Thirteen hours

A rare conjunction of coursework came this weekend, and I found myself with four substantial assignments to grade over the last 48 hours. On Friday Calc I students submitted their sixth set of homework problems and their first team projects; 280 students turned in their fourth homework sets and their first take-home exams.

It took thirteen hours to grade it all: a couple of hours on Friday night, roughly eight hours yester(Satur)day, and another three this morning.

Highlights?

My first section of Calc I clocked in with their highest homework percentage yet.

The team project write-ups were uniformly good, with two or three exceptions, including one project that was so stellarly composed I felt it warranted a few extra points on top of the full measure. (This team was the only one of sixteen that had me look over a rough draft of their project before revising it to create the final version. I have two words for this team's leadership: "you," and "rock.")

The 280 exams were hit 'n' miss, the hits coming more frequently on the more difficult questions people spent the most time on ("Prove that every year has a Friday the 13th" and "obtain and prove carefully by induction a formula for the nth derivative of sin(x)"), the misses coming through careless errors in translating English propositions into quantifier notation. I'm hoping that with the opportunity to revise, students will be able to put a high shine on these rough drafts.

The 280 homework on induction was quite good, I think people are understanding the mechanics of an inductive proof pretty well. I've made a mental note, however, to spend a little bit of time tomorrow reminding them of the direction arrows must point when "reducing" the desired proposition to an "obvious" one. This tricky point was the number one cause of errors in the HW set.

The 280 folks are writing well. I'm much more conscious of the writing going on in the course this semester than I was last Spring, and I sense that the students are already stronger (or at least more self-conscious and self-aware) mathematical writers now than last year's students were by the semester's end.

Tomorrow?

Tomorrow, two of my departmental colleagues will be sitting in on my Foundations class in order to gather material to write letters on my behalf for the my reappointment file and for the teaching award for which I've been nominated. Woo hoo. I must say I'm sorry that it's nothing tremendously exciting we'll be discussing tomorrow, just some basic combinatorics (the Pigeonhole and Addition Principles, most likely). Too bad the committee reports won't come until Wednesday.

Now?

Now, I'm decompressing.

Meh.

I think I've forgotten how to relax.

Random thoughts, volume 2

I had a few interesting conversations today, with colleagues and with students. I also found myself unfairly piqued during my second section of Calc I, and I feel an apology is in order to my students.

Let's start there: Quiz 4 came today, asking for a brief rundown on the two fundamental interpretations of the derivative. As I would soon forcefully point out to my students (post-quiz), it's not as important to me that they memorize the formula for the derivative, nor that they master every one of the rules for differentiation we will soon study, as it is that they understand what derivatives mean, and how it is that we can see them in nature, and put them together to help us understand natural phenomena. As I put it to them, Mathematica can do all the derivatives for us, and much more quickly than we ever could. What Mathematica can't do is study a natural process, recognize that there is an interrelationship between two or more dynamic quantities, chart those interactions over a long enough course of time to posit a model that describes the way they depend on one another, and use that model to put together the derivative that gets fed into Mathematica at the end of the line. Mathematica, in this sense, represents the mathematics of the past, when it wasn't yet the case that there was a handy formula that one could apply to find the derivative of a given function. That was then. This is now, and the future is yet to come. The mathematician of the future needs to know more than a mechanical rules for finding derivatives (as important as they are to be able to apply well); she needs to know how to use derivatives, how to interpret them.

Of course, when I said all this to them, it came out ne'er so fluently as it did just now, above. Figures, huh?

Performance on the quiz was...meh. It wasn't horrific (I've given harder quizzes), but it was by no means stellar. A couple of my best students cornered me after the second section and asked if they were going to be okay from this point on. Tallulah: "because I didn't do well on that quiz." "I doubt anyone did," I told her. "The folks in the first section pretty much biffed it. It was a hard quiz, largely because you're not used to being asked questions about concepts rather than computations." She's a fantastic student, she'll recover splendidly.

I realized even as it was happening that I was (unfairly) letting my frustration with my students' conceptual misunderstandings get the best of me for a few minutes during that second section's class. I threw markers and punched the board like I always do, but I did so with more vigor than is typical, partly to dissipate my frustration. "How dare they not get this? Damn it, what, they think they can coast in this class if they spit up a formula or two?!"

My righteous indignation subsided as the class went on, and I realized by the end that if they'd not focused on the concepts over the calculations, it was as much my fault, and my colleagues' faults, as it was the students', for not asking them to refocus their attention elsewhere in the first place. I've got to try harder at that myself. For some reason, I have to admit, Calc I has proven the most resistant of all courses to redesign along the lines of discovery and application-based learning. Only now am I beginning to understand what a truly problem-based Calc I class might look like, and I admit that this semester I'm falling far short of that mark.

I promise a less angry, less frustrated tone tomorrow, folks: you really are a great bunch of students, and I enjoy working with you very much. Let's make tomorrow's class a good one, huh? I'll bring some donuts tomorrow morning, and we'll start off with a couple of conceptual exercises to get our creative juices flowing. Sound like a plan?

Good.

