Thank you

It's been a good week, and as the week winds down I want to pause, to reflect gratefully.

Thank you, Irene and Iphigenia, the Honors students who volunteered to person the registration table for today's Teaching About India conference, held all day in the Laurel Forum, just next door to my office.

Thank you, Oswaldo, Frederica, Kent, and all my other Linear I students who grapple tirelessly with the crazy conceptual problems I pose them with week after week, patiently accepting my feedback on their imperfect algebra and rambling grammar as they labor to ensure that each draft is better than the last.

Thank you, Lula, my 479 student who spent several minutes this morning talking with me about the problematic nature of the Honors Program: how do we countenance such expenditure of financial and human resources on students who, honestly, don't need the extra assistance to excel while many more of their peers could use all the academic help they can get just to pass? The program is a walking, talking equity issue, a logical extension of the same K-12 "tracking" systems my 479 students just yesterday derided as iniquitous and unfair.

How do I sleep at night? I overcame white guilt years ago; I think I'm still working through my issues with other forms of power and privilege.

Last, but certainly not least, thank you to Queshia, my wise and warm and indefatigable program assistant. Her able handling of administrative nonsense, her unending supply of good ideas, and her tactful dealings with students expressing every need imaginable all make my job a helluva lot easier. I can't imagine a better right-hand person.

Now. Let's make next week a good one too, huh?

Not a private inquiry

My MATH 365 (Linear Algebra I) course this term has been designated "Inquiry ARC," the "ARC" standing for "Apply, Reflect, Communicate." "Inquiry ARC" is the theme of the university's Quality Enhancement Plan (QEP), a component of our reaffirmation of accreditation through the Southern Association of Colleges and Schools (SACS).

Alphabet soup.

Briefly, every ten years, as a condition of keeping our accreditation, we're asked, as a campus, to formulate a plan for enhancing student learning. This time around we came up with "developing critical thinking," roughly, manifested in the Inquiry ARC program. For the past few semesters we've begun running Inquiry ARC courses, courses which focus intentionally on developing skills relating to one or more steps in the Inquiry ARC process. Last spring, once I knew what I'd be teaching this year, I thought, "what the hell, I'll apply for Inquiry ARC status for Linear I, since the way I teach the course, it's already an Inquiry ARC course." (Not-so-dirty little secret: as any teacher worth her or his salt will tell, you every class should be an Inquiry ARC course...it's just good teaching, folks.)

So far, so good. Communication is always front and center in my courses, given my focus on writing, and I've always been big on reflection. As I've written at length in this blog, I teach this course primarily through applications...and the inquiry? I'm really just going to be able to leave that up to the students.

On Wednesday we spent about 15 minutes on an example I'd meant to take 5, simply because the students kept asking fantastic questions about the example. "What if we changed the number in the last column?" "What would have to happen for us to have a unique solution?" "How does Mathematica know that it's an augmented matrix with three variables and not a matrix with four variables?" and so forth. Not I-have-no-idea-what-I'm-doing-so-I'm-asking-simple-questions-trying-to-get-a-grip-on-this-shit questions, but I've-got-the-basic-operations-down-so-now-I'm-trying-to-take-it-apart-to-see-why-this-shit-works questions. Good stuff!

If they keep it up for the rest of the semester, I'm gonna be a happy man.

Day One, revisited, or, Patrick very quickly gets off into pedagogical theory

Back to the grind. Today hardly felt like a school day at all, as my only MWF class is an 8:00-to-8:50 section of Linear Algebra I that was over nearly as soon as it started. (At least on Tuesdays and Thursdays I won't be done in the classroom until noon-thirty.) As first-days-of-class go, it was a good one, though. I've had better, but I've had far worse.

Plus ça change, plus c'est la même chose. Both other times I've taught this course (including the first time, the iteration of the course that occasioned the founding of this blog) I've started with some variation of the same game, a simulation of a Markov process in which the students shuttle some sort of token back and forth at each iteration of the game. The first run (Fall 2006), the students themselves were the tokens as the class participated in a great big single instance of the game; I switched to pennies (and smaller groups) the next time I taught the course (Fall 2010), and I stuck with that latter version today, though leaving a bit more room than I did before for students to discover and speculate upon the patterns their own damned selves. This time around I also asked the students to take bolder and more unassisted steps toward the next conceptual mile marker, solution of the linear systems that arise from the Markov process we investigate together: not only must the students experiment and then speculate on the outcome of their experiment, they must then find the appropriate mathematical model (a simple linear system in two unknowns) and then back-solve the "run the model in reverse." All in 45 minutes' time!

All in all, it went remarkably well. No one seemed lost ("one in a row!" as my colleague Tip would say), and everyone participated actively. I'm aided this semester by the fact that I've only got 23 student in the class (yay), though they're packt like sardines in a crushd tin box (boo), sitting at single-person-sized tables (yay) bolted together and arrayed in orderly rows (boo) in such a fashion as to discourage all but the most anachronistic teaching techniques (boo hiss).

Interesting facts (yes, there is a train of thought that took me from the previous paragraph to this one): recently, while reviewing the literature on the effect of class size on learning, I discovered that (1) said literature says almost nothing about college-level instruction, most research having been done at the K-12 level, and (2) a number of studies do not, strangely enough, suggest small class size improves student learning in mathematics. It was only after a bit of reflection that I realized why this might be: such studies, while controlling for class size, do not (and, methodologically, cannot) control for instructional method. Thus what I suspect is happening in these studies is large-section lectures are being pitted against small-section lectures, lecture being, until recently, about the only viable instructional paradigm for large-section classes. Of course, it is pedagogically retarded (in the literal...well...until recently literal...sense) to assume that one's instructional method remain the same when smaller class size permits more effective application of student-centered learning strategies: pit large-section lectures against small-section IBL and you're sure to see a difference.

Maybe more about that in a post soon to come (why on Earth was Patrick researching this topic? Edge-of-your-seat action!). For now, I've got reading to do for my first meeting of HON 479 tomorrow!

What have we learned?

At some time in my youth I learned that I loved mathematics. Its precision and exactitude suited my budding rational empiricism (though I couldn't have thought to say this at the time). I decided that I'd be a math teacher, because what else could one do with math?

As an undergraduate math major, I learned that I'd most likely teach math at a university, and not at a high school. "Don't waste your talent," my adviser told me. In those words he said it (albeit sotto voce); being a mathematician himself, he likely lacked the social grace to say it less bluntly. I was led by him to believe the dictum that "those who can do; those who can't teach." Though I've learned that it's often the case that our future teachers struggle more mightily with advanced mathematical concepts than do some of their more pure-math-minded colleagues, that struggle is often a fruitful and maturing one, and I no longer keep stock in this saying. Besides, knowing college faculty the way I know college faculty now, and recognizing the importance of a rock-solid middle school and high school education, I'd trade a dozen run-of-the-mill university faculty for one fantastic high school math teacher.

As a grad student, I learned that I loved both teaching and research...in other words, I was born to live a life in the traditional academy. I loved the energy in my classrooms, the excitement of my students. I loved helping them to their personal epiphanies and "aha!" moments that make discovery so fulfilling. (I remember some of my experiences there with remarkable clarity, as in this old post.) As much as this, I loved holing up in my office or in the library and plugging away at the pure mathematical puzzles presented me by Coxeter groups, braid groups, and Artin groups. I loved confronting the unknown, clad only in the armor of socially-constructed axiomatic systems (though I couldn't have thought to call them this at the time), armed only with a few blunt theorems. My doctoral adviser urged me to get a postdoc, to prepare myself for a career at a top-notch research institution, where I'd surely be happiest.

