Moore ain't less

Back in January at the Boston Joint Mathematics Meetings my frolleague Stanislaus told me my name had come up in a conversation about plenary speakers for this year’s Legacy of R.L. Moore conference, an annual celebration of inquiry-based learning (IBL) sponsored by the Educational Advancement Foundation and the Mathematical Association of America. I was honored: this conference is well-known and reasonably high-profile. I wasn’t sure I was the best person for the job, though, for although I practice IBL in every course I teach, I generally do so in moderation. Only rarely do I use techniques that even closely approximate all-out Moore method, as I did this past semester in my two sections of Calc III. (Not having taught in Moore’s style for several years, I was a bit rusty at it, and I think the results were mixed.)

Nevertheless, I accepted the invitation.

I warned Stanislaus that I felt like something of a charlatan, for not only did I use Moore’s method infrequently, I had never even attended the conference before this past week. Stanislaus and others on the conference’s program committee reassured me and insisted that I might have something to say about inquiry and undergraduate research, something about which I do know a bit more.

So I set to work on my talk. It took me a while to decide how to pitch it. Should I focus on the act of research itself, and the role that inquiry plays therein, or should I try to tie research back to the classroom, where we’re more used to finding IBL more explicitly articulated? I settled on the latter approach, putting together an interactive presentation that would, I hoped, call attention to the parallel learning outcomes we encounter both in classroom teaching and in authentic disciplinary research, and highlighting the ways in which IBL helps us achieve those outcomes in whatever setting we might make them.

Early on Thursday afternoon, not an hour before my talk was scheduled to start, I was chatting with another frolleague, Ephigenia, whom I’d met during my postdoc at Illinois (she’d been a graduate student then). “I’m not sure I’m going to be saying anything new to anyone here,” I admitted. After all, I was smack-dab in the middle of IBL central. That research takes inquiry, and that research is in many ways no more than an extension of an interactive inquiry-based classroom, are not new ideas.

“This is sometimes a bit of a feel-good exercise,” Ephigenia assured me. (Boy, have I been a needy Nadine!) Makes sense: many of the folks at the conference are coming from colleges where no one else practices any sort of intentional IBL, and these folks need to find some kind of community. Hey, I’m not one to pooh-pooh the role that affect plays in teaching and learning.

I went ahead with my presentation, and as far as I can tell it was pretty uniformly well-received. It’s not likely that someone’s going to come up to my face and tell me that it sucked, but I had many tell me quite the opposite. I still don’t think I said anything new, though I hope it helped to give concrete examples of inquiry activities that don’t quite fit the Moore-shaped mold (the birdhouse exercise from last fall’s precalculus classes and the conversation on claw-free graph powers that took place between me and this year’s REU students about a week ago now). I might not be justified in feeling like a fake.

So now I’ve been to the Legacy conference. Will I go again? It’s good people, and I always like an excuse to get to Austin. But this time of year’s a busy conference season, and I’m pulled a hundred different ways these days.

We’ll see.

Shameful self-promotion

One of my editors, Bethany, has urged me to blog more frequently. So here I am.

I understand her point: she’s legitimately concerned that I might not be doing as much as I could to promote my work. Student writing in the quantitative disciplines: A guide for college faculty (Jossey-Bass, 2012) has been out for a few months now, and though I’ve every reason to believe that sales are quite good, they could probably be better, given more ambitious self-marketing. I’m not sure I feel up to this, though.

It’s not that I feel that self-promotion would be beneath me, or would constitute “selling out.” That attitude would be intellectually elitist and unbecoming. Believe me, I’m not against garnering a little fame (and a somewhat smaller fortune) from the book. There’s nothing wrong with showing a little pride in one’s work. It’s just that I’m not sure this blog is the appropriate venue for that self-promotion. Others are doing it, why can’t I?

Bethany mentioned the blog my good friend Erdrick writes, and the one that Maryellen Weimer, who helped me tremendously as my consulting editor for the book, updates regularly. Erdrick’s a wonderful colleague and a superb teacher, and his blog is superlatively good. Maryellen’s blog, too, is a wonderful periodical piece, and a wide-open window on current best practices in teaching at the university level. But I don’t think it’s fair to compare these blogs with my own; they serve different roles. I’ve never meant this space to be an intentional documentation of best practices, or a how-to manual on pedagogy. Though I don’t doubt my own excellence as a teacher (I’ve a great deal of evidence to suggest that I am quite accomplished as an educator), I have been, and I remain, reluctant to take on the task of systematically codifying my thoughts on teaching.

Rather, I’ve always thought of Change of Basis as a safe place to unpack my own teaching activities (and not, though they may frequently coincide, best practices in teaching more generally) in real time, keeping tabs on what goes on in my classroom, in my REU, at my school, in my mind. It’s more made up of notes-to-self than it is directives-to-others. Though I may cite books on teaching, I don’t do so as a careful and intentional review of the literature, but rather as an indicator of what I happen to be reading at any given time. Though I might bring out all the buzzwords (problem-based learning, inquiry-based learning, Moore method, writing across the disciplines, writing-to-learn, etc.), I don’t treat them methodically but only as they come up in my own work. How else to put it? I try to teach by example and not rote lecture. My tone is more anecdotal and less comprehensive and directive. It’s “I tried this trick out, and it worked out well” and not “studies show that this trick will reach students most effectively.”

I mentioned to Bethany that one of the reasons I’d not updated lately was that I’d not had much time lately to write. Between near-constant travel to present at conferences, seminars, colloquia, and faculty development workshops; leadership on the Curriculum Review Task Force (a full-time job in itself lately), preparations for the REU (now done with its second day), several ongoing research projects (in both math and composition and rhetoric), assumption of the Honors Program directorship (my administration began officially a few days ago), and teaching a full load of courses, I’ve not had the time I once had to dedicate to this blog…and when I have the time, it’s often directed into other writing projects (notably, 3x30 and poetry).

Honestly, I’m too busy being a dedicated educator to write about being a dedicated educator.

It’s thus that I offer my apologies to you, Bethany. I’m sorry I’m not posting as often as I might, and that my posts aren’t as pointed or focused as they might be. Please know that I wish I had the time and energy to post once, twice, thrice a week, offering some digestible and downloadable 750-to-1000 words of wisdom each time. Please know that I’m not angry with you for asking more of me, and that I do understand, and appreciate, your concern. This just ain’t that kind of blog.

That said, please consider giving Student writing a read. I’m proud of it, and I feel that it’s a very good book. I feel very strongly that writing has much to offer to students and scholars in the quantitative disciplines, and that we do well to pay attention to writing’s potential. I will walk the Earth from end to end to say so, again and again. If you’d like to talk about it, let me know.

