

Ethelred: What was missing? Patrick, you've always been a good teacher. Why change?

Me: I've always been good, but why not be better? And I'm not sure I've been as good a teacher as I should have been. "What's missing?" you ask. Fair question.

Ethelred: And?

Me: Substance. Structure. A coherent overall frame. A big picture.

Ethelred: Overrated.

Me: Really? You say I want to teach critical thought. Fair enough. What does that mean?

Ethelred: And?

Me: No, really, what does that mean, to teach critical thought? It's something I've taken for granted, something I've assumed myself capable of, something I've assumed I could recognize, that others could recognize, when they'd achieved it. It's something I've always thought inhered in mathematics, so fundamentally so that you couldn't get out of a math class without having exercised it at least somewhat. But that's not so: you can make it through Calc II without even so much as rubbing your elbows up against critical thought, though you both sat side-by-side in a crowded classroom the whole semester. The problem is that the students don't recognize critical thought. And I've grown so accustomed to thinking that I recognize it that half of the time I don't, either.

Ethelred: What, so you have to tell them that they've had a "spiritual experience" in a math class in order for them to actually have one?

Me: Not necessarily, but you may at least have to set them up for one. I'm not sure I was doing that before, however much group work I was doing, however many cupcakes I got them to cut up, however much fun and interactive my classes might have been.

Ethelred: And what's so grand about this Linear class?

Me: The other day Livonia was in my office, hangin' fire, chewing the fat. We were talking about the class, and all of a sudden she said something like, "I have a feeling this class is really going to help me with my upper level math classes. It's hard to do, but I'm going to need to know how to learn math on my own."

Ethelred: So?

Me: I pulled up the syllabus on-line and scrolled down to the learning goals: "be aware of your own learning styles, and of how to make effective use of those styles."

Ethelred: Hmm.

Me: Just a few minutes later, she was talking about her team's research proposal, and she mentioned how important understanding of the issues involved in their top choice would be later in her career: "that's the kind of thing I'm going to have to do, take a problem and take it apart to understand it, and then put it together again." I pointed back to the syllabus: "dissect a real-world problem in order either to analyze the way in which linear algebra can be applied in order to solve the problem, or to explain why linear algebra might not be applicable."

Ethelred: She's a bright kid.

Me: Oh, yeah. You can bet dollars to donuts she is. That's a roomful of smart people I've got in there. Don't think I'm not afraid of letting them down, of not helping them to find the best damned education they can get.

Ethelred: And your list of learning goals the way to do that?

Me: It's not just a list. It's not just a touchy-feely component of a lifeless document that becomes irrelevant as soon as it's printed. It's the shadow of something far more important, of an underlying structure, of an edifice that I built before the class even began. That list is a continual reminder to myself of everything I've got to do in the classroom before I head over there at 2:35 on Mondays, Wednesdays, and Fridays.

Ethelred: And what do your students have to do? Do you think that they're thinking about these things, too? Haven't they got enough to think about, with all the reading they've gotta do, all of the writing, all of the crap you've got them doing in the class itself?

Me: I don't know. I hope I've set it up in such a way that they don't have to think about the list, that every one of those fifteen goals is met without them knowing it. That's my intention, anyway, and whether or not I'm succeeding...well...we'll see how it turns out. Edward was asking me the other day about teaching one of the freshman colloquia: did I think I'd want to do that at some time soon, did I have any plans, any ideas for such courses? I said that yes, I had an idea that I'd been kicking around for a while that would make a perfect LSIC course, but that I wasn't ready to teach it yet. "It's probably a pretty good idea to get settled into the department before branching out like that," he said. Or something to that effect. That got me thinking: is that what I'm doing? I don't think that it is.

Ethelred: You're not getting settled?

Me: No, that's not it. I think I've done a fantastic job of getting settled into the department. I don't think that I'll ever be done doing that, of course. In fact, you're probably in trouble if you ever start to think that you're done getting settled. I mean that I'm not just sitting around in the math department, waiting for someone to give me permission to take a step outside Rhoades-Robinson: 365 isn't just another math class, and I have to say I took a little offense at the implication that that's all it is.

Ethelred: Is that what he meant to say?

Me: Maybe, maybe not. But that's the way I heard it. This course is so much more than that. You want a writing intensive course? You got it. You want an information intensive course? You got it. You want a course that provides a legitimate research experience? You got it. You want interdisciplinarianism? You got it. It's a more integrative course than just about any LSIC course I could ever dream up. This course is the perfect example of everything UNCA claims to be about.

Ethelred: Hmm. What are you thinking now?

Me: Now? Not much. It's late.

Ethelred: Get to bed. Tomorrow's here already. You need some sleep.

Me: I know.

Ethelred: Your students'll be there tomorrow. They're good kids. You've got 'em hooked now. They'll come back.

Me: I hope so.

Ethelred: I know so.

Me: We'll see. Thanks for talking this out. It's helped.

Ethelred: No problem. Good night.

Me: Good night.