As a postdoctoral scholar, I learned that I didn't want to spend the rest of my life at a research-intensive university. As much as I loved (and still love) research, I could not see myself doing it to the exclusion of almost all else. I had hearty colleagues; they were wise, intelligent, and in their own way fun. But they sacrificed virtually all for the sake of their research careers, publishing so that they might not perish and openly disdaining teaching, for it took time away from their scholarly pursuits. As I grew to be a better and better teacher and came to learn the fulfillment it gave me, and as I came to recognize how much more of an impact an exceptional teacher can have than even the best researcher in pure mathematics, my personal future in academia came more clearly into focus.

As an untenured faculty member, I learned that yes, indeed, teaching is where my passion lies. My research was (and is) fruitful and fun, but had (and has), for me, less meaning than the moments I get to spend working with my students, talking about teaching, and living as fully immersed in an authentic community of learners as I can. I learned that math wasn't all there was, and that my longstanding love of writing could play as big a part in my working life as it did in my personal one. I opened the door to poetry, composition, and rhetoric, and discovered rich new worlds I'm only just beginning to explore. (Much more about that in a forthcoming post on my personal "community of scholars.") Math was moved aside to make room.

As a tenured faculty member, I'm learning more and more each day that I can do what I want to with my career, and that I can follow my ever-changing passions. I'm learning that my most meaningful collaborations are most rarely found among people in my own discipline. I'm learning to adopt and adapt ever more reflective and integrative practices into my classes. Some of these practices are decentering and unsettling, both for the students I work with and for me...but ultimately such practices are the most rewarding. I'm learning that I stand to learn the most from those who are "supposed" to learn from me, and that only at a liberal arts institution like UNC Asheville would I have the chance that I have to engage in that learning.

I've learned enough to know that I don't know much at all, but that that's okay, because no one really does, and I know as much as anyone I'm likely to run into. We all know different things, though, and when we meet at the crossroads and shelter at the inn there for the night, we'll have wonderful stories to share with one another before we move on to make our way in the world once more. Sharing travelers' tales: that's how we learn best.

Here's a tale. This semester I'm working on an outside project with two of my favorite students, Ino and Ned, with whom I've shared several courses in the past. They've signed on as "undergraduate research" students because initially we expected that the project would involve a good deal of digging into linear programming and other linear algebraic methods of optimization, requiring us to perform original research at some point. (It's an offshoot of a miniproject Ino and Ned worked on with a couple of other students in my linear algebra course last fall, in which they performed a feasibility study of several different diets from both a nutritional and a budgetary standpoint. Wonderful stuff!)

The deeper down the rabbit hole we go, the more it becomes evident that we're unlikely to break any theoretical ground, but the more we realize the transformative potential of the work we're doing in the larger sphere. The work these bright kids (and they are among the brightest our school is privileged to serve) are doing will positively impact our region by helping community outreach organizations assist low-income families in planning healthy and affordable meals. Our off-campus partners are excited about the budding collaboration my students are forging, and I predict great things coming of our engagement with one another.

You might still call this "undergraduate research," writ large...more accurately still, you might call it "service learning." Whatever it's called, it's learning of some sort. Moreover, it's authentic, it's transformational, and it's real. It represents the sort of learning I hope to do more of as I move forward from here. It's the sort I hope to help my students and colleagues do more of as well.

Let's wait until morning and then move on, and make up more travelers' tales to tell at the next crossroads we meet.

Observations

A few random observations about my courses this semester:

1. Ever since our department implemented a writing component in the Senior Seminar, the overall quality of the students' oral presentations has gone up as well. Of course, this makes perfect sense for anyone who understands the concept of writing-to-learn: the students are using writing as a means of engaging meaningfully the topics on which they'll later present...moreover, they're making that engagement earlier in the semester than they normally would, asked as they are to complete a rough draft of their written report by the halfway point of the term, at least a week or two before they must present.

2. As in every semester that's ever been and every semester that ever will be, the "nontraditional" students in my Calc I class are outperforming the young 'uns. The more mature folks (juniors, seniors, and returning students) are the first and most frequent askers of questions and the strongest showers on exams, and they make up the majority of those still completing the homework regularly, despite the fact that it's been optional for a week or more by now.

Though there are a couple dozen Cs and Ds scattered throughout my two sections of Calc I, none of the 15 (out of 65) older students has a grade below a B going into the final exam. While they might not have quite as fresh of math skills (or even intrinsic mathematical aptitude) as do the first-year students, the older folks have vastly superior time management techniques, study habits, self-direction, commitment, and sense of purpose. Those skills and attributes (among the most important ones learned or acquired in college) help them to be, by and large, far more successful than their younger peers in most of their courses.

3. This semester's end-of-term presentations in Linear (the first three of nine given today) are far and away better than those the students put together for the course the last time I taught it...and they students have had less time to prepare them than they'd had the last time, too. Though this is a particularly strong class, I humbly give myself some credit for doing a better job overall in leading this course than I did last time. The exercises and activities I've put together are more authentic and robust, and the ways in which the various components of the course were fitted together simply made more logical sense.

Incidentally, one of the presentations will very likely lead to a rich research project. Ino and I have already discussed (at great length) our plans to parlay her team's presentation on linear analysis of nutritional data into a full-scale publishable project. The sky's the limit, and I'm really looking forward to directing what will likely be my first-ever truly applied math research project.

4. As early as I can next term I need to make a point of helping students get past their own pride when it comes to asking questions in class. Students (especially younger ones) often have a morbid fear of "looking dumb" in front of their peers, and of course asking questions makes one appear ignorant. (As opposed to remaining silent, which leads not to appearing ignorant but simply to being ignorant.) I've got to more actively help students overcome that fear.

To that end, I made some remarks in my second (the quiet) section of Calc I today that seemed to have a positive effect on students' querulousness: before proceeding from a specific example of a definite integral computation to a general one, I said something along the lines of "any questions before we move on? This idea is a crucial one, and it's very important that you have a good grasp on it before we proceed. [Silence. Pause.] How many people are there in here? [Count out loud.] Thirty or so? In a class this size, I fully expect more than half of you, probably 15 to 20 of you, don't understand something about what we just did. I expect that. Typically at this point half of the class doesn't fully know what's going on. I expect that. I'm sure you've got questions. I'm just not sure why you're not asking them. But I really can't do anything about it if you don't ask, so we'll just move on."

Move on I did, and within a minute or so two or three people who almost never ask questions posed a few. I was ecstatic! It's the first time this semester that some of these people have come out of their shells.

I've got to remember that trick.

5. I will never ceased to be amazed by how wonderful are the students I work with on a daily basis. Our students are incredible. They're intelligent, devoted, hard-working, honest, and down-to-earth. They're smart, sassy, funny, and fun. They're some of my favorite people on the planet, and I cherish every moment in working with them. I am the luckiest man on Earth for getting to do what I do, and getting paid for it.

Shift, paradigm, shift!

This morning was painful.

I had the uncomfortable task of watching 30 of the students in my morning Calc I section puzzle their way through a difficult (but fair, I feel) exam, their last "mid-term" exam of the semester and a traditionally hard one (on applications of the derivative). This awful duty was the last straw.

The penultimate one came yesterday: I was called upon to help adjudicate a case of "academic dishonesty" (to use the lovely euphemism) in a colleague's class. This colleague wanted to know if the students in question had indeed "cheated" (to stop beating around the bush). After cursory (and then more in-depth) inspection, I agreed that yes, indeed they had.

However, to me the incident said more about the culture of the academy than it did about the "dishonesty" in which the students were involved. More specifically, the students were clearly guilty of "cheating," but to me the more crucial issue concerned why it is they felt the need to "cheat" in the first place.

To "cheat" requires that there be a "game," and that it matter that people follow the rules of that game, and further that it matter that in order to succeed at the game one must "do better" than anyone else playing the game. This was definitely the case in this particular course: the students had been given a (very) high-stakes exam, which to them was more than an assessment instrument; it was moreover one of a very small few means of receiving feedback on the degree to which they were mastering the concepts of the course the exam was given in. To them, "cheating" on the exam was a natural response, given the way in which they've been acculturated to consider the exam a must-win game in which success is measured by high marks.