Bounce

In a recent post I fretted a bit about one of my sections of Calc III, which section seemed to me a bit underprepared for class this past Monday. I worried that their apparent lackadaisicalness (if that is indeed a word) regarding Problem Set 4, on which we were working in class on Monday would lead them to be unready for today's class, in which they would be presenting their solutions.

I stand corrected. That section bounced back, showing themselves up to the challenge. Every single student called on to present did so, and did so with aplomb. I was particularly impressed by Dionne's willingness to work all of the way through the dreaded #61, which asked for a proof that two non-parallel vectors in the plane span the entire plane. Dionne, one of our promising young majors, has some exposure to linear algebra and is currently enrolled in Foundations, so she's no stranger to the proof genre. With a little help from a couple of her colleagues, she beasted that problem.

Yes, they bounced back, but not before I exhorted them to keep up with their work outside of class. Don't just come ready for the problem you think you'll be presenting (padded with the one or two preceding problems for insurance); come ready to present any one of them...and ready yourself as soon as you can so that when you're offered time in class to hash out the details, you can do so without delay.

Good work, everyone! I have to admit to a bit of nervousness at running my first Moore-method class in four or five years, but so far you're all making the most of it. Thank you for that, and for all that you do.

Feedback, as ever, is appreciated.

Worth a repost

This morning I received a brief but touching comment on my most recent blog post: "I miss Patrick teaching." I responded to this anonymous post in a manner which I repost here because I think it's worth wider readership:

Please know that I'm organizing this class in a non-traditional manner not because I want to avoid "teaching" (though, believe me, I'm doing as much teaching, in a non-traditional sense, as I would in any other course), but because I truly feel that the Moore method is the best way to approach this material. By asking you all to explain your ideas to one another, it firms up your understanding of those ideas. By asking you to take responsibility for your work, you become the authors (quite literally) of the ideas you're presenting to one another. It's much more learner-centered, and ultimately (I believe, and the literature on pedagogy bears me out) more effective.

Thank you for your kind sentiment! I've not totally disappeared from the scene; as you've noticed, I hope, I'll take my turn "on stage" from time to time.

To elaborate briefly: I know I'm a good lecturer, and I know that I explain things well. But seeing something done and doing it yourself are two different things, and you stand to gain much more from actually solving the problems yourself and explaining your solutions to each other than you do listening to me do it for you. It's a bit more work on your part, to be sure, but the time you spend on that work will be time well spent. Meanwhile, please know I'm still doing a lot of work behind the scenes, arranging problems in a manner I think is effective to help you work your way through the new ideas, including the definitions and theorems I think are most critical to us in our work, and working with you in class as you develop your solutions.

This I promise you: my explanations are still here for you if you need them, and I will be delighted to help you work your way through any problem you might struggle with. All I'm asking is that you give it all you've got to come up with solutions on your own first. Believe me, you'll get much more out of it that way.

So, let's stay the course, y'all. I'm enjoying class so far, and so far you're doing a marvelous job. Keep it up!

A tale of two sections

Today it was evident that over the weekend, most folks in one of my sections of Calc III took the time to work through the problem set they'll be presenting on Wednesday. It was equally evident that most folks in the other section didn't.

We'll see how things go on Wednesday. Most of the problems are pretty straightforward, but there are a few that might give pause. If Wednesday goes as I suspect it might, a few folks might learn the hard way that though it always pays (no matter the class) to keep on top of the work, but in a course structured as ours is, it pays double.

Moore is more

Three weeks into the semester, my Moore-method Calc III class has made it through three problem sets (50 problems), treating a substantive review of topics from Calc I and II and five or six sections of the textbook. It's been a few years since I've taught a course in this fashion, so there's been a bit of adjustment as I've gotten back into it.

So far, so good. The students are getting much better at explaining their solutions in front of a large audience (one section has 27 students, and the other 35), and they're becoming more relaxed, visibly. Yesterday's second section was particularly laid back, assiduously focused on finishing their tasks but willing to joke around and have fun in order to set the solvers at ease.

I've been very impressed with students' ability to be wrong in front of each other, and similarly impressed with the audience's willingness to ask questions. They're getting better at asking each other for clarification or elaboration, and not turning to me to ask. I'm letting minor errors slide, perhaps adding a little "does everyone agree?" if the solver's slipped up somewhere. Generally this has been enough to prompt one or two to express disagreement.

How's it helping the students? Hard to say. Several have said they get a lot of the course's design, though one or two have admitted "it's not what I'm used to, and I'm having a hard time adjusting." I've reminded them a couple of times now that in this sort of course they're expected to take on a bit more responsibility than they might in a more traditional course, preparing well and keeping up without my continual exhortation for them to do so.

I'm going to poll them more formally on the course structure at the end of the coming week, after we finish off the fourth set of problems. We'll see where we are.

Meanwhile, if anyone in the class is reading this and would like to comment, please feel free to do so, anonymously if you'd like.

Swimming!

Two class-days into the semester, and things are going swimmingly. I'm putting the course together with a modified Moore method, cycling (roughly) through the following steps:

1. handing out problem sets,
2. giving the students time in and outside of class to work through solutions in groups,
3. asking the students to present their solutions in class,
4. asking the students to write up solutions to selected problems as homework, and
5. quizzes the students on completed problem sets.

As of now I've met with the first section of the course twice and the second section once, but every meeting has been lively and energetic, and I've needed absolutely no coaxing to get students to work in groups and to come to the board. The atmosphere in the course has been supportive and friendly. I'm delighted that this morning's class gave us an opportunity to note that, as I'd promised, no lightning bolts came down from the sky when a student made a mistake at the board.

It's been a long time since I taught course using anything close to the Moore method (my special topics course in graph theory, run in Spring 2008), and I notice that I've grown considerably as a teacher since then. I'm more confident, and that confidence has enabled me to feel less awkward taking a peripheral role. In particular, I find that I'm much more able to sit in silence than I was in the past. Silence in a crowded classroom is disconcerting, and it's all one can do to keep from saying something after ten or twelve seconds of quiet have elapsed. I've grown accustomed to such silences, though, just as I've grown accustomed to (or, more accurately, enamored of) the thrumming of three dozen voices trading tricks as the students work in groups in class.

I'm confident. It's going to be a good semester.

On deck

It's a new year, and we're only a few days away from a new semester (beginning Monday, January 9th). There are big, big things in the works: my book comes out in a couple of months, the work of the Curriculum Review Task Force should be coming to a head this term (with concrete recommendations to the Faculty Senate due by April), and...well...other news items about which I'll be able to say more in a few weeks' time.