From the second section of Calc I, it was off across the quad, to the second of the semester's Writing Intensive meetings (the first was this past Monday) for me. As I anticipated, I'm enjoying working on this committee, conferencing with a group of peers who feel as strongly and as passionately as I do about writing-to-learn and writing-across-the-curriculum and writing in general. I'm starting to get a good sense for the way writing is integrated academically, campus-wide, rather than simply in my own courses and in those of my math colleagues. The bar is quite high; the quality of writing instruction university-wide is solid. Nevertheless, there is room for improvement, and I found myself in a heated exchange of hallelujahs with Lexington, the WI committee's acting chair, as we walked back to our shared building after the committee meeting.

We agreed that the university has made tremendous strides forward in terms of embracing writing-across-the-curriculum, undergraduate research initiatives, outcome-based curricula, discovery learning opportunities, and so forth...but that there's also a lot of work to be done before perfection is reached. "If we're going to advertise that we're using discovery learning," Lexington said, "we've got to start doing just that, and to do that we're going to have to get serious about giving people the resources they need to do that." We agreed that we need to try to drive class sizes down (I mentioned my conversation with my own Chair last week regarding getting my Calc I classes capped at a lower level), we need to offer kids the opportunity to engage in alternative classrooms early and often, we need to make a focussed, directed, campus-wide effort to provide these opportunities to students from the get-go.

After a brief stop at my office and a moment in the Math Lab to unstick the stuckness of a few of the Calc I kids in computing the derivative required of them in the team project, I was off across the quad again to 280. Davina caught me before class with a few concerns about her service on one of this week's homework committees. For one, she wasn't sure about what to write on a person's submission if he said something like, "I'm stupid, I can't figure this out," or something along those lines. "That's a hard one," I agreed. More substantially, she wasn't sure she was giving the right kind of feedback, and she felt like she was being hypercritical, telling people to reword this, change that, and so on. I suggested that she might try to balance positive and negative feedback, and to offer comments like, "I'm having trouble understanding this, could you make this more clear?" or "This is a really good insight, it really helped me to see this point more clearly!" I later reiterated some of these ideas to the whole class, and wrote on the board: "Recall that the purpose of the committee work is not to homogenize, but rather to help people to clarify their own individual ideas."

The subsequent committee reports were good ones. I was particularly impressed with Davina's discussion during the presentation she and DeWayne gave on the homework problem they'd been assigned. I admired the way she was willing and able to come out and say that her serving on the committee definitely helped her to better understand the concepts involved in the problem they'd reviewed: "seeing how other people did it made me see how I could make my own writing more clear and more concise." I'm glad she came out and said that, and I hope her sentiment is shared by the others. I'm certainly going to ask the students about their committee experiences explicitly when I pass out midsemester evals in a week or so.

Came then (after another half hour of set theory) the trek back to Robinson Hall, where I'd spend a few more hours before heading home. I finished grading the second section's Quiz 4s, on which they did marginally better than the first but still not wonderfully. I also got a chance to work with a number of the teams as they struggled through their projects (they're all doing quite well, from what I can tell), and I met up, one-on-one, with several of the 280 students, helping them to polish various drafts of homework problems. They're definitely developing an appreciation for more and more subtle nuances, meanings of stereotypical mathematical phrases ("thus...," "for every...," and so forth), and clarity, clarity, clarity in writing. (A funny, and very heartening note, if I may: at the close of today's committee reports, I reminded the students to keep an eye on the rubric I'd handed out last week as they worked on their math writing, and I asked them if they could remember the "four Cs." In nearly complete unison, they intoned: "correctness, completeness, clarity, and composition." I didn't have to say a goldarned thing. I was a happy man.)

While I was finishing off those Quiz 4s, Cuthbert, Chemistry colleague of Lexington and a big, big man in undergraduate research, came by to ask me if I wouldn't mind providing him with the titles of the projects the REU kiddies worked on this past summer: he was putting together material to take to a regional conference on UG research, indicators of the sorts of things that can be made to happen in a liberal arts science curriculum. I was happy to oblige, and not an hour later I'd send him a list of the topics. We then got to talking about the inclusion of research components in UG courses, and he asked if I'd done much of that. "Depending on how you defined research," I told him, "I do that in nearly every class I teach." I told him a bit about the course that spawned this blog (MATH 365, Linear Algebra I, taught last Fall semester), and he asked if he could have information about the projects those students worked on. I obliged him there, too, sending him the prompts for a few of the research projects (Monopoly, Traffic Patterns, and Come on, Feel the Noise).

We hit a few more points in our short conversation, enough to give me a sense of déjà vu, feeling as though I was reliving the conversation I'd had earlier with Lexington. We talked about the increasingly interdisciplinary nature of scientific study; the need for broader, deeper, more meaningful implementation of discovery learning in our courses; the need for more robust interdisciplinary course offerings than an occasional and disingenuous cross-listing: we need team-taught classes, classes that provide a true interface between one field and another, like one Cuthbert mentioned that took place at his previous institution, in which beginning chemistry students learned their ways around a lab while generating real, unprettified data that could be fed to statistics students who would put it through realistic rigorous analysis, the results of which analysis could be funneled back to the students who ran the lab experiments in the first place in order to help refine their techniques. We agreed that a course that combined the concepts of physical chemistry with those of linear algebra would be a tremendous boon to both the Chem and Math departments; I have no doubt that we'll be talking about this again soon.