The point I'm getting at is the following: the more we as educators eliminate, or to the greatest extent possible downplay, the competitive aspect of education (high-stakes testing, rigid and number-driven grading schemata, individualistic learning paradigms, etc.), the less likely we are to find our students "gaming" the system by engaging in "cheating." In some regards, "cheating" will cease to exist, as it simply will have been defined away.

As I hinted above, most "cheating" (I truly believe) is undertaken as an act of desperation, a means of coping with failure as measured by receipt of lower-than-average academic grades. "Cheating" is a means of striving to succeed within a system which provides extrinsic rewards for optimal performance rather than intrinsic rewards for authentic mastery and authorship. I cannot but believe that the vast majority of students would welcome an academic system in which the goal is not to earn high marks but rather to learn, and that students accustomed to this system would see no need to game the system by "cheating." I'm not so naive as to suppose that every student will respond well: there will always be those so acculturated by thirteen-plus years of a largely competitive educational system that it's in their blood to fight tooth and nail for every last percentage point that might tip them from a B+ to an A-...but I believe that even those who are very comfortable with this traditional system will abandon it if given the chance to do so.

All of this gets me back to this morning, and the thoughts I had as I watched my Calc I students (even the strongest of them) wiggle and squirm in completing their in-class exam, stressing out not over whether or not they'd really learned the concepts we've been working on together for the past few weeks, but rather over the grades that will unmercifully adorn their papers when they get them back tomorrow morning.

My thoughts can be boiled down into two short words: no more.

No more in-class exams. Ever. I'm through with them. The one I gave this morning (and will give again in a couple of hours) will be my last.

For quite some time now I've not given in-class exams in my upper-level courses, feeling that little meaningful could be asked on such exams, aside from requiring students to parrot already-proven theorems or give short answers to requests for definitions. In these courses, in-class exams offer none of the opportunity offered by take-home exams to ask authentically engaging and probative questions, and therefore I've found them pointless time-sinks.

For quite some time now I've resisted the banishment of in-class exams from my first-year courses, thinking, I suppose, as I've heard some of my colleagues to think: "there are certain computational techniques the students will have to learn to perform, and to perform quickly." True, but even the most straightforward computational skills will be as well developed (and much more readily understood) if performed in the service of completing the more meaningful, authentically engaging exercises included on take-home exams. This is as true in Precalc or Calc I as it is in Abstract Algebra I or Topology. Even in lower-level courses take-home exams offer a much more meaningful sort of assessment, and even if those exams are still meant as individual exercises and not collaborative ones (of the sort with which I've been experimenting in Linear Algebra this semester) they go much further than do in-class exams in encouraging a culture of collaborative engagement, simply by downplaying high-stakes individualistic assessment.

Keep in mind that I'm not boo-hissing exams entirely: exams offer students a means of reflecting on the ideas they've learned for the past ___ weeks, and if properly responded to they're fantastic tools for giving students feedback. Well-designed and well-delivered exams give students a healthy way of furthering their learning and assimilating their knowledge as they think critically about it. Exams are here to stay.

In-class exams, however, for me will soon be a thing of the past.

I've already responded to a couple of concerns I anticipate colleagues (and even some students...very bright ones, in fact) might have about this decision of mine, but let me respond to a couple more hypotheticals.

Colleague/student: "If you de-emphasize high-stakes individualistic exercises like in-class exams, some students will game the system and prop themselves up on others' work without really learning anything themselves."

Me: "No matter how you set the system up, students are always going to find a way to game it. Gaming the system I propose means cheating themselves out of learning. Gaming the system as it currently stands involves engaging in a behavior that's viewed as sociopathic but is really little more than a symptom of deeper systemic problems. I find the former course far less pernicious. Sure, there'll always be students who frankly don't give a shit, but we're not going to serve them well in any system, and with a system more conducive to authentic learning, they might just pick up a thing or two along the way."

Colleague/student: "You can tilt at as many windmills as you'd like to in your own little class, but you've got to assign grades at the semester's end, anyway. Won't the system you propose result in massive grade inflation?"

Me: "My response to this concern is twofold. First, in courses in which I've begun doing away almost entirely with individualistic activities (begun in Foundations, Topology, and Abstract Algebra I and II in previous semesters, and taken to the extreme in my Linear course this semester), I've seen little noticeable change in the final grades for the courses. This was true even in Topology, in which students had unlimited opportunity to revise and resubmit all work, with no constraint on collaboration. I simply don't see evidence for grade inflation. Second, even if there were grade inflation, so what? It would be incredibly difficult to determine whether students' grades were made higher because they were bracing themselves on each other ("gaming" the system in the sense expected by my first colleague above), or simply because they'd actually managed to more richly and more fully understand the ideas addressed by the course. That is, maybe the grades are higher for a reason: the students are actually, for the first time in their lives, getting it."

I truly feel this way.

Moreover, I truly feel that I've become proficient enough as an educator that my courses offer the sort of rich collaborative learning environment wherein in-class exams no longer serve a meaningful purpose. They're simply anathema to my teaching philosophy, and they're counterproductive to my goal of establishing a safe, stressless, and supportive setting in which we can all learn from one another in robust and authentic ways. From here on out, they're gone.

Before I leave, let me pass my apologies on to the students currently enrolled in my Calc I course: I'm sorry that you had to suffer through the last iteration of this practice. I know how hard the topics you're being tested on are, and I'll keep that in mind as I respond to the exams tonight. I'll respond to them not with an eye toward giving you a grade, but with an eye toward letting you know how well you're learning. I hope you'll receive them back from me with the same thoughts in mind.

One last note: I should point the interested reader in the direction of Alfie Kohn's No contest: the case against competition, about which I've blogged before, and which over the years has probably proven to be the single book which has exerted the greatest influence on me as a teacher. It should be required reading for all educators, at every level.

Great expectations

It's been a bit since I wrote anything truly original on this here site; I fear things have been rather hectic in Patrickland.

I've had a few straight weeks of mind-boggling busyness, but every one of them has been good, almost to a day. Seriously: things have been chugging along furiously but swimmingly, and I've been enjoying all of my classes immensely.

As you can tell from the two most recent posts (this one and this one), Newton v. Leibniz came off without a hitch. Both sections of my Calc I class prepared well and performed well. Though the debate wasn't as heated as I've seen it in the past, both sides were ready and the back-and-forth was steady and confident. I was particularly impressed with the woman playing John Collins in the second section, and the woman playing Henry Oldenburg in the first. The lead attorneys all did splendidly, with well-prepared arguments and clear questions. Objections were at a minimum, and everything was civil (for the most part...there was one somewhat heated exchange in the second section). I enjoyed it a lot. Judging from a preliminary reading ("skimming" would be more apt) of the students' reflection papers dealing with the project, they got a lot out of it, too.

Since then (that was a week ago this past Tuesday), my primary occupation has been getting ready for the Conference on Constrained Poetry. Everything's in order now, with all of our speakers in town (aside from my colleague who's driving over from the Sylva-Cullowhee area tomorrow morning), almost 80 conference-goers pre-registered, recently-run articles on the event in the campus literary mag and the local free weekly, and a well-known journal (Critiphoria) ready to print attendees' creations. Not bad for a first time event. I'm quite sanguine about our chances of repeating next year.

Mimicking our guest Michael Leong's construction of occasional constrained poems in honor of our conference (check out his chapbook, The Hoax of Contagion...also, my thanks to Michael for linking to a handy on-line n + 7 generator), I applied my finite-state automatatistic (what a wonderful word!) constraint to the list of last names of students in my Linear class, obtaining the following automaton:


The word list I obtained from this constraint was short enough to offer a challenge and long enough to give me a rich source of words to describe the joy I've had in working with the students in this course, one of my favorites since coming to this school five and a half years ago.