What's in store, teaching-wise? I've got three preps this term, one for two (large) sections of a course I've not taught in almost six years (Calculus III), one for a single (small) section of a course I've never taught (a Masters of Liberal Arts [MLA] course on the cognitive psychology behind mathematics), and one for my section (of two) of our senior seminar.

In the seven years I've been at UNCA we've not taught two concurrent sections of that last class, so I'm not sure exactly how we're going to manage it. I've yet to talk to my colleague Timon, who'll be teaching the other section. I imagine we might hold many activities together, splitting when it comes time for the students to present. There's simply no way we'll get through 26 student presentations in the five or six weeks we can offer them to speak. We'd have to have an unprecedented four talks per period to make it work, and that's simply unworkable. Thus splitting into separate sections for that part of the course, though not ideal, is about the best we'll be able to do.

Meanwhile, I've got plans for the other courses. Calc III, which I've not taught since Summer 2006 (the last summer I wasn't running the REU), I'll be running with a modified Moore method: one day each week will be devoted to discussion of new definitions and discoveries, a second day to small group work on the current problem set, and the third to problem presentations. Both sections of this class are big enough (roughly 30 students apiece) that I'll probably ask students to "present" simultaneously (two to four at a time) whenever feasible. This'll ensure that we make it through problem sets in a somewhat timely fashion, and that each individual student gets more opportunities to present. I've already worked with about half of the students in both sections, which familiarity will help me ease into the new term.

I can't say the same for my MLA course. It's been nearly a decade since I taught a graduate-level course (a special topics course on Coxeter groups and related groups which I led at UIUC back in Spring 2004), and this course differs dramatically from that one. We'll be exploring the learning and cognition of mathematics, and I plan to inject a good deal of philosophy and sociology into the mix as well, drawing on a number of sources to paint a picture of mathematics most people never see. We'll begin with Stanislas Dehaene's marvelous book Number sense: How the mind creates mathematics, about which I've blogged a bit before (see the "Dehaene" tag at the right), surveying the psychology of mathematical discovery, before moving onto Imre Lakatos's Proof and refutation, a philosophical treatise designed to lay bare the workings of what might be called the "mathematical method."

I'm not sure what to expect from this course. I suspect there'll be a week or two of me feeling out the students (currently there are seven students enrolled) to see where their interests and aptitudes lie. Likely none of them are straight-up mathematicians; I'll be curious to learn what they're hoping to get from the class, and I'm certain they'll help me give it more direction.

Ah, well...one week to go. Before then I'm off to Boston for this year's JMM, at which several UNCA students (and a few past REU students) are presenting. I'm particularly excited to see how far Ned's and Ino's work on nutritional data has come since their presentation at Kennesaw State in November. (They're presenting in a special session on mathematics and sustainability.)

Further bulletins as events warrant, likely soon.

State of mind

A few months ago Zima, one of my grad school colleagues who now teaches at Kennesaw State University in Kennesaw, Georgia asked me to present at an undergraduate research conference for which she'd just received MAA funding. (Said conference is this coming weekend; it's the one Ino and Ned are presenting their findings at.) I'll be giving a run-of-the-mill plenary talk on some of the graph theory I did with a couple of REU students this past summer, and I'll be presenting in a workshop on inquiry-based learning (IBL) at the outset of the conference.

I offered Zima a title that's so generic I really could talk about anything: "Guided discovery in the mathematics classroom." I feel confined by this generality. Indeed, when I actually sat down a week or so ago to try to figure out what in the hell I needed to say about IBL, PBL (problem-based learning), Moore method, etc., I had a hard time coming up with much to say other than expressing my feeling that all too often these techniques are too "formalized." That is, I get the sense sometimes that the people who apply these techniques look on them as an all-or-nothing process: "if it ain't straight-up Moore method, it ain't anything at all" or "I use guided discovery every single day to address every one of my students' learning outcomes." So I put together a half-hour laundry list of things to say along these lines: be open to using guided discovery in moderation; it's not the be-all-end-all any more than any other pedagogical paradigm may be.

Then just now, while lying in bed unable to sleep (though admittedly probably needing to), I realized that I can say more, for I realized of a sudden why I've had such a hard time trying to come up with something practical (and original...I suspect that the folks I'll be addressing in this workshop are going to make up a choir to whom I won't really need to preach) to say about guided discovery: in my mind, guided discovery is not so much a pedagogical process as it is a state of mind.

I find more and more that in teaching it's not so much what I do with my students as how I do it that matters most. Put another, perhaps more practical way, effective teaching comprises a gestalt-like complex of actions and not a single action individually. Guided discovery is what might be called an emergent operation which cannot be broken down into its constituent parts without losing much of its energy and effectiveness. So it is that I don't necessarily engage my students in singular activities, each of which forces students to lead themselves to original, new-to-them, conclusions, so much as I try to treat them in every way, in everything I do, as co-learners, co-discoverers, seekers of authentic knowledge.

Practically, this realization makes it possible to grow opportunities for genuine discovery in the most infertile academic soil, for every simple textbook problem becomes, if viewed from the right angle (like anamorphic art) a chance for authentic "research-like" engagement. Guided discovery is an "angle" from which these problems may be viewed.

I'll try to say a bit about this on Friday when I'm leading my portion of the IBL workshop. Are these views original? Meh...perhaps not. But they're more original, and, more important, more meaningful, than whatever else I will have to say. We'll see how they're received.

More face time

"I don't know if you knew this," said one of my Abstract II students to me just moments ago, "but I'm planning on being a teacher."

"Excellent!"

"Yeah, so...the more face time as we can get in front of class, the better."

"Excellent!"

"I know I griped a lot about the way graph theory was taught, but that was right around the time I was beginning to think about teaching."

I hope more students appreciate Moore's method!

Chapter 5. Blueprint

My first exposure to Moore method instruction came during my first year of graduate school at Vanderbilt University, back in 1998. My graduate-level topology course was taught using the Moore method, and as there were only four of us in the class, our going was painful: often every one of us would be called on to present a problem on every single day.

Of the four of us, two had seen a good deal of advanced theoretical mathematics already: I came having finished a Masters degree and having taken just about every math class offered at my undergraduate institution, and my colleague Oleg came from a European nation where his undergraduate training had been much more focused and intense. We had a decided advantage on our less-experienced colleagues. Erdrick and Tara were fresh out of American undergraduate programs and had had little more than the standard course of study one would find in such a program: both were smart, but relatively fresh.

As one might expect Oleg and I had a far easier time than our two friends: we came upon our solutions more quickly and we had an easier time presenting them. Although this advantage dwindled and then disappeared almost entirely by the time we got to algebraic topology, a topic none of us had dealt with before, for a few months Oleg and I were able to set the cruise control and drift along breezily while our friends struggled.