Thence, from my office, to dinner at Sorrento's (offering the best Italian in town in a cozy out-of-the-way den halfway to Oteen that sadly never seems very busy), and then home. Once home, I finally had a chance to call Bedelia, my colleague late of Harvard, now of Lesley University, and catch up with her. As it was bound to be, our conversation turned to...teaching! Imagine that! We talked for a bit about a program she's like to get off the ground, a sort of summer prep course to get local (in her case, Bostonian) college-bound kids up to speed on the math skills they might need to succeed in a challenging entry-level math course, while providing them with some useful advice on making the transition to college more smoothly. I compared the program she was describing with UNCA's own SOAR program, and she seemed to agree with the comparison, only she emphasized the local nature of the plan she envisioned. Then we talked shop for a bit, I about my own six courses (if one counts the senior seminar and my three independent study students), she about her Quantitative Reasoning and Pre-Calc classes.

We're both having a good time.

Hey, I've rambled on long enough, it's time to stop. I'm going to make a smoothie, maybe watch an episode of Mr Bean or some other such nonsense, and call it a day. Thanks for reading, I hope you'll free to comment, I always appreciate your feedback!

Random thoughts

No time to get much coherent down, but before it slips my mind...

1. Faculty Learning Circles are where ideas come to life. Before today's meeting I spent a good deal of time thinking about what self-authorship would look like in mathematics students: how could it be assessed? In what manner would a self-authored math student behave? Are there warning signs? Once one knows what to look for, how can one go about designing the appropriate activities to facilitate and promote self-authorship? I raised these questions with the small group that convened this afternoon, and it was decided that it might not be a bad idea to start thinking about a conference on Self Authorship Within and Across Disciplines. (No good job goes unpunished!)

2. A few minor homework committee woes creep in: after it came to my attention that a few people had felt steamrolled by forceful personalities, I felt it necessary to send an e-mail to the 280 folks reminding them that there is almost always more than one correct proof to any given proposition. When serving on a committee, this must be kept in mind so that one doesn't turn a blind eye to alternative correct proofs one isn't expecting; when receiving feedback from a committee, this must be kept in mind so that one doesn't feel obligated to thoughtlessly undertake a committee's suggestions: if you're pretty sure your proof is right, perhaps the committee misread your argument, or misunderstood your intentions. Stand by your proof, and take it up with one of the folks on the committee. They're human, too, and every one of us is capable of error. (God knows I've demonstrated that over and over and over and over and over and over and over and over and over...)

3. ...I could have sworn I had a 3. Never mind.

Everything else is groovy. The Calc I folks are off and running with their team projects on specific heat, those are taking shape before my eyes (love those Mathematica graphs, huh?). In Foundations it's sets, sets, sets, and we're getting ready for Round Three of the newly-rechristened WNC (to includ Western Carolina University and Warren Wilson College as well) Mathematics Problems Group, tomorrow evening at 5:30. Pizza 'n' Putnam, what better combination?

This week in college mathematics

Where to begin?

The week got off to a rough start with an uncharacteristically stern lecture on my part to my Calc I students. (Musta worked: their homework for this past week, graded this morning, was far more complete and correct. Well done, y'all!) They came back that night for a pleasant review session in preparation for a relatively tough test I'd dish out to them on Thursday. I always enjoy review sessions: the students are awake, receptive, responsive. I wish students would bring the same eagerness and vigor to class as they do to those evening sessions. They finished off the exam with an overall class average of 75.9%; one student nailed it with a clean 100%, and there were several other As. With revisions (I told them to think of the in-class exam as a first draft), they've got the chance to bring the class average up to 84%, and I promised a 3% cherry on top of that if they manage to make it within a couple percentage points of that goal.

I spent a good deal of time (most of it while running) thinking about how I'm going to put Calc I together the next time it falls to me to teach it. I've already created almost all of the resources I'd need to make the course decidedly more student-centered: the plan would be to pare down and tweak the existing class notes to the point where they could be used as effective worksheets for the students to complete outside of class and present to one another in class, much as I'm currently doing with 280. I'd break further away from the text than I have already by eliminating all textbook homework problems (they'd instead be "recommended" as practice problems, alongside illustrative examples from the relevant sections of the text) and replacing them with problem-centered applications worksheets (similar to the first team project I've just passed out to my current classes) which would be handed out and collected on a weekly basis. Completing these worksheets would require students to master all of the concepts discussed in class during the previous week, and would force the students to integrate content with application and to produce realistic technical writing. Exams and quizzes (both individual and team) would continue as at present. Classes would be focussed on student presentations and student-led discussions. With the exception of a handful of appropriate weekly projects, I've got most of the materials made up, the transition wouldn't be too hard for me. I'm ready; it all comes down to one issue.

Class size.

To realistically expect freshpeople to speak up and participate in class to the extent that this course scheme would require, I'd need to establish a relatively small and tight-knit community of learners; this semester has reminded me just how difficult that task is when working with a class of 30 students.

Further bulletins as events warrant.

And then there's 280, with another round of committee reports. They did a bang-up job on Wednesday, raising a number of crucial issues, including appropriate choice of notation, simple vs. short in the context of proofs, writing for a given audience, and the fact that there may be more than one way to skin a mathematical cat. So far I've been impressed with how well the committees have appeared to work. Beyond the great conversations we've had in class, I've no doubt that homework has been made immeasurably stronger as a result of input from the committee members. That's my take on things, and I encourage any of my students to give me their side of the story: how are things going on behind the scenes, folks?