I'll leave you with the resulting poem, but I'll be sure to check in again soon to let you know how things go tomorrow (because I know you give a damn).

We are one

Here we were married.
We were wedded,
welded
one to one
on a vessel on a charted curve
etched in every memory
in marked meter.

We were battered hard
at intervals,
in imagined rain,
in grey tones.

We are free, we veterans,
sores patched, keels buried,
meet, reserved, nerves set.

We are free, we veterans.
We are one.

What's it worth to me?

19 hours, apparently.

19 hours, 7200 words in response to students' drafts, several follow-up e-mails looking for students' sources (ah, unintentional plagiarism!), countless handwritten notes (sorry for the scrawl, y'all), and, it must be said, a good number of sighs and head-shakes...but a few smiles as well.

This weekend was the perfect storm of grading. I had a look-see at the roughly 25 first drafts of Newton v. Leibniz papers my Calc I students produced, the 9 "sections" from Linear Algebra for Dummies the Linear students wrote, as well as two problems sets from the former class and one from the latter, and assorted exam revisions and older homework sets. There were times during the past 48 hours at which I cursed myself for setting a precedent for such quick turnaround, at which I thought, "is it really work it to me?"

The answer, I think, now that I've finally got a few hours of free time before things start up again tomorrow, is...yes.

That's a simple answer which masks the earnest reflection I've done over the last couple of days on a number of weighty pedagogical issues: (1) meaningful instruction of authentic disciplinary writing (or even basic academic writing at the collegiate level), (2) the role played by homework completion (and feedback received on same) in the learning process, (3) the relative uselessness of computer-generated/computer-graded homework in providing meaningful conceptual instruction, (4) the role of numerical grades, especially as they pertain to the establishment of an extrinsic rewards system that encourages students to become number-crunchers at the expense of real learning.

Et cetera.

How've these issues come up?

Newton v. Leibniz offers students an imposing and unanticipated challenge: most of my students don't expect to do much writing for a math class, and I'm quite certain that (whether they admit it or not) they don't expect their math professor to be a stickler when it comes to writing. My suspicion is that if they've ever had to write much for a math class in the past (and they likely haven't), whatever writing instruction they've gotten from their math teacher was half-assed and half-baked.

So maybe I shouldn't be surprised when I receive inchoate two- or three-page drafts in which ideas are scattered and half-formed, or reference lists peppered with websites whose authorship is unidentifiable and with textbooks which are (literally!) over a century old. Four or five of the drafts were marvelous: well-researched and well-organized, clear, correct, and easy-to-follow. Five or six others have obvious potential, and rest on a foundation of appropriately-chosen references. These might make a few slips compositionally, and they may have a few logical lacnuae here and there, but they'll be solid with a few more sources and some good ol'-fashioned elbow grease. The rest (another five or six) have a ways to go.

I've been playing this game long enough that I know almost exactly what's happened as the students put those last few together. Eight days into the nine-day period they've been given to get their shit together, the members of the group putting one of these papers together met after class (seven hours to the due date) and said, "hey, when are we going to work on this?"

By this time it's too late to make use of the fifteen-odd recent and well-reviewed books on Newton and Leibniz and their fellow-travelers I put on hold at the library, so they opt for the next-best...er...well, still a halfway-decen...um...well, at least an okay...well, all right...a pretty piss-poor stopgap solution: Google.

Pulling up the first two websites they can find when they put "Newton versus Leibniz" into the Google search field, they read.

Of course, one of those websites (this one, written by someone I can only refer to as Mr. Angelfire) is utter crap:

1. It's impossible to tell who wrote it, although it's likely a term paper written by a high school or college (I'm betting on the former) student with limited skills for selecting references:

2. the only print references it cites are at least (not kidding...wish I were!) 40 years old, including one that's over a hundred, and

3. the only relatively recent source is a website...it's a good one, but it's a website nonetheless, and unless you're familiar with that site (I am; my students are not), you wouldn't know this from the way it's cited (incorrectly).

4. It's a perfect example of that boilerplate "five-paragraph essay" nonsense they teach kids how to write (for some godawful reason) in high school these days. It takes no stance, it offers nothing real. It has no voice. I want my students to take a stand, stake a claim, and fight for it tooth-and-nail. This essay is a lousy model for this sort of behavior.

The second-most-cited website is this one. It takes a little effort, but you can find the author of this site, one Robin Jordan, Professor Emeritus of Physics at Florida Atlantic University, whose website (featuring marvelous animated .gifs written, no doubt, circa 1998) makes him seem to be a pretty decent teacher, actually. Not only is Jordan's paper much more well-written than the other, but it's richer and relies on stronger sources. I'm okay with my students drawing on Jordan's paper, but I'd still rather they use him as a stepping stone to get back to the print sources on which he himself draws.

All of that aside, what's the next step in our hypothetical students' last-minute writing process?

Reference (singular) found, they devour it, in a matter of...well, minutes...because that's all the time that's left to them during the lunch period on this last day. A few choice quotes plucked from the "paper" they've just read, they begin writing.

At this point it's too late to develop a thesis of their own, so the students opt for that old standby, "we just can't tell who it is who invented calculus, doncha know?" Asserting that there's just not enough evidence to tell which man has the greater claim (or a claim at all), the students hammer out two or three pages in which they eventually get around to saying that Newton did it first, but maybe Leibniz did it on his own anyway, who's to say?

Well, you're to say.

One thing these last-minute Larrys and Lauries don't do is say anything, at least not anything meaningful. But I so much want you to do this, my young friends! More than anything, this is what I want from you: I want to hear your voice.

Make a claim...make a bold claim. If you feel like putting it in boldface letters 18 points high, then do that. But make a claim, and make it your own. Make it your own by finding the sources that help you say what you want to say, that help to prove that, by goodness, you're right. Find the sources which lend support to your claim, and lead me through an analysis of those sources, step by step. Prove to me that you're right, sentence by sentence, page by page.

I don't want to know what Dr. Robin Jordan thinks, I sure as hell don't want to know what Mr. Angelfire thinks...I want to know what you think, and why you think it. That, my young colleagues, is the essence of academic writing (and, indeed, academic thinking of any kind): saying something intelligent, and saying it in an intelligent way as you insert it meaningfully into conversation with all of the other thinkers who have come before you.

Is it worth 6 hours of poring over drafts and 7200 words (that's 27 pages of 12-point, double-spaced text, by the way) of responding to those drafts, if it helps you all become better academic writers?

Hell, yeah. I'd do it again in a heartbeat. And I'd be delighted to look over further drafts if any of my students care to hand me some between now and the due date next Friday.

The rest, the other 13 hours? Problem sets, problem sets...yeah. The Linear problem sets were fine, and those for Calculus I...were great, actually. I put 'em through the wringer, computationally. I've no one but myself to blame if I go blind from having to puzzle through the first six or more derivatives of e^x sin(x). The extra work is worth it to me, if only so I don't have to read through thirty or forty poorly-transcribed copies of the solutions manual because I made the mistake of assigning problems from the textbook (a pedagogical practice I'll never again adopt as long as solutions manuals are readily available).

Indeed, in the end the students did really well on the two problem sets (one on the Product and Quotient Rules, the other on trig derivatives), given their relative difficulty. The only gripe I might make about them concerns, as above, evident procrastination: if you don't get going until Friday, a few hours before they're due, you're not going to do that well.

All in all, though, these problem sets too are worth the time I put into grading them: I feel strongly that feedback (frequent and full) is essential at this stage in the students' engagement of higher mathematics, and I feel strongly that graded homework is the best way to provide that feedback. (Students know this, too: almost without exception my former students remark to me how helpful graded homework is once they've gone on to a class with one of my colleagues who doesn't require it...and when given a chance to assign their own grading weights to the various activities we take part in in my classes, they always give homework a substantial boost.)