But about three months into the Fall 1998 semester, during one of Erdrick's more protracted proofs, I noticed something striking: his presentations were richer than mine were, they involved themselves with more details, and they provided a thicker description of the underpinning concepts and computations than my own presentations did. Moreover, they were clearer, and contained a more cohesive and coherent argument and were therefore more well composed. Even as he professed his lack of understanding of this or that topic (often one with which Oleg and I had dealt in earlier classes), it was evident that he was taking the time to dig more deeply into said topic, uprooting it and holding it up to the light to better see it and better get at its meaning. And as he held a topic up for his own examination, so he held it up for all of us to see and analyze and understand.

His relative lack of knowledge was making him a better teacher than I could ever be.

If one is able to unashamedly confront one's ignorance in front of others, one is acting in the manner of an effective educator: teaching is little more than a practiced and public examination and remediation of ignorance.

Fast-forward ten years, nearly to the day.

While I was talking on the phone with Griselda a few weeks ago, the conversation turned (surprise, surprise) to our teaching and our classes and our colleagues and our colleagues' teaching of their classes. Not shockingly, we've both had colleagues with whose methods we don't agree, and about whose teaching we've heard students say decidedly negative things.

"Isn't it funny," Griselda said, "that when students complain about a teacher not being very good or not being able to explain something at all, they take that to mean that that teacher is really smart? 'I can't understand Professor X at all...she's really smart, but I can't understand her lectures.'"

"I know!" I agreed. "The assumption is that if you can't explain something it means that you're operating at such a high level that you're unable to 'bring it down' to the level of the students."

We all know the stereotype of the absentminded mathematician whose dwelling in the realm of functions and formulas precludes any meaningful interface with the world around him. This hapless researcher wanders about the "real world" clumsily, muttering apologies to those he jostles and jabs with his elbows, even as he crunches numbers ceaselessly in his head. This professor may have difficulty in "bringing it down" to the level of his students, simply because he's often incapable of understanding just how little his students understand. (Incidentally, you'll find this type of teacher most often at research-intensive universities where the publish or perish mentality ensures faculty need not give a rat's patoot about teaching, and where even if good teaching isn't actively discouraged it's certainly neither actively encouraged.)

There's a different sort of delinquent educator, though, and one that's much more common at schools like my own: the one who has become detached from her own discipline to the point where she's no longer engaged in active research; who has ceased the active pursuit of new knowledge in her discipline and so has lost the ability to discern, to analyze, to interrogate the stuff that is the quintessence of her field; for whom academic inquiry has become static and devoid of new and novel points of view...for this one teaching will be an arduous and sometimes insurmountable task, as she will find it difficult to show her students anything more than an unchanging road map. If a new and more convenient highway is built, she may stubbornly drive on down the old routes as though unaware of the new one's existence. Who knows what sights she and her students might miss?

For this reason I believe that the highly active researcher who values teaching above all else will make the most effective educator. Let research dominate and one risks becoming an ivory tower-dwelling hermit; let research die away and one risks becoming a theorem-spewing robot stuck on autopilot.

This is an oversimplification, to be sure, and as there are exceptions to every rule so we each will be able to identify examples that defy the schema I've laid out above. Nevertheless, the rule is proved by its exceptions, and time and again I've noticed that my colleagues who excel in the classroom are generally those who excel as researchers in their respective fields.

The above observations suggest the following blueprint for teaching excellence:

An excellent teacher is one who

1. is unafraid of professing ignorance and who can confidently confront that ignorance publicly with

2. a rich skill set developed in the course of active disciplinary inquiry, and who

3. values her students' understanding above all else.

Let's see how well this blueprint poses a solution to the following question: "How can someone not trained in writing instruction provide effective writing instruction to his students?"

Following the blueprint mechanically in lockstep, were I to wish to teach my students how to write well, I would

1. admit to myself that I don't know a whole lot about teaching writing and be honest with my students about that fact,

2. take it upon myself to learn what I can about writing (in my discipline, or to learn, or to meet some other end) and the teaching of writing, engaging in the creation of new ideas on writing should the need arise to do so, and

3. keep in mind that the ultimate goal of this procedure is not to publish a paper on writing but to instill in my students a greater understanding of writing so that they may make use of the tools I've helped myself develop and so that they too can help themselves and each other to develop their own tools.

Note how process trumps product, particularly in this last point.

Does this seem a satisfactory solution to the problem I posed above?

I think so.

I have to admit that I'm really thinking out loud here, but it seems to me that the best teacher is a courageous learner, and that as each of us is fully capable of learning should we dare to, so too each of us fully capable of teaching.

To the point I'd originally meant to address, at the risk of belaboring it: any person well-trained to perform in her discipline (whatever "performance" may there mean) will be well-trained to instruct her students in that performance, so long as she keeps her own performance skills sharp. As "performance" in nearly every discipline involves the creation of a textual record of some sort, any able and active disciplinary performer will be well-trained to teach her students to write about her discipline. Indeed, who could do a better job than she could? That is, who could better teach math writing than a mathematician who is skilled in the use of writing to perform mathematics?

Put this way, the whole question of "qualifications" becomes a silly one on its face, and the idea of "writing in the disciplines" makes a hell of a lot of sense.

(As an aside I might note that perhaps the theory above could be used to prove qualifications to perform other disciplinary duties as well: who better than a sociologist to train students to use statistics as they would be used in a sociological setting? Who better than a chemist to train students in the physics that describe the commonest molecular interactions that occur in a chemical lab? How far off-base am I here?)

Perhaps one reason "qualifications" came to be questioned in the first place is that all too often we hold to very narrow and circumscribed notions of "writing" and "writing instruction" in the first place. If we demand that all academic writing take the form of five-paragraph essays whose construction is governed by a rigid set of rules, we might then consign all writing instruction to the realms ruled by composition theorists and rhetoricians.

But not all writing, not even all academic writing, takes this form. There are many more modes of writing than there are disciplines in which one can write, and we forget this at our own peril: one would never write a policy paper where a lab report is required, and a poem will generally not do for a mathematical proof.

Nevertheless, it's not always easy to remember writing's rich diversity. Even the "experts" may forget. I'll have more to say about this in this series's next chapter as I ponder the steps and missteps my colleagues and I took as we attempted to design a rubric for assessing students' mastery of writing in a particular discipline.

For now, as ever, I invite your feedback on my rambling thoughts. I came further than I'd originally hoped to in this post, and I'd really be interested in hearing others weigh in.

Leaps and bounds

Two days in, and going strong.

We've now wrapped up two days of intensive seminar-style introductions to graph theory and group theory (to say nothing of mountains of paperwork, campus tours, and bureaucratic snafus of various orders of magnitude), and the students seem to be taking it just fine.