All in all, I'd characterize this semester's 280 class as a friendlier place to work than last semester's was. Not only have the students had little problem in communicating with each other, they've shown willingness to speak up and let me know when I'm full of it, too. This past Wednesday a handful of them objected, quite openly and strenuously, to the way in which I'd worded one of the examples in a worksheet, and sure enough, a subtle semantic oversight I'd made in designing the sheet last Spring came to the fore, and I was forced to change it before we reconvened on Friday. Bravo!

We'll be continuing with set theory for the next couple of meetings, and by the end of the week should be making our way into the realm of relations. I'm eager to see how they handle the first take-home exam, due this coming Wednesday.

Finally, I ought to mention that the first meeting of this semester's Learning Circle, on self-authorship, went down this Monday. I have a good feeling about this group, we had a great discussion concerning the basic idea of self-authorship, and how it fits into our various philosophies. In particular, I mentioned that I appreciate (among other effects) the way in which the concept of self-authorship effectively displaces "content ownership"; my colleague Thibault from the Drama Department concurred and added that he's happy to say farewell to the term "development," a word that simply connotes passivity, as though students just happen to turn into learners, magically, mysteriously. (One of the contributors to Meszaros's volume addresses this mistaken view of development.) I'll likely have more to say on these issues as we continue to meet.

Well, it's after midnight, the Badgers have just beaten back the Hawkeyes (on, Wisconsin!), and it's time for me to head for bed. Until next time...

Bleh.

Everyone's got a limit.

I'm not sure what it was that got me going this morning just before the start of my first Calc I section, especially since by and large the students are turning in the homework regularly, and, by and large, they're doing pretty well on it. Despite this, I felt the need to take a few minutes and "lecture" on efficient time management.

What set me off? It was probably one of the students' willful misunderstanding of the point of the Conundrum Conversations exercise ("It's explained quite clearly on the assignment sheet." "I lost the assignment sheet." "The assignment sheet, like everything else for the course, is downloadable from a meticulously well-maintained website."). That, and the excessive dilly-dallying I'm getting from a couple of the students as they hold off on installing Mathematica 6.0 (a several-hundred-dollar value, available to them for free) on their personal computers, thus taking the installation disks out of circulation to their classmates.

The worst part is that the kids in question are so obviously intelligent. I just want to take 'em by the shoulders and shake 'em awake, screaming at them, "Moses on a pogo stick, people! Have some self-respect! Take some pride in the work that you do!" Oy.

"How long do you think I spent in grading y'all's homework this weekend?" I asked. A couple of people proffered somewhat accurate guesses, in the 5-10 hour range; the true figure was about 6.5 hours. "At 15 minutes a problem [a copiously generous allotment], the 11 problems I assigned last week shouldn't have taken more than about three hours. Maybe four hours, if you're having particular difficulty with these concepts, which is perfectly understandable. In any case, I wouldn't think you'd need more than four hours to complete the homework. Spread over seven days, and you're not even talking half an hour a day. If I can devote over six hours to your assignment, do you suppose you can devote three?"

To my students from Section 1: I'm sorry for lecturing at y'all, but I feel it was overdue. Please remember that I love you guys as a class, and that I'm happy with your homework, for the most part. Most of you are finishing it, and many of you are doing so very well. Please understand that I only want the best from you, and I don't think I'm quite seeing that from many of you yet.

But with a little more carefulness, and a little more time, I think you can all shine a bit more brightly.

So let's do it, huh?

Walled in

Sorry, no update yet on that teaching philosophy. Give me a few days.

For now, movie reviews.

I just finished watching The Paper Chase, James Bridges's 1973 film featuring John Houseman in his Oscar-winning role, Charles W. Kingsfield, Jr., the tyrannical Harvard Law professor.

Which question does the film answer, "How does one teach well?" or "How does one not?" Kingsfield is bombastic, insulting, abasing, flippant, and shows utter disregard for his students' persons and personalities. Nevertheless, his embrace of "the Socratic method" (loosely construed), an inchoate problem-based learning method, over and above flat-out lecture; and his remarks on the uselessness of memorized yet unanalyzed data, show a genuine concern (if not fully expressed) for his students' intellectual development.

What I find most interesting about the film is the use of space: fittingly both the classroom and the library become places for reverent worship of knowledge and learning, while in the end even a hotel room is transformed into a makeshift chapel of sorts, its walls repapered, its floors recovered with note after note on case law and legal commentary.

On at least three occasions James Hart, the film's hero, finds himself alone in the lecture hall in which his course on Contract Law (Kingsfield's course) is held. His demeanor is always one of awe: the hall is a place of solemnity and numinousness. There, the wisdom of the Judges is passed down, from generation to generation; there, the law is received worshipfully.

The sanctum sanctorum is the "Red Set Room," a special section of the Law Library reserved to hold the personal notes of the school's faculty members, including the notes Prof. Kingsfield himself made in the class he took on contract law nearly fifty years before the film takes place. To gain entry to this hallowed place, Hart and his sidekick must resort to simple breaking and entering, stealing peeks at their professor's notes by flashlight in the dead of a dust-covered night.

I won't go on, there's probably a film school dissertation written on it somewhere: Paper walls, paper halls: the academic setting in the work of James Bridges. I'd only like to draw a parallel with my own experience.