This post, which began with a gripe, now draws to a close with acceptance and contentment. I've lost a weekend, in some regards (although I did take in several really good college football games yesterday), but I've come through to the other side a better teacher, having reflected a bit more carefully on, and asked myself to remember, the reasons I do the things I do.

Before I go, here's a postscript for my Linear students (in particular, for Ino and Iris, to whom I was complaining on Friday, about having to assign numerical grades to their written projects): I plan on asking you all to assign your own grades to your Linear Algebra for Dummies sections. FYI.

What a difference a day makes

Today was much better than yesterday: much less stressful, much more fun.

I had a blast in all of my classes, working heavily with splines in both Linear and Calc I: it's lovely to find a project that's meaningful to both groups of students! The Linear students are learning how splines are actually constructed and investigating algorithms for building arbitrarily complicated cubic splines...the Calc I students too are going to be able to get in on the action when they look into the construction of some very simple quadratic splines in a week or so. It's all spiraled out of a question a Calc I student posed to me a week ago regarding the "interpolation" problem set I'd given them. How wonderful that such good ideas come from working with students!

Today I also managed to find something about which my second section students are delighted to talk: designing their own grading scales. Even the shyer students were happy to speak up when it came time to ask them whether quizzes should count for 5% or 10% of their overall grade. After ten or twelve minutes of debating the issue, both sections decided upon tentative weighting schemes for their grades:

Section 1
Homework: 35%
Quizzes: 5%
Projects: 25%
Midterms: 25% (total)
Final: 10%

Section 2
Homework: 20%
Quizzes: 10%
Projects: 25%
Midterms: 25% (total)
Final: 20%

The first section's scheme is closer to the one I would typically use (back when I assigned the weights myself), but the second section's scheme is within acceptable tolerance. I can hang.

We'll see how they work out; they were well-arrived-at (after a good deal of earnest give-and-take which considered amount work, locus of learning, revisability, ease, and so forth). I'm always impressed with the students' maturity when they're trusted to make decisions that affect them meaningfully. I'm glad that they don't often abuse the trust I give to them.

Yes, it's been a good day.

Before I call it a night, my thanks must go to my colleague Dolores (and her husband Ken), for driving all the way down from Virginia to give a lovely MATH 480 Senior Seminar talk on Catalan numbers. Thanks, Dolores! Very well received! We'll have to have you down again sometime soon.

The importance of being earnest

Doomsday averted.

This morning's Writing Intensive Subcommittee meeting was wonderfully productive (we plowed through a TON of tasks), and the pre-meeting conversation I had with the Student Government Association (SGA) rep who's been in conversation with me for the past few weeks (let's call him Kenyon) was even more productive.

I think we're on the same page now. I had a bit of a "come to Jesus meetin'" with this young man, and we were both very honest and open about our concerns. I realized early on that he's more the messenger than the source, and that his intentions are good ones. "We have to be open with each other," I insisted. "We can't sneak around, we can't take part in romantic revolutions are crusades. We have a number of common concerns, I'm sure, and we can work on them together...if we choose to talk to one another about them." He agreed.

He sat in on our meeting, and though I think he's got a thing or two to learn about note-taking, I admire him for following fairly well the course of a rather convoluted proceedings. We hit everything under the sun, from the minutiae of WI proposal wording to the philosophical underpinnings of the WI mission itself...and faculty development and assessment in between. A real trouper!

We agreed by the end of the meeting that it would be worthwhile to establish some sort of "liaisonship" between SGA and ILSOC (or some of the willing subcommittees thereof) in order to open, maintain, and benefit from a dialogue between faculty and students on issues pertaining to ILS and other academic affairs which affect us all. He invited me to attend this evening's meeting of the Academic Affairs Committee of SGA (on which he serves, and for which he's been running his little end-runs).

So I went. I'm glad that I did.

The meeting, held in the SGA offices in the student union, was attended by six members of SGA and me. It was unassuming and informal, as Kenyon assured me it would be. The students took turns reporting on their progress on their individual "homework assignments" from the previous week. One had been sent to data-mine various sets of statistics concerning the ILS Clusters, in the hopes of finding correlation between students' choice of topical clusters and their majors. (Undoubtedly such correlation exists...and as it happens this is one of the students' primary concerns, to which I'll return in a bit.) Another reported on the SACS (Southern Association of Colleges and Schools, our accreditation agency) meeting he had attended. I commended him for his ability to bust out all of the buzzwords.

Kenyon discussed his meeting with me and his attendance of the WI meeting, and I took a moment to explain our feelings about establishing a "liaisonship" between SGA and ILSOC. I'm not sure that everyone at the table was sold about the efficacy of establishing such a dialogue, but one of the student leaders (the Vice President, Samantha), was totally on board. She pleaded vigorously and eloquently for our case, and I'm glad that she did. I think her advocacy helped the case considerably.

As the meeting went on I became a little more annoyed by the continued reluctance of the students simply to come out and say what concerns they were having, and it soon came to the fore why it is they have this reluctance. Historically, it appears, every time they've brought complaints to faculty, they've either (a) been blown off or (b) been told that they need "data" to back up their claims.

So they've been gathering data.

I let them know that they were likely talking to the wrong faculty back then, but that they should feel safe in talking to me and to the other members of ILSOC. At this time, ILSOC consists of very active, engaged, and motivated faculty who are heavily invested in meaningful interdisciplinary learning and fully committed to the spirit of the ILS program. They won't need to sell their story; we've bought it already, and we're eager to talk. I assured the students that we are the people they need to talk to. Samantha was resold, and once again Kenyon put himself forward as a liaison.

After the meeting I lingered a bit and talked some more with Samantha. At long last I learned a bit more about their specific grievances. For instance, it became evident that their primary concern with the LSIC Colloquia is strongly related to our own: uniformity of the quality of instruction across all colloquia. Knowing this, we can talk about it openly and brainstorm ideas together.

It also became evident that the students' primary concern with ILS Clusters is that students aren't getting enough incentive to be daring in their choice of clusters: because they're so pressed to finish their majors with the number of credit hours they're given (lest they pay egregious overage charges), they find they have to select topical clusters focused on topics cognate to their majors. These students would like to see greater incentives (more flexible rules for "double-dipping" courses? Elimination or lowering of overage fees?) offered to students to try out clusters more distant from the safety of their majors, thereby engaging in a richer interdisciplinary learning experience.

Honestly, this perspective is far more mature than that of many members of the faculty, who simply want to scrap the clusters altogether. (Admittedly, this contingent of curmudgeonly academic extremists is getting smaller each year, as the fogyish stalwarts retire one by one.) I was thoroughly impressed with the students' position, and I told them so. We can definitely work with them on this.

Ultimately, as I said above, I'm glad that I went to the meeting. A lot was accomplished, and I'm excited to see the directions in which this heads.

Before I go I should mention one more (not wholly unrelated) incident. Late this afternoon, while sitting in my office, I overheard a couple of my Linear students (sitting in the near end of the Math Lab) quietly voicing their frustrations about not being able to keep up in class because of the way we'd plowed through the definition and derivation of eigenvalues and eigenvectors. I guess we'd been moving a bit too fast. Although I felt a bit uneasy about admitting to my eavesdropping, I sneaked across the hall and joined in the conversation.

"You've got to let me know," I told them, "if you're having trouble with something."

"I'm just not one of those people who can pick it up really fast," one of them said. "I have to think about it and let it sink in before I understand it."

Although at first the conversation was a bit strained and awkward (I had been eavesdropping, after all), after a bit it warmed up. I agreed to keep tabs on the pace, and to throw in a few more examples and explanations here when needed. They agreed to let me know if things get moving too quickly again.

"I understand that you can't change the way you teach the class for just one person," the more outspoken student said.