Example: after three and a half hours in which we plowed through four weeks' worth of graph theory (we worked off of modified versions of my Moore-method notes for last semester's 473 course...the out-of-class homework problems adapted well to their new niche of in-class exercises), their "homework" was to strike out on their own and track down definitions, examples, and applications pertaining to 30 different graph theoretical concepts. The kids came through in spades, spending the first four and a half hours of our working time together in taking turns presenting the material they'd come up with. Although I suspect there was a good deal of division of labor (certain students "claimed" certain problems and broke up the workload along clearly demarcated lines) it was just as clear that many of the topics had been multiply-researched, and thoroughly, at that.

They kept each other honest, too: they weren't shy about comparing divergent definitions, asking questions to reconcile apparent contradictions, demanding clarification on points that weren't so obvious at first. One of the students is particularly bold about asking for elaboration, it's marvelous to have her there since her boldness and the elaboration it requires are no doubt leading to greater understanding on everyone's part, including my own. My thanks to you, I hope you know who you are!

Indeed, few of the students are shy about taking part. Two or three are ever eager to strut their stuff on the board, and a few more are just as comfortable in directing the action from the cozy quarters of their desks. A couple are quieter than the others and are therefore harder to read, but I have a hunch they're all following along pretty well. (Two of the students have yet to take an abstract algebra course and today's dessert course included and introduction to geometric group theory, so those two folks might have felt a bit pinched at the end. I hope they roll with the punches and persevere, I know they're both highly intelligent and will weather this first week's storm if they stay the course.)

I've got very high hopes for this group of young mathematicians. They're already top-notch scholars, and I hope we can effectively take them to the next level.

In other news, I've just received the copy of Oulipo: a primer of potential literature (Warren Motte, trans. and ed., Dalkey Archive Press: Champaign, IL & London, 2007) I ordered. Oulipo is a French acronym, standing for Ouvroir de Littérature Potentielle, a consortium of artists (most French) who in 1960 began a movement dedicated to the production of rule-based works of literature. The generative rules that govern the poems they create are highly mathematical in nature, and any serious study of math and poetry demands that Oulipo's work be considered.

Meanwhile I just this morning sent "interview" questions to seven of my students from last Fall's Calc I classes, asking them to reflect not only on the poems they created for that course, but also on the process that led to the poems, and on the thoughts and feelings that governed that process.

I have to admit I'm not entirely sure what I'm going to do with the data I glean from these interviews. I think I'll only be able to find that out once I've got the student responses in front of me.

For now, I'll put down my pen and away myself to bed. It's late, and I've got another long day that begins early tomorrow (at 8:45 in the OneCard office, where I must meet with our Excellent Eight to sort out a minor matter involving their inability to check out library materials on their cards).

Feel free to let me know if you're out there!

Overdue notices

I'm overdue to post (so saith a few faithful readers, including my mother-in-law, my wife, and two of my most diligent students). Enough, already! So here I am, I'm writing.

I've been here, I've been busy. The last few weeks have been exciting ones, and next week promises its own supply of hecticness (hecticity?).

What's new, pedagogically speaking?

Well, yesterday was the deadline for applicants to this year's REU. This afternoon I counted up the number of distinct applicants from whom I've received at least one document (application, statement of purpose, transcript, or letter of recommendation): 64. Not bad. That's about what we had last year, though I believe the ratio of men to women is higher this year than last. I've yet to break it down geographically, but I think this year's pool is more widely distributed throughout the country than last year's. I'll probably begin looking at the applications in earnest over Spring Break. No sooner: next week's going to be a bear. Tuesday brings a colleague from Davidson College to campus (hello, Twyla! thanks in advance for making it out! and sorry again for the confusion in scheduling!) for the Research Seminar. Then comes the Math Literacy Summit we've been planning for several months, highlighted by the public lecture and keynote address given by Dr. Robert P. Moses. That's Wednesday night and Thursday day. Saturday sees the start-up of Super Saturday once more (I'll see if I can enlist some student helpers...the downside is the onset of Spring Break, taking many students away from campus). By Saturday afternoon I'll be beat.

What else is new?

This past Friday my Graph Theory gurus had their first exam, an in-class ditty that represented my first attempt in almost two years at writing an in-class exam for an upper-division course. It's known far and wide that I'm not a big fan of such a format for seminar courses, and it was hard for me to write an exam that I felt made fair demands of the students' knowledge and was doable within the alloted time. The end result, I feel, was a hair (and no more!) on the long side, and a tad too easy. (I'd be interested in knowing how the students feel about both of those appraisals.) As it is, the students went down to the wire time-wise, several staying for an extra ten minutes to finish up, while the average was high, around 86%. Maybe I didn't ask for enough proofs? Maybe I was a little light in grading? I don't know. The only question that seemed regularly to ensnare the unsuspecting was the problem asking the students to compute the number of homomorphisms from the star with 3 vertices to the star with 4 vertices: there was a broad array of answers to that question.

How's the day-to-day activity in that course been? The students' presentations of solutions are getting tighter, more succinct, more precise. For the most part their diagrams are more descriptive and intuitive than they'd been during the first few weeks, and their proofs are more straightforward and understandable. Most interesting are the differences in presentation styles between the various members of the class. Some are silent until all has been written on the board; their presentations then consist of little more than "voilà! Pas de lacune a remplir!" Some are so verbose they can barely put chalk to board to draw a single tittle without prefacing it with a megillah of mathematical exposition. I wonder to what extent they find others' presentations are affecting the style of their own? (This sounds like a perfect question for a mid-term evaluation!)

We've gotten to the point where the students have a grasp of the basics, I can probably branch off in whatever direction I'd like to in terms of the ground material. The most recent problem sheet (the seventh, available here), deals primarily with components, paths, and chromatic polynomials. I believe I'll make trees the focus of the next sheet, unless someone has a better plan. Lorelei mentioned the other day to me that she'd like to see more applications, so I'll do what I can to work those in (colorability has many applications, and they'll soon be ready to take off in that direction). Markus came to me on Thursday, indicating that his relatively light schedule is granting him plenty of time to take on some independent study in graph theory, so I gave him some reading on graceful labelings, maybe he can join Trixie and Sieglinde in their pursuit for new graceful trees.

Speaking of which, we'll soon have our strongest showing at an MAA Southeast Sectional meeting since my arrival at UNCA: at least five faculty members and four students have indicated interest in going to the meeting, and I'm trying to get three of these students (including Trixie) to present in the student poster session.

And speaking of Trixie, how's Calc II? They too completed their first exam this past week, and overall the grade distribution was pretty fair, with a course average of about 76% after corrections were made. Oddly enough, though, there was a profound difference between the two sections of the course: the first section's post-correction average was about 68%, the second's around 86%. I kid you not. After corrections 12 out of 16 students in the second section got either an A or a B (8 As, 4 Bs), while only 10 of 30 students in the first section earned that bragging right.