On the evening before the first meeting of the first course over which I had complete control (calling all the shots on classroom activities, homework, quizzes, exams, projects, grades, the whole shebang), I walked from my apartment about a half-mile from Stevenson Center at Vanderbilt University and let myself into the building. At our building's far east end lay my classroom, a spacious room walled by glass on three sides, looking out onto the South Quad to the north, towards the library to the east, and to another of Stevenson's wings on the south.

When I arrived that night was ten o'clock, perhaps, and it was dark outside. I hit the lights.

The room was empty, clearly. I was alone. The board, two sheets of brownish slate, one which could be rolled upward on a wheeled track, was clean. The floor was uncluttered, every desk untouched.

The silence was electric. I was in awe. Learning (I was convinced) would soon happen there.

Even now, this very semester, when I slip into the calculus classroom in Rhoades Hall at 6:45 in the morning in order to set up for my eight a.m. section, I pause to appreciate the immensity of what will transpire in that room in little more than an hour: learning (ideally) will soon happen there.

Why do I feel this way?

It's a humbling feeling: it's part of my job to cause learning to transcend those walls. It's important to help the students to see that learning cannot be confined by any barriers of plaster or brick, no more than it can be locked away by an enlightened cadre of practitioners with strings of letters after their last names.

But transcending the classroom, that Holy of Holies, is a tough task. Why is such a powerful pull exerted by the classroom? Is this gravity merely a remnant of the human penchant for compartmentalization, or is there more to it, a psycho-philosophic connection drawn between religious devotion and academic erudition?

To be continued, perhaps.

Readers: thoughts?

Philosophy 101

Time to rewrite my teaching philosophy.

Or at least give it a good ol' tweak.

I was just renominated for a teaching award, and they're asking for a copy, so's they can have a record of how I claim to teach in my classes. (I'm happy to claim that my theory and my practice follow convergent paths, but I'm realistic enough to realize that teaching philosophies are typically highly idealized documents which record not so much the way one acts as the way one would like to act; it would be hypocritical of me to claim that I'm above hypocrisy.)

The "official" philosophy I've circulated as my own is roughly six years old now, first written for the sake of my first academic job hunt, begun that long ago. (Apology to the morbidly curious: I no longer have this document posted on my website, I'll likely remedy that oversight once it's rewritten.)

I find it interesting that despite the six pedagogically pregnant years that have passed since I wrote it, my philosophy still reflects my beliefs and my practices with a good deal of accuracy, all without the benefit of more than a slight rewording now and then. The most recent retooling came a year ago, the last time I was nominated for a teaching award, and, I may as well say now, as it's now pretty broadly known, at the outset of an abortive new job search.

"Interdisciplinarianism" is the keynote struck in the first paragraph, and subsequent paragraphs build on that theme by making impassioned calls for "relevance" and "holism." I then indicate a need for well-articulated course goals, and follow this exhortation with a long paragraph giving lip service to a diverse array of teaching paradigms (lecture ["used effectively and not exclusively"], discussion, group activities), all of which I'd used extensively by the time of the philosophy's initial writing, and all of which I'd implemented quite effectively, but none of which I'd truly understood or reflected on until very recently.

After this comes a couple of brief paragraphs on assessment, with a laundry list of assessment tools used to keep track of both my students' progress and my own. There's the obligatory spiel, included no doubt to appease hiring committees, on teaching with technology. I then end with a paragraph on the nurturing of round-the-clock learning experiences and a bland reassertion of the primacy of pedagogical diversity.

I'd give my own philosophy, as it stands, a B: it says a lot, and it says it articulately, but incoherently. Rather than writing to inform, I often get the sense that I'm writing to impress. (But who's to say the latter infinitive isn't the true goal of this document?) More than anything else, I feel it shows that I know how to teach well, but not that I know just why it is that teaching in the manner prescribed can be construed as teaching well. In the terms of my soon-to-be-posted writing rubric, it's correct and well-composed, but not incredibly clear or complete.

What is to be said? asks the revolutionary in me.

I don't have all of the answers in front of me (and likely never will), but there are a few words and phrases that ought to appear and don't at present: "problem-based" comes to mind. "Discovery learning" is another. Beyond the buzzwords, more needs to be said about the nurturing of the student's sense of authority, about encouraging the student's nascent autonomy.

I'll probably work on that philosophy a bit this afternoon, so you might get me checking in again later with running commentary on the process.

I'd also like to begin working with my colleagues here in the UNCA Math Department to craft a coherent department-wide teaching philosophy; I suspect there's a high degree of agreement between most of us on a number of salient points, though there may be a few holdouts.

Topic for another post: I've been kicking around ways of making my Calc I course (the last stand of somewhat traditionally-structured courses) more problem-based and student-centered. What better time than in their first semester of college to introduce students to discovery learning?

More on that later, too.

Tips of the trade

The results of my survey of successful students' homework tips (see this post) are in, and available for your viewing pleasure here.

I'm definitely going to make this reflective exercise a regular part of the first few weeks of my semesters; I hope it'll prove useful not only to the students benefiting from their peers' helpful hints, but also, by encouraging awareness of learning styles and study habits, to those providing the hints.

Carolina dreamin'

I had an odd teaching-related (so Maggie says) dream this morning.