"True," I admited. "That's what makes it hard, hearing, as I do, from all of you all of the time. It's awfully hard to teach a course at any sort of pace when I don't want to leave anyone behind. On the other hand, though, I can take your perspective into account and use it, along with everyone else's, to come up with a sort of 'normalized' perspective. If I only hear from the people who are chugging on ahead, I can't help but think everyone's all right with the way things are going. I need to hear from folks like you."

I know it took courage for them to have that conversation with me, and I'm really impressed with that courage. I told them how much I appreciated their earnestness and forthrightness. I think that conversation, a difficult one for both sides, was a fruitful one.

Students, please remember this: there's no shame in taking a little more time to learn something than some of your peers. It's okay to be confused. If anything, there's shame in not owning up to your confusion in the first place.

I guess the moral of the story (by now a twice-told tale) is: if you've got a tale to tell, tell it. Someone will be willing to listen.

Bragging on my students is a full-time job

Before I scooted off to the 2010 Carolinas Writing Program Administrators conference (also known as "the best damned conference on the planet") at Wildacres the past few days I entrusted my Linear students with a task to perform in my absence on Monday, and all signs show that they pulled it off wonderfully.

Their job was to meet as a class, brainstorm topics from linear algebra which would be placed under broader "general headings" (also brainstormed), and assign themselves, in whatever way they felt effective and appropriate, to teams each of which would be tasked with writing a "section" of a review manual on one of the general headings decided upon earlier.

Apparently Ino, never one to let chaos cramp her style, enlisted Iris's help in leading the class through the brainstorming exercise, and they had it all done in short order. She later e-mailed me the list of headings, each with three or four students assigned to it.

I'm delighted that I was able to trust my students to meet without me and get the job done, and I'd like to think that the effort they showed in finishing this job was payback to me for showing them that trust in the first place. I think there's something to be said for the benefits that mutual trust can bestow both on teacher and on students. I trust this particular batch of students fully. They're great.

As I hinted above, I got back in town early this morning (in plenty of time for my 8:00 a.m. Calc I class) from CWPA. As ever, it was a fantastically productive experience. This year's get-together wisely eschewed rigid structure, offering instead ample opportunities for participants to meet in whatever groups and subgroups they felt they needed to.

I spent Tuesday morning talking assessment with folks from Elon, Mars Hill, and Charleston Southern. Our conversation helped me to tease out the issue at the root of what was puzzling me most about our current writing-intensive assessment plan. Namely, what do we do with the assessment data once we've got them? There must be more in store for them than a place in a forgotten file cabinet or a SACS reviewer's dossier. Yet for assessment data to mean much more, they must be highly esteemed by the faculty to whom they're given.

I realized about halfway through the day that faculty need to be helped to see the intrinsic value of both the assessment data and the assessment process itself. If faculty can learn to see assessment as a formative, reflective, and dialogic process involving many inter-interested parties, rather than as a summative, unidirectional, and punitive process instituted from above, they may come to understand its usefulness. As I said to a few of my colleagues at CWPA, it would be great if we could get them to the point where they'd say, "hot damn, data!"

I can dream, can't I?

Although I'd like to think that faculty development workshops can solve all problems, it would be great to come up with a more exciting means of changing faculty perceptions regarding assessment...any thoughts?

Speaking of workshops, I've got a great idea for what I think could be a fun faculty writing workshop, but I'll save that for another post. Now, it's bedtime: tomorrow's another long one.

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?

Week One: done!

Ugh.

In Linear Algebra today I used rampant tidal locking as an explanation for the fact that this past week has easily been the longest one ever.

It's been a long week, but a good one, and I've gotten a lot done. More importantly, I feel I've set the appropriate mood in all of my classes. The Calc I kids are slowly starting to come to life, and Linear's just getting better and better.

I've now had "get to know ya" meet 'n' greets with about 20 or 25 of my new students, and these have been uniformly fun. As always my students very in every conceivable way: age, prior education, academic background, major, motivations for study. I've got young, old, quiet, bold, math-minded, math-shy, self-directed, and diffuse. A couple share my love of poetry, one my fascination with early human evolution. Some are sarcastic, like Dwight, the student in my second Calc I section who showed both bravado and cojones by merely modifying the work I'd done on the board to solve an earlier problem, instead of transcribing his own solution from scratch. Others are retiring, like Bertie, whose description of greatest common factors was practically unintelligible owing to the quiet of his voice.

So far, aside from a little bit of hesitation on the part of a couple of the more outspoken students, all of the feedback I've gotten on the structure of the courses (both highly student-centered, Calc I to an extent I've not taught it in the past) has been positive. A couple of the Calc I students have even commented appreciatively on the use of writing. One thanked me for asking them to write in order to more deeply understand the difference between two oft-confused rules for exponents (when do you add 'em, when do you multiply 'em?): "I didn't do so well the last time around in calculus, but I feel like I'm going to do a lot better this time."

One of my nontraditional students (I'm always happy to have them in my classes: they're generally so much less shy about speaking up) expressed some concern in my meeting with her on Tuesday morning. "Am I going to be prepared for Calc II by the end of this semester?" I assured her that we'll do all of the computations we'd do in a theorem-driven calculus course, and we'll learn all of the concepts we'd learn in that setting...we'll just approach them all from a much more useful angle.

"Don't worry," I said. "We'll get there."

In fact, freed from having to put together a punctilious list of soon-forgotten limit laws, we're making good time: just a week in and we've already computed the slopes of several tangent lines, having figured out that that's what we need to do in order to solve the problem posed to us on Wednesday morning.

Yup, classes are going well.

In other news, I've finished the first draft of an introduction to my book. I'm quite satisfied with it. It charts my personal history with writing in the discipline of mathematics, highlighting the ways in which I've grown as a writing instructor as I've progressed in my career and indicating the empty spaces in the writing literature which need to be filled. Tomorrow I hope to start work on either the first or the fifth chapter (these are the two whose structure I can most clearly envision right now).

Now, bed calls. Students: please let me know what you thought of this first week. Where do we go from here?

Oh, and: 400th post!

Dialectical driver's ed

How's it going?

Three days into Calc I, and it 's difficult to say which way the class is going to break this term: the students are still strongly engaged in class and contributing well to the problem-centered classes, but I think a few are hesitant about the rather non-traditional format of the class. As yet we've not done a great deal of computation, haven't derived or delved into any formulas, haven't stated any theorems. A number of the students are used to doing all of these things within the first couple of days of a math class and so they've got a sort of deer-in-headlights, "WTF?" sort of look in their eyes.

The formulas will come soon, I'm certain, and I hope that with them a sense of familiarity and comfort: we're about to enter into a discussion of tangent lines and secant lines and their slopes, a discussion will include a great deal of computation and review (and practice) with algebra.

I'm not sure quite how quickly we'll get there: it really all depends on the pace at which the students end up driving the course. As I mentioned to my first section this morning, they're sitting in the driver's seat, working the pedals, holding the wheel. Every now and then, as instructor, I'll reach over and grab the wheel with one hand to steady it or add a few degrees to a particularly too-tight turn.

So far the structure of the course is "dialectical," an exercise in turn-taking: I've proposed an initial course heading, and now we've walked for a while. Now, at the end of the first day of hardcore hiking, having surveyed the site at which we've made camp, I've worked out an itinerary for our next day's travel together. We'll see where that takes us. We'll make camp again, make some next-day plans, and set out again in the morning. I know the stops we need to make, and we'll make them all, but it's up to the students to figure out which we hit when, and in what order.

We'll figure it out.

Meanwhile, I am loving Linear. The class is huge (the largest I've yet taught at UNCA), but has great cohesion. The students are eager to work and are making great progress, and I feel that I'm doing a much better job of structuring the discovery process than I did the last time I taught the class. I've put a lot more thought into precisely which skills are needed to solve which problems, and we're making a (long) beeline toward our first major goal: analyzing the long-term behavior of a simple Markov process (like the game we played on the first day).