What, as they say, is up?

Could it be class size? Admittedly I find it much easier to engage the second section as a whole and as individuals, owing largely to the fact that it's got about half as many students. Moreover, the students are less intimidated by speaking up in front of one another, and by presenting on the board. They're also much less reluctant to get into groups and work on problems together. Whether any of this has anything to do with the size of the class, I don't know, but I can't help but think that class size plays at least a small role. (Incidentally, I'm happy to report that I'll be teaching two sections of Abstract Algebra I next fall, each considerably smaller than the traditional single sections that have been run in the past. Our program has been so successful in courting majors that we're having to run two sections of the upper-division courses. America gonef! [You'll have to excuse the Yiddishisms, I've been making my way through Leo Rosten's delightful Hooray for Yiddish: a book about English. I'm easily influenced].)

Could it be the time of day? But one might think that the sluggish 9:00 a.m. class would be more ideally situated in that regard than the post-lunch but usually-punchy 12:45 p.m. class. Or maybe I just think it that way because I'm a morning person. Perhaps there's something to it: the morning section's usually slow-to-rile and torpid, the afternoon section's much more get-up-and-go.

Could it be...the luck of the draw? I've got great students in both sections, but they're just more highly concentrated in that smaller second section. Maybe it's just coincidence that the second section's so much more lively.

Whatever the cause, the difference between the two sections is as that between night and day. I love both of them, but I find myself often wishing wishing wishing that the first section would wake up and stop dragging its heels!

What else?

Faculty talks have ended in the Senior Seminar, I capped them off with a presentation this past Wednesday, on open problems in graph theory. I'm proud of the fact that we had three speakers from off campus, and that we'll soon have at least two more visitors coming to speak in the Research Seminar. I truly believe that our department should attempt to cultivate a more active research environment, and I think we're well on our way towards achieving that goal.

Student talks begin in the second week after Spring Break, two-by-two they'll fill up the last six weeks of class. I'm looking forward to those talks, the topics look to interesting.

What else?

The Writing Intensive committee (well, technically it's a subcommittee, but who's keeping track?) has sprung back into life, continuing our analysis of WI applications and beginning our conversation on the assessment of the success of already-WIed courses. This is a sticky wicket of a tricky schtick: How are we to judge whether a WI course has met the goals it laid out for its students? What materials must we demand of the course instructors in order to perform a proper assessment? How many of a course's learning goals relating to writing must be met in order to call the course a success? And if a course is less than entirely successful, what consequences do we as a committee mete out? It's unrealistic to aim for the ideal right out of the gate (assuming the ideal can be articulated from the outset anyway): all but the perfect instructor is going to stumble here and there, and no course is flawless in design and execution. Therefore it's pointless to pull someone's WI away should perfection not be attained. We don't want to smite those who fail in providing this or that element of their class's purported learning experience. Instead, we wish to encourage the instructor to look carefully at her course's goals, to look at the students' products in attempting to meet those goals, and say, "this was done well. But this, when asked of the students, proved unrealistic. Better I ask that they reach for the moon with their hands at their sides!"

How many of us are so reflective? I'd like to say that I am, but who am I to say?

In the first of what will be several meetings of the writing assessment project this past week (didn't I say I've been busy?) I told my colleagues that last semester's 280 course taught me to be truly conscious of the role played by writing in my own particular discipline...indeed, I think I learned more in that class than my students did. My approach to writing as a tool for learning has changed because of that class.

Writing is playing less of a role in my Calc II class this semester than it did in either of last semester's classes, and while I've not shone a spotlight on writing in Graph Theory, it's ever present. (The work I've done in 280 over the past year is most evident in the structure of the students' proofs on the blackboard: I'm thrilled whenever I see clear statements of hypotheses, an explicit indication of proof technique, summarizing sentences that indicate when and why a proof has been concluded, and so forth. In all only two or three of that class's students didn't have 280 with me, and almost daily I see elements of my own idiosyncratic style that have rubbed off on them.) I'm going to take a few minutes during this coming week to refocus the students' attention on writing and encourage them to keep an eye on the criteria for solid mathematical writing as they put together their solutions to the problems selected for written submission.

What else?

Um...hmm...giving a recruitment spiel on the upcoming REU and speaking ongoing graph theory research to a wonderfully receptive and warmly inviting crowd at Morehead State University in Kentucky (y'all were great, thank you so much for having me!), writing about a dozen or so REU rec letters for current and old students, joining a couple of colleagues in a presentation to the university's Foundation Board, agreeing to help organize this coming May's Writing Intensive workshop, and shaking off a nasty cold that took me out of commission for a few days...see? There's a reason I've not been around!

If you'll now excuse me, it's Maggie's birthday (which one, I will not say, though I doubt she'd mind if I did, she's not embarrassed by her birthdays), and we've got to go celebrate it in the manner of her choosing.

As usual, all comments, questions, queries, suggestions, insinuations, epiphanies, innuendi, graffiti, scritti politti, revelations, retorts, ripostes, and recriminations are welcome on the comments page.

Until next time, live well, and try to learn something new today.

Oh, hey!

Hey, sorry I've not checked in for a bit!

I think about writing, I really do.

And then something else gets my attention. Some small fire pops up and needs putting out, someone comes by with a ten-minute diversion, or I just say to myself, "gee, I'd like to finish reading that Singer story I started this morning before the sun came up."

Some of my favorite of his stories involve the framing device wherein a motley crew of wayfarers, scholars, beggars, etc., find themselves holed up in a snowbound Hasidic study house somewhere in semirural fin-de-siècle Poland. There's a coziness to those tales, an intimacy, that makes them more believable, more real than they already are. You get the sense from that device that Singer himself told that tale by the flickering light of tallow candles, or at least overheard the story as it fell from the mouth of some unnamed wanderer who spoke of the spirit who haunted his second wife and caused her to suffer horribly and cavort wildly and brought her (and him with her) to shame in the eyes of his town's most devout Jews.

But I digress.

I've meant to say that we've done away with the soccer ball (mercifully!), and the last few classes of 473 have recovered much more of that sense of excitement with which the semester began. People have been better in not speaking out of turn, though, and it's led to more polite exchanges with less cross-talk and more consideration for others' rights to have a say. All in all, it's been an improvement.