I dreamed that I was house-sitting for Maggie's Aunt Susan (if you know Susan, this whole dream is inexplicably funnier). In her dream house she had a titanic tropical fish aquarium...we're talking floor-to-ceiling here, occupying the entirety of a room or two. As you might guess, she had several large exotic fish in this aquarium, and though I seem to recall one or two were sufficiently weird to try to remember what they looked like to describe them to Maggie after I woke up, I've since forgotten all about them except that they were rare and rather large.

At some point in the dream, a dog (not one of ours) jumped into the aquarium and in some fashion broke through the glass, causing all of the water inside to wash away, dissipating through the dream's drainage system without causing a lick of damage to the house's furnishings.

The fish, however, were gone. There ensued several dream-minutes of minor panic as I careered around the house, vainly seeking the fish that had disappeared. I was in a bit of a bind; I seemed to think that Susan's return was imminent (if you know Susan, you know it's probably best not to be on her bad side). After a thorough search I could find neither fin nor scale of even the largest fish, but I was saved from almost certain death by waking before Maggie's aunt returned.

I told Maggie about the dream later this morning. "It sounds like you're worried about failing in some responsibility," she said matter-of-factly.

"Thank you, Dr. Freud."

"Probably in your work or your teaching, you feel like you've let someone down."

Funny, after Friday's 280 class (see yesterday's post).

Thinking a little bit further about what went on in 280 (I promise, I'll stop over-thinking this once I've done writing this post!), I realize that I most definitely should have just shut up and let the committee report play itself out.

Oh well.

I'm going to muster my courage to do just that from now on. I'm pretty certain that this semester's class is on average older than last semester's, and as I've mentioned recently, more cohesive, more mature, more ready for the responsibility I'm entrusting them with. For my part, as well as I led the class last semester, I feel that I've done a superlative job so far this semester in getting the students ready for the responsibility I'm expecting them to fulfill. They can handle it.

I'm going to throw them a sop, too. Well, it's not really a sop, it's simply an opportunity to add another layer of revision in the construction of their written work, but in addition to giving them the chance to improve their writing, it'll give them the chance to improve their grades as well (and a little extrinsic motivation doesn't hurt, right? I just happen to get paid fairly handsomely for doing a job that I'm good at and that I love). To wit, I'm going to give them the chance to revise two problems per homework assignment, after I've given them feedback. Grades'll be handled as with the exams, into which I've already built in guidelines for revision.

I'll put it to the class tomorrow, but I don't foresee major objections.

For now, I'm off to enjoy a lazy Sunday evening. Tally-ho.

Friday

Boy howdy, was my brain ever in Friday mode yesterday. I was muffing this and that, one thing after another, and until late in the afternoon my to-do list was growing longer far faster than I could cut it back.

I've had a good weekend so far, though, so I guess things are evening out in the end.

So, yesterday, what of it?

The first misstep of the morning took the innocuous form of a forgotten stapler. I'd meant to make a clean start and begin bringing it with me to class Fridays so that my Calc I students could properly assemble their ever-unattached homework pages in some manner other than messily crimping the corners together in a sad and useless little lump. Of course, the stapler found itself left behind on the corner of my desk.

No biggie. 'S all good, 's all good.

Then I started handing out the Mathematica installation disks (finally). Six students in each section got a disk, and I calmly went through my little spiel about the installation process...completely omitting three crucial points: (1) they'll need passwords to register, (2) they won't get the passwords instantaneously, but rather those'll be sent to them by e-mail within a day or two, and (3) they must use their school e-mail addresses when registering, otherwise our license manager won't know any one of them from Moses on a pogo stick when it comes time to kick a password studentward. (I later remedied the oversight by sending the entire class an e-mail about the correct procedure.)

I soldiered on. Saturday, after all, was but a half-day away.

It was 280 that was truly frustrating for me, though I know it probably shouldn't have been.

The second committee report of the season (regarding the construction of a truth table purporting to demonstrate a certain tautology) went off without a hitch, and there was robust, respectful discussion on a number of salient points: "Do you have to include all relevant columns in the truth table, or can you omit a few if you can perform some of the operations (like simple negation) mentally?" (It was then determined that one ought to write not for one's own understanding, but for that of the reader, and error should fall on the side of liberality in column inclusion.) "Even though it's obvious if you're looking for it that the first column (containing statement P's truth values) and the last (containing the truth values of a logically equivalent compound statement R) are identical, so we're meant to conclude that R is true if and only if P is...but wouldn't it also be correct to answer the question 'what can you say about when R is true?' by comparing R's truth to some other column, if you didn't notice the equal columns?" (It was agreed that though technically another answer might be correct, indicating the tautology P <=> R would be a "more correct" response to the question.)

The third committee report (concerning a tricky proof by contraposition, asking for a verification that [not Q] implies [not P] in order to prove P implies Q) went a bit more roughly. Tamar and Cornelius led the charge, and all went well in negating the given statements P and Q. But things got a little dicey when it came time to prove the desired contrapositive. Perhaps due to the subtle nature of the negated statements (one was a conjunction requiring a DeMorgan Law), the team got [not Q] and [not P] turned around. I'm not sure they were completely comfortable with the proof they presented, though: Cornelius correctly indicated that he wasn't sure their proof would handle one of the three cases which might arise in [not P], but they weren't able to salvage the proof once it foundered.