Today we figured out how row operations mirror the operations needed to solve a system of linear equations. Next, after posing and solving a couple of realistic problems requiring the solution of a linear system, we'll motivate and move toward matrix multiplication. From there, it's on to matrix inverses, invertibility, determinants, eigenvalues, and, at last, an analysis of Markov processes like those we began with. At every stop we'll solve another real-world problem or two, get some practice with computation, and write a bit about what each concept really means.

I figure it should take us something like 5-7 weeks to get where we're going. After that, we'll talk about linear transformations, vector spaces, changes of basis, etc.

Exciting!

Meanwhile, I'm reading some fascinating works on writing and rhetoric in preparation for various parts of my book. Most recently I've begun Deirdre McCloskey's The rhetoric of economics (2nd edition, 1998, Madison: The University of Wisconsin Press), a delightful rhetorical analysis of economic discourse in which she dissects the ways in which economists convince one another and others of the truth of their assertions. According to McCloskey, much of what many economists consider unassailable scientific truth is really rhetorical "smoke and mirrors."

Not that there's anything wrong with "rhetoric," a word which McCloskey takes pains to save from those who would use it only in a pejorative sense: rhetoric is ubiquitous and unavoidable, and McCloskey merely asks that we be intentional with the rhetoric we use. We all use metaphors and other rhetorical devices; it's simply important that we be aware of the devices we employ. As one might put it, extending the tried-and-true "lens" metaphor, it's no crime to have poor eyesight, as long as you can admit that you need to put your glasses on in order to see properly.

Her opening chapter, which begins with a close reading of passages from canonical articles in economics, has made me think about the way in which I begin some of my own mathematical papers: what sense of authority am I conveying though my choice of words? What world am I building? What agent is acting? What am I asserting about the truth? I'd like to look into my own writing to do a careful reflective analysis.

But with all that I've got on my plate right now, such an analysis has got to wait for a little while. I hope to finish off the first draft of the introduction to More Than Numbers tomorrow, and set to work on the slightly slanted history of WAC, WID, and WTL which will make up much of Chapter 1.

Much to do, much to do, and so little time...but so much fun!

Day One, Part III

Well, Linear went wonderfully, too.

I am tremendously happy about the way this first day went, and in all three classes I think I owe the smoothness and success to several different sources:

1. The students (duh). They were almost without exception engaged, focused, and excited to be where they were. They clearly came ready to learn, and I hope I didn't disappoint.

2. I came ready to teach, not to recite a bunch of bureaucratic nonsense from the syllabus. I truly feel that the act of delaying the handout of the syllabus by 40 minutes (at the end of the first period instead of at the beginning) does more than any other single act to place the class's focus on the proper point (learning) and not on the improper ones (grades, lateness and attendance policies, etc.).

3. Student-centered, student-driven first day activities. Both courses began with activities whose pace and ultimate outcome were determined by the students. I provided an overarching structure, but the students led in the act of discovery. Of course, this is what happens all semester long in my classes, but it's important to do this on Day One. Do NOT put it off. The more you're able to get the students working together, helping each other, and sharing ideas on the board on the very first day of class, the more comfortable they'll be doing these things all semester long.

4. Knowing their names. It cannot be overstated: no single act more quickly earns the students' respect and admiration (by showing them respect and admiration) than learning their names as soon as you can, as many as possible on the first day.

It's going to be a great semester. Dare I say...best ever? We shall see.

Students: thank you for making today a wonderful one. (And a welcome to all of you who are joining one of my courses for the first time! Post a comment, if you'd like!)

Day One, Part II

Two down, one to go. My second section of Calc I was more wide awake, but a bit more unfocused. They did a great job in the "memory dump" of review topics from algebra, trig, and precalc, however.

It's going to be a good semester, that's for sure.

Now I've got to prepare for my 35-student (!) Linear Algebra I section.

On your marks, get set...

It's been a while, huh?

As you might suspect, I've had a lot going on, keeping me from posting here for the past, oh...two months? Ouch.

What's going on?

The REU's over, but the work has really just begun: the students this year were fantastic. They were smart, fun, and funny, and they worked tremendously hard, producing a ton of interesting mathematics. (We should get at least five papers out of it in the next couple of months, and maybe more beyond that in the follow-up.) And this year it was "buy eight weeks, get one free": six of the eight students ended up traveling to MathFest in Pittsburgh, PA to present on their summer research.

The REU took up much of my time, but I've had a number of other things going on, as well, among them: (1) working on student learning outcomes for both the Mathematics Department and the Writing Intensive program, (2) putting together the ILS Oversight Committee's report to the Academic Policy Committee (still not submitted, but finally almost finished), (3) planning my contribution to the short course I was helping run at MathFest, (4) working with my College of Charleston peeps on our paper on the rhetoric of mathematical writing, (5) looking ahead to my round table discussion of program assessment for this year's CWPA (coming up in a few weeks), (6) laying the groundwork for the book I'm now slated to write for Jossey-Bass (w00t), and (7) trying to get ready for my Fall classes, which start on Monday.

(1) Student learning outcomes: although I see the purpose, ultimately, of programmatic assessment, too highly institutionalized assessment tends to become bureaucratic and corporate. I don't want to say more about my role in this university-wide process this past summer other than that it was frustrating at times, had its minor joys, and though it was ultimately fulfilling in that I feel like some good will come of it, I worry about the intentions some people in administration may have for codifying the university's student learning outcomes as rigidly as is being done. 'Nuff said.

(2) The ILSOC report to the APC: this is more of the same. I see the purpose, and the writing of this document is a worthwhile activity, but one which runs the risk of being overly parliamentary and pro se. 'Nuff said.

(3) MathFest short course on Sage (an open-source computer algebra system): I actually wrote to my colleague Oscar (one of the co-organizers of the short course) in mid-July, a few weeks before the course was to take place, expressing my insecurities about putting together an hour or two of meaningful material.

"Nonsense," was the substance of his reply; "you'll do fine. Write about what you know." So I threw together several worksheets full of Sage code purporting to implement a number of higher-level graph theoretical algorithms and called it good. The worksheets actually went over really well in the short course and filled a rather comfy niche in the program. I felt good about what I'd done.

The moral of the story: just do it.

(MathFest, by the way, was a blast. Pittsburgh's a nice city with lots of pretty runs and nice architecture, the REU students were fun to hang out with, as always, and the MathFest program was replete with fun stuff to see and do. A good time had by all.)

(4) Math rhetoric: as I mentioned in my last post, oh so long ago, Bella and Damian came up from the coast to spend a couple of days with me, hammering out a plan to finish the paper we'd begun (with Nicola, who couldn't make it) on the rhetoric of the writing my REU students had produced in 2008 and 2009. They had a lovely time while here, visiting with and interviewing the REU students from this past year, working on our draft at the time, and just hanging out. I'm happy to say that as of last night we have a draft that's nearly ready to submit to the comp/rhet journal Across the Disciplines. It's the first of what we hope will be...well...at least more than one article on the rhetoric of mathematics and its learning, and the role writing plays in inducting students into the scholarly mathematical community.

(5) CWPA: I'm already looking forward to the Wildacres retreat! It's going to a be a full house this year, several dozen of us crowding into the nooks and niches of the Wildacres North Lodge to join in round table discussions of assessment with various forms and functions. I plan on presenting a brief "natural history" of the assessment that's gone on in our WI program during the past three years, exposing to public view the layers of rock comprising the "pilot" assessment Lulabelle headed up in AY 2007-8, the refinement she and I performed in the following year, the plans for assessment she and I laid out in the lead-up to the university's reaffirmation of accreditation (notice I haven't mentioned SACS yet in this post...), and the modifications those plans underwent as I limned the WI program's assessment of student learning outcomes this past summer.