One our class's quietest students led us off with the very first presentation after the soccer ball's eternal banishment, and it made for fifteen minutes of silence as he very meticulously wrote most of his proof (of the fact that the subgraph relation is an order relation) on the board before explaining it. (I can't help but think that a week before, there would have been a half-dozen interruptions during this time, by onlookers eager to offer their 34.5 cents on the problem's solution, but all of us did a remarkable job of sitting on our hands and biting our tongues.) The explanation was solid, and though not quite complete it was almost entirely correct. One or two others interjected helpful suggestions to move the proof to the finish line. It took about half the class, and it made for some tense moments, but it was well executed.

On Wednesday Joachim "solved" the first of the "review and discussion" problems I've begun adding to the problem sheets, at the suggestion of one of the students. These problems ask the solver to recap the definitions, theorems, and examples considered in the given problem sheet, providing the class with a "where are we now?" moment. I think these'll be useful in focusing the class's attention on the highlights and reminding them of key definitions and results.

I'd like to see the students improve their ability to interpret definitions; there was a bit of confusion over the definition of "bounded degree" on Problem Sheet 4. Or has it been that I've not made the definitions as clear as I might have? It's likely a combination of the two, we could all stand to do a little better. I have to remind myself that (a) I'm not writing to my research peers when I write these definitions, and (b) I'm not going to take extraordinary pains to describe these definitions to the students in person; it's up to them to interpret, draw examples for themselves, understand. I'm happy to help them over the hump if they come to me with questions, but I expect them to make the effort alone to understand a definition and apply it properly. After all, one of the learning objectives of this course asks the students to develop an ability to read a mathematical article and interpret and understand it, alone. I'd like for them to be able to read a fairly low-level math paper unassisted by the end of the semester, and that'll more often than not entail wading through a few new definitions on their own.

Nevertheless, I've got to insist on absolute clarity on my own part. I'm going to pay special attention to my definitions from now on, to make sure they're clear as crystal. Students, if they're not, please call me on it!

Meanwhile, Calc II has been chooglin' along. My morning section is a soporific one, but the early afternoon section, a bit smaller, is more lively, more engaged. I've only lost one student from that second section from the start of the semester, and two from the morning section. We're in the middle of methods of integration right now, about two weeks away from the first exam of the semester. So far the students have been really good about getting homework in, with only a few stragglers. Aside from a couple of folks whom I've carried over from last semester who look like they're crusin' for a losin', most everyone's eager to do well, a phenomenon that's a welcome change from Calc I, in which there are always a handful of folks who don't really give a rat's ass and are just drifting along until the end of the semester.

News flash, by the way: I found out that I'll be teaching Precalc (!), of all things, this coming Fall, along with two sections of Abstract Algebra. Woo hoo! This'll be the first time I'll have taught Precalc ever, and the first time I'll have taught Algebra since coming here. I'm excited on both counts.

The time has come for me to say adieu, as I must away to dinner in Greenville with our grad school buddy who now teaches at Furman U.

Farewell, and have a wonderful weekend, what remains of it!

Vox populi

The students have spoken!

Some of them, anyway. I thought I'd post the feedback I've received so far on this last Friday's class. The core issue is class structure: soccer ball or no? A few students brought up residual issues, but this single one remained at the center.

Saith one:

I find the best form of government is a benevolent dictatorship. Think about that premise and I think you'll be able to maintain an iron fist over the class without compromising the spirited environment.

In response to this point of view said I in an e-mail: "Point taken, kindly. I'll put my iron fist in a velvet glove and see how things go."

This student's take was echoed by a colleague:

I definitely felt like my thoughts were being forced to remain in my head, during Friday's class. I do agree that maybe some of those thoughts might be better off there, but what's most disappointing about that feeling is that it felt like much of the passion and the fun that existed in the first two class periods was sucked out of the experience.

I think that the soccer ball should be eradicated from the world of Graph Theory 473. What should be put in it's place is some self awareness, and some consideration on the part of those who are speaking, a few rules giving the 'presenters' more authority while they are in front of the class, and possibly a comment from the prof. when things start to get a little out of hand.

Man, I just went back and read your suggestions and ideas for next class. [See the excerpted e-mail from my previous post.] I didn't realize that you had already written the same ones that I did. Oops. Oh well, that's how I feel.

One student waxed a bit more philosophical:

Even though it was painful, I am glad we had the class we did on Friday. For me, two big ideas came out of what occurred during class. The first is, that in setting up time to ensure that we build a solid foundation for what we are learning, I believe that we are avoiding some long term pitfalls that might only have come up in the last weeks of the class. I feel like what really came out of the horse that we beat to death from problem sheet #2 was that carefulness leads to the deeper meaning that we are trying to glean. I know that there are Algebra and Calculus ideas that I learned only well enough for testing purposes and now wish that I understood more principally. I like the idea the idea of the review problem and the carefulness in answering problems.

Second, I think some good things came out of the 'structure discussion.' While I do not like the soccer ball, I do like the metaphorical one. I don't think that things were out of hand before Friday, in fact I love this class. I do think that we might not have had this discussion until things were out of hand though, and that would have been more painful. I like the 'sitting in a circle' idea. I believe that half of what we were concerned about will be fixed by that one addition. Also, because our Friday talk was not based in failure but in improvement, I think it gave us an early opportunity to be a little more conscious about how we do want to shape the class. I would guess that most of us gave that a little more thought after the discussion. So I guess the second 'big idea' for me was that awareness that came out of Friday. How could that hurt us!

Ultimately, some kind of order is necessary, as acknowledged by the folks above, and by our last commenter:

My thought is that there definitely needs to be some type of system, because the first few days it did get kind of crazy and was hard to really understand people's thoughts. But I also think we are all college students and should be able to respect one another enough to listen when they talk and then put in our thoughts (without interrupting). And people should be able to ask questions!

The plan from here: let's abandon that awful soccer ball, let's grant the presenter the authority to open or close the discussion, let's allow for open conversation while discussion is "on," with the understanding that that requires respect for one others' points of view and rights to have a say as well, and let's let me have another go at better moderating the discussion should the need for moderation arise (see my "velvet-coated iron fist").

Friday was painful for me, too. As the second commentator above pointed out, it felt a lot less fun, and when it comes to research (let's face it, folks, we're doin' research here) fun shouldn't be undervalued. I want this class to be fun and engaging, and I think we can manage that without the help of the soccer ball.

I'm glad we've had this conversation, I think it's helped us all to understand better what it takes to make up a healthy learning environment, and I'm glad it's happened in Week 2 instead of Week 11, allowing us twelve more weeks of organized, respectful, blissful interaction!

I'll check in again tomorrow and let you know how things go down.

Chaos

I've got mixed feelings about today.

Calc II felt all right: the first section was a little sleepy, but the second was more engaged and seemed to enjoy the sugar fix provided by the shortcakes used to illustrate the method of cylindrical shells.

Graph Theory?

Hmmm...