At this point, I stepped in for a few minutes to try to patch together a proof of the corrected implication [not Q] => [not P], but my own hastily-assembled argument was a weak one. It was technically correct, but smacked of proof by contradiction, something we had yet to discuss (and in fact would begin discussing a few minutes later), and didn't explicitly use the DeMorgan Law I hinted at on the homework sheet.

By the end of the report, I think everyone (including me) was a little confused and wearied, and we had only twenty minutes to finish a direct proof from the previous class period and to begin tackling contradiction. We made our way through an outline of the proof technique, and now stand poised to construct our first contradictory proof, a feat we'll undertake on Monday.

After class had ended, Dewey came up to me and asked me to take a look at his solution to the contrapositive problem.

It was beautiful.

Aside from a small boo-boo in the negation of one of the statements, his proof was error-free, elegant, and made full use of the required DeMorgan Law. Best of all, it didn't have a whiff of contradiction about it.

Ah, c'est la verification.

So why was I "frustrated"?

Because I know what it's like to muff something in front of one's peers: it ain't pleasant, and I hoped that the folks on that third committee didn't feel overwhelmed by the problem they'd been given.

I also felt frustrated that at the time I'd felt obligated to step into the ring, when I probably should've just stayed the hell out. After all, the whole point behind the use of the committees is to (fittingly) commit its members to take authority over the task they've been delegated. I see my role as that of an ex-officio, advisorial member of each committee formed: though I might provide a little input behind the scenes if it's asked of me, it's not my place to usurp the committee's authority in the classroom. If I keep doing that, how can I expect them to grow more comfortable in wearing the crown? Until yesterday, I feel I've done a really good job in reining in my own reign, and my frustration is probably born from the fact that yesterday, improperly, I let slip my own authority.

In that regard, I definitely fucked up.

On the other hand, the experience provided an object lesson to everybody: we're all going to miss a few now and then, and as I've said many times before (and as many smart people have said before me), learning how to prove things and learning how to clearly record those proofs are iterative processes that generally make their progress (sometimes) painful fits and starts. The students all contributed their versions of the desired proof, the committee undertook the thankless (if you're reading this, thank you, all!) task of collating these versions and producing their own, incomplete version, I made a half-assed attempt at cleaning this up, and Dewey succeeded in showing me up with a nearly flawless feat of mathematical legerdemain.

And hell, isn't this sort of imperfection the essence of discovery learning?

Yeah, I'm learning, folks, I'm learning. Slowly, maybe, but I'm learning. I've certainly got a rough road to travel to my own mastery of that method. In the past several years I've come a long way down that road, but the had sun set on Friday before I could take another step.

Author! Author!

Nearly three weeks in, and still going strong!

The past week has been a good one, and not just for the vacationlet in Virginia Beach, where after the half-marathon Maggie and I and friend/ex-student Mariposa (now teaching middle school in Fredericksburg, VA) hit a local pizzeria called Pi-zzeria, whose theme is the letter pi and from whom I bought a wickedly cool shirt with a pi on the front. I've brought a few fun activities into all of my classes, and I'm particularly happy with some new ideas I've incorporated into Calc I.

There, last week, we pieced together a mathematical jigsaw puzzle, an exercise I thought up on my way into campus that morning. Here's the recipe:

1. Print out a somewhat familiar picture (I used Mona Lisa for one section, and a detail from the ceiling of the Sistine Chapel for the other).
2. Subdivide another sheet of paper into a number of rectangles, grouped in fours, equal in total number to the number of students in your class.
3. In each rectangle so created, write an unreduced expression involving exponents and logarithms, in such a manner that the group of four rectangles contained in any given "region" of the paper holds equal values.
4. Photocopy the picture onto the backside of the grid you've just created.
5. Cut the rectangles apart from one another and shuffle 'em up.
6. Distribute them to the students, and let 'em assemble the picture by first piecing together the local regions with similar values, taping these together, and then fitting these regions into one another.

Of course, the exercise can be modified to provide a review activity for just about any concept you can imagine: compositions of functions, derivatives, integrals, you name it.

The first section took about 9 minutes and change to put their picture together, while the second section came in around 9 minutes.

That was last Thursday. Then they had their first team quiz on Friday, and everyone did very well (between the two sections only a couple of teams missed a perfect score, and even those got 4/5). For the quiz I gave them a problem which likely would have been rather hard for my Calc I students from last semester, and these kids just ate it up. I can tell I'm going to have to challenge these folks with some tougher open-ended problems. From what I could tell, most of the teams collaborated smoothly, too: as I walked around the room, I heard a good deal of explaining, cooperating, clarifying. I don't think there are any truly indomitable personalities in either section. (I do have to say, though, that one student, Tallulah, did mention that she was a bit disgusted with the nattering negativity coming from a pair of her peers in class the other day. I hope this was just a blip on the radar, not to be repeated. I'm doing all I can to create a classroom environment in which people can feel free to pose possible solutions to the problems we discuss, even if they're not entirely sure of their answers; careless critiquing of those brave enough to venture such solutions is hardly appropriate. I don't know of whom Tallulah was speaking, but if you're reading this and you recognize your own behavior, shame, shame!)