It's sure to be riveting.

Actually, I'm more looking forward to hanging out with Damian and Bella, and with them planning the next phase of our study of the REU students' writing.

(6) Book: Oh yeah...I'm now under contract to write a text (working title: More Than Numbers) on the role of writing (specifically, writing to learn and writing in the disciplines) in mathematics and the mathematical sciences. Jossey-Bass received well the proposal I pitched to them several months back (in March, I think?), and after some fine-tuning of the project, I was brought on board. I'm excited, but at the same time terrified. I'm confident it'll all work out well, though. I've already done a ton of enlightening reading (if you get a chance, check out Scott L. Montgomery's The scientific voice...it's a fascinating read that, along with several other sources, has helped me put together a theory of the schizophrenic nature of undergraduate mathematical writing), and I have some great ideas. Furthermore, my consulting editor for the project is someone a few of whose books I've read and whom I respect very much; I'm sure she'll be very helpful to me.

I will say this, however: if between now and next March I start randomly babbling to you about reader response theory or the role of writing in the mathematical finance classroom, just smile and nod your head.

(7) Fall 2010: yup, we get underway in two days. Yikes. I'm actually really excited: I've got the first week mapped out in all three of my courses, and I've plotted out what I think are clear and navigable problem-centered paths for both of the lower-level courses (Calc I and Linear Algebra I) to follow.

Calc I is going to begin with a brief review, and then the students will work toward developing the skills needed to solve an optimization problem from economics. They'll need to "invent" the method of secants and tangents, the definition of a limit, and the definition of the derivative; and they'll need to learn how to compute some simple derivatives in order to solve this problem. Once all of that groundwork is laid, we'll spend some time working with the textbook and picking up some useful formulas for limits and derivatives, but everything we do is going to be application-driven and learner-centered. I hope to not have to lecture for more than 10 minutes at a time for the whole damned semester.

Linear's going to be the same: Day One features a variation on the "Markov Dance" that started the semester off the last time I taught this course, in the semi-mythical beginnings of this blog. The limiting behavior of this system is going to be the motivating problem for everything we do in the first half of the course, as we work our way through everything we'll need to do to get to the fun and useful stuff (eigenvalues, diagonalization, and Markov processes): solving linear systems, inversion of matrices, solvability criteria, and finally eigenvalues. I'm going to downplay theory and upplay (what a cool word!) functionality. I hope to get to the fun stuff by the sixth or seventh week, at which time we can spend almost all of our time on applications. It's going to be fun.

I should mention that I've modified my grading schemata somewhat, but not substantially, from last semester: students in both of the above courses will still have an opportunity to perform potentially unlimited revision on most things they hand in, but I'm tempering the grading scale so that it's no longer possible to earn full points back with each revision. I think the system I've set up is a generous and reasonable one, though, and I believe it'll still serve to motivate students to learn rather than to grub for a good grade.

Meanwhile, the Senior Seminar's speaker schedule is chocked full, with four visitors (all versed in speaking to undergraduate audiences) coming to share their knowledge and spend a little time with the students. It's going to be a good semester in that course.

Okay, that's all for now...I'm going to try to post more regularly during the coming semester, although I can't promise it'll be more than anxiety-tinged stream-of-consciousness rambling about this or that idea for the book.

Fare thee well!

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!

Spring Break!

Well. Yeah.

Here we are.

Here in the South (where they wouldn't know real winter weather if it smacked 'em upside the head with a two-foot blizzard and subzero wind chill) it snowed yesterday, and though there are two weeks left before spring actually begins, we're on Spring Break.

For me this means it's a good chance to get ahead in class prep, since March and April are going to be busy travel months for me (two conferences, two colloquia on the schedule so far), and I'm not going to want to fall behind while gallivanting about the eastern United States. It's also a good chance for me to post here for the first time in a looooooong time...

The funny thing is, I'm less busy this semester than I was last semester, despite the fact that I'm doing three preps this time around (and for the first time ever). Most of the slack is due to the fact that 365 is not one of the classes I'm teaching...I put so much of myself into that course, and I let so much of myself (including sense of self-worth, I fear I must say) get wrapped up in how well I pulled it off. Even when I was done getting ready for 365, I was never done worrying about it.

I have to say that I'm enjoying this semester a lot more than I did the last, perhaps to a large extent because I've managed to distance myself personally from my classes. That's not to say I'm not being myself in class, as I am, and it's not to say I don't care about the students, their learning, or their welfare. I mean only that I recognize that the success or failure of the class, however that might be measured from day to day and week to week, reflects in no way on me as a person.

And the funny thing is, I think I'm doing a much better job with both of my upper division classes this semester than I did with 365 in the fall. Things are running a bit more smoothly. I've found in both 280 and 368 a good balance between me standing at the front and yammering like a talking head and the students working the entirety of the class period with minimal direction from me.

368 is purring along particularly nicely. The text is fantastic, and eminently suitable for the manner in which I'm teaching the course. I've found that by distilling each chapter into a short worksheet I can ask the students to do most of the computations and the bulk of the simpler proofs, leaving me to stand off to the side to lend a hand on the trickier arguments as they arise. The presentations are rotating nicely in that class, and I've had no trouble convincing people to volunteer to present. Right now we're off-text, working our way through a couple of weeks of analytic number theory. We spent the last week on divisor sums and Dirichlet convolution, and the coming week brings an estimate of the average number of divisors for large numbers (a value that tends to ln(x) in the limit). After that we'll return to the text to do a little cryptography before heading off towards elliptic curves and more about Fermat's Last Theorem.

280's easin' on down the road, too. Freeing myself from a text was the right thing to do for this class, I believe. Though it's meant that I don't have a ready reference immediately at hand, it's also meant that I can run the course using the in-class worksheets without having to defer to a text that covers topics in just such an order, that provides at-best weak explanations or overly difficult exercises. I'm happy with the day-to-day goings-on, though people aren't nearly as eager to present HW solutions as the 368 students are. (I'm chalking that up to the relative mathematical inexperience of the 280 crowd; it's nothing unexpected, nor is it to be sorely lamented.) I'm very pleased with the improvement I'm seeing in a lot of the students' work. I'm thinking particularly of folks like Neville, Sylvester, and Una, a trio whose first homeworks were lackluster, but in whom there was definitely potential. In the past few assignments from them, I've seen much stronger structure, clearer arguments, cleaner logic, better use of notation. All three of them have put in long hours in the Math Lab, and it shows. Kudos! Of course, I'm getting stellar performance from people like Fiona and Elmer, folks I knew I could count on to do well. From everyone, there are struggles to be won, but I think overall we're at a good place to be by midsemester. Right now the order of business is combinatorics: combinations and permutation rule the day, and by the end of next week we should begin talking about functions and relations.

Oh, yeah, and the University's Writing Intensive Committee has ruled: 280 is now officially a WI course, from here on in!

Finally, I'd be remiss if I left unmentioned my Calc I class. They're a laid-back bunch, and after 280 and 368, which are often hectic and fast-paced, it's often nice to wind down the day with the 191 folks. We're moving a bit more slowly than I typically do, but I truly think it's the right thing to do. There are a number of people in this class who haven't had math for quite a while, who aren't so confident of their math skills as they might could be, and the slower pace is letting them absorb the material more meaningfully than if we were simply blazing through it. We're just now working our way through the Product and Quotient Rules, and by next week's end should be ready to consider some interesting applications.

So that's the score.

Big-picture-wise, I'm still waiting to find out whether or not I'll be getting this REU picked up for the coming summer...the fact that I've not yet heard could be a good thing. Ever the optimist am I. If we don't land that big fish, I'm going to try to rustle up a couple of research assistants for the summer so's I can have a crack at a couple of problems from geometric group theory I've had on the back burner for several months.