What else? Yesterday towards the end of class I asked each student to provide me with a pair of topics discussed so far in class, one of which she or he understands thoroughly and a second on which she or he feels fuzzy. I took some time last night to match each person up with someone else from the class, pairing people off who expressed the same uncertainties in understanding: two folks who felt iffy on inverse functions might have gotten grouped together, or two who reported feeling lost with logarithms. For next Friday I'm asking the pairs of people so matched to work together to construct a dialogue in which they help one another through their mutual difficulties with the topic with which they both expressed confusion. My hope is that in addition to understanding the relevant mathematical concept more clearly, they'll all uncover something about their own learning styles as they examine what it is they're unsure about. Moreover, hey, it's a great way to get them to do a little writing. (Boy, I am the WAC nerd, aren't I? Speaking of which, I've still gotta finish up an abstract for Austin...)

Finally, before and after class yesterday I approached the students who had done particularly well on the most recent homework sets and asked each if he or she wouldn't mind sending me a short e-mail indicating the way he or she completes the homework assignments, in the hopes that I can glean from the ensuing comments some helpful hints I might compile in a handout to give to these students' colleagues who are having more difficulty with the work. I hope they might answer questions like: what do you do when you do your homework? Do you work alone, together with friends, in the Math Lab? Do you make use of a solutions manual? How do you use it, if you do? Is there a way you approach certain problems, a particular way of viewing them? Do you have any specific techniques you recommend, tips for your peers? I didn't ask these questions specifically, hoping to receive unprompted and unfiltered responses. So far I've heard from Magdalena and Xavierina (whose homework, by the way, is some of the most beautiful I've ever seen: it's clearly written, organized, well-documented with an appropriate amount of work shown, and almost entirely correct; Xavierina, if you're reading this, kudos!), and I've had hallway conversations with a few of the others who promise to send me their comments soon.

Meanwhile, there's 280. I'm still having a bit of ball in that class. As well as last Spring's 280 was received, I feel better still about this most recent installment. I can't put my finger on it, but I feel there's a healthier dynamic in this group of students than there was last semester. It almost feels as though the class is significantly smaller, even though there are only two fewer students now than a few months back. It's cozier, comfier, somehow.

The first committee report was made last week (Monday, I believe?), and the three members of that first deliberative body seemed to work well together. At least, I heard no complaints. Their report was a brief one, doing little more than illustrate a couple of the superior responses the committee received (by the way, participation was salutarily high, with about 2/3 of the class submitting solutions). I think the students might have felt a little uncomfortable about indicating others' errors in front of the class (even anonymously), so they avoided outright criticism, but I hope future committees (two more reports tomorrow!) will feel it's okay to indicate common pitfalls, especially if many people fell into them.

Voluntary committee involvement has been strong, I've had no trouble getting people to offer themselves up, and both committees received submissions from over half of the class yesterday.

I have 280 components like the homework committees at the front of my mind as I make my way through the latest in a long line of teaching-related reads, Peggy S. Meszaros's (ed.) Self-authorship: advancing students' intellectual growth, Jossey-Bass, San Francisco, 2007, the focal text of yet another university learning circle I'm taking part in this semester. So far though I've not found the book thoroughly engaging, it's served to reconfirm much of what in the past few years I've come to know and believe about progressive pedagogy at the university level.

In the opening essay, "The journey of self-authorship: why is it necessary?," Meszaros takes the definition of self-authorship offered up by the now-canonized Marcia Baxter Magolda: "the capacity to internally define [one's] own beliefs, identity, and relationships" (p. 10, from Baxter Magolda, Making their own way: narratives for transforming higher education to promote self-development, Stylus, Sterling, VA, 2001). (Justifiably this concern takes center stage in many of today's progessive college classrooms: time after time we hear that what students in today's universities most need to learn is indeed simply how to learn.)

On the facing page in Meszaros's essay, we find the following snippet: "Becoming the authors of their own lives involved reshaping what they believed (epistemology), their sense of self (intrapersonal), and their relationships with others (interpersonal)" (p. 11). As I read this, I jotted some notes in the margin regarding the role played by a discovery-centered approach to proofs and proof-writing in helping to affect changes of all of these sorts:

1. By being encouraged both to construct their own proofs and to thoughtfully critique others', the students gain a deeper understanding of the nature of mathematical knowledge, particularly of the fact that it doesn't inhere in any one person, no matter how intelligent that person is. Knowledge ceases to be "out there, somewhere," but rather "in here."
2. By allowing students to take command of both the proof-writing and the proof-reading (in a literal sense) processes, as I'm attempting to do in our class by establishing the homework committees, I challenge the students to take on the role of the mathematical authority: mathematically speaking, anyone who can grab hold of the governing rules of math logic can stand in judgment of the correctness of a given proof. No longer are the students simply vessels for knowledge not yet bestowed; they are the bestowers themselves, they are the experts. They are participants in the mathematical process, not merely spectators.
3. By cooperating and collaborating in the proof-writing and proof-reading processes, the students come to appreciate that mathematics is a social enterprise, that it is conveyed in a transmittable medium, that it is a part of our shared heritage, ultimately constructed by human beings working in concert with one another.

How successful will this class prove (no pun intended) in easing my students down the road to self-authorship?

I don't know.

Do my colleagues think as deeply about these issues as I do?

I don't know.

I hope so.

I'd really like to see my department develop a more coherent pedagogical philosophy.

But that's another story.

And it's late.

I'm going for now. I'll let you know how tomorrow's committee reports go.