After Wednesday's class got a bit rowdy, with lots of cross-talk, overdubbing, interruptions, and just plain ol' mayhem, we decided that maybe we ought to try out a means of directing the discussion. I suggested the possibility of getting a small plush object to toss around: she/he with the ball was the one who got to speak. It seemed a bit puerile, but I didn't want discussion to get out of hand, lest people start zoning out, not understanding what's going on, what's being said, what's being proven.

So Theodoric brought in a plushy soccer ball, and we tried it out.

The atmosphere was...

...well, to me it seemed a bit dead. I'm not sure the deadness was completely unwelcome, but I don't want to kill off the natural enthusiasm that folks are having for the course.

I think some people had difficulty getting the attention of whoever it was who had the ball at any given time, others felt like they weren't going to stoop to "playing the game" of getting the ball before speaking and so said nothing...for the most part we stuck to the plan and didn't speak until given the ball...but it felt stilted, juvenile.

Having to choose between lively and occasionally cacophonous debate on important mathematical topics and stultifying silence, I'll take the debate, even if it means a little chaos every now and then. As I put it to the students in an e-mail exhorting them to write to me and let me know how they felt (their comments will be posted here as they trickle in):

My own two bits, for what it's worth: we've got a room full of 16ish smart, eager people, and I know that there are time when we've all got something to say. I want to keep the class lively and the discussion excited, but I also don't want it to descend into utter chaos. I'm not doing my job well if I let the class devolve into a kindergarten class. That said, if people are overwhelmingly for it, I'll be open to trying to use the soccer ball again (thanks for bringing it, Theodoric, by the way), but my feeling is that (a) you're all mature enough to not interrupt one another and to not crack jokes when other people are trying to explain something, and (b) I can do a better job at moderating discussion should it need moderation. I'd like to come in on Monday and try to go without the soccer ball, we'll let the person at the board lead the discussion (opening it up once he/she is through presenting), and if things begin to get rowdy, I'll exercise my authority and rein it in.

We'll see what they have to say.

Mathematically, we finished off a single problem today, proving that a subgraph of a simple graph is also simple. Our proof, built up in bits and pieces, was ultimately a careful one. One person starting things off with an intuitive explanation, a second tag-teamed with a more solid justification, and a third stepped in to nail it down with some clear notation. The result was a pretty clean proof, and I'm glad we took the time to make it rigorous. Remember, folks: I'd like you to be able to understand these theorems, but you should also be able to prove them.

That's all for now. I'll post student comments on the Great Soccer Ball Fiacso of 2008 as those comments come in.

One down, a whole bunch to go

Day One.

A difficult beginning?

Not really.

The day was tiring, but pleasant.

I felt uncharacteristically (for a first-day-of-semester) comfortable in my first section of Calc II, and hardly more perturbed in my second section. There was a bit more nervousness in Graph Theory, but overall my ordinary first-day jitters subsided quickly.

As I suspected would be the case, I had a hard time getting to sleep last night, and once asleep I couldn't stay asleep. More than once I awoke to find the room still black in deep night. The early morning hours dragged, and I swore that the six hours or so hours I'd allotted myself were among the longest I've ever lived, wide-eyed and ceiling-staring.

My alarm went off at 5:30, and I was strangely refreshed. I showered quickly, had breakfast, and headed out, puzzling over various schemes for constructing expander graphs in my head as I walked into campus.

It was just growing light as I arrived, and it was near enough to the start of the first class period (not mine, this semester) for the earliest of the students to be poking their overly-punctual heads into their 8:00 classrooms as I entered Rhoades Hall.

The next hour or so was spent putting together a few odds and ends I'd need for my first Calc II course of the day; organizing my notes, syllabi, handouts; putting a couple of finishing touches on the website; placing the Skittle-filled candy machine in the Math Lab; responding to a few early-morning e-mails.

Then came class. By the afternoon's end there'd be 32 people in the class, only 9 of whom I've not had the pleasure of working with before. (Belladonna, chagrined, pointed out that there are only 6 women in the section. I mentioned that there's often a precipitous drop-off from Calc I to Calc II, that women more heavily populate the biological sciences than the mathematical ones.) Class seemed to go rather smoothly, despite the massive amount of crap I wanted to get through by the first day's end. After an exercise in which I asked the students to compile a list of dos and don'ts when constructing a safe and effective learning environment, and after a brief review of the essentials from Calc I, I left them with the assignment sheet for Confectionary Conundrum.

Trixie and I had a chance to catch up after class and spend a few minutes talking about the graph theory she'd been working on over break. One of her friends, a fresh face to me, lingered too, and she took a few minutes to explain to him, very well, the problems she's been considering. She's made great progress, and if she manages to push it much further, I don't think a presentation at MIGHTY would be out of her reach.

The next few hours saw me doing all manner of busywork, and by 12:45 it was time for Round Two, duked out with a smaller section consisting of a mere 18 students (half of whom are women!). The smaller section is sure to make for a more intimate environment, and already I can tell that people are more comfortable speaking up in front of each other than are the folks in Section 1. I have high hopes!

Graph Theory, having shrunk by a student, now has 16 students. After obligatory welcomes and niceties, I explained to the students the structure of the course: they'll be in charge, hands on the wheel and the feet on the gas, directing the flow and the pace of the course. I'll be there with a road map if they need it, but I'll try to keep it tucked away in the back of the glove compartment, beneath a pile of oil change receipts and a pack of 10-year-old once-minty chewing gum. Their presentations of problem solutions will dominate class time, and through their work with one another I hope that they will learn to become colleagues in discovery. (For a complete description of the "Moore method" means I'll be utilizing, please consult the syllabus.)

I apologize for the highly simplified, blow-by-blow account of the day's proceedings. Honestly, I don't have much left in me to make the day sound any more poetical than a pedestrian succession of events. It was a good day (superlative, as first-days-of-semesters go), I'm glad I've lived it, and I look forward to many more like it this semester.

I'm just beat.

Well, Calc II continues tomorrow, I'd best be off to get some R 'n' R before beddie-bye.

Giassou!

Problematic

The last couple of days (as the coming couple of weeks will also) have seen preparation for my courses this coming semester. While Calc II is something I could do with my eyes closed, I'm spending a bit more time in getting ready for Graph Theory.

I spent a few hours yesterday afternoon putting together the first problem sheet for that class.. It asks for a nice mix of examples and proofs, all interspersed throughout a series of definitions that flesh out the basics of graphs. I'm quite pleased with it. It's got a little more than one question per person currently registered for the course, so I hope that everyone will have a chance to get involved through this first set of exercises. I don't think it should take us more than the first week or so to get through it. I'm going to get started on the second sheet soon, but it won't be made public immediately; I have a hunch I'll want to make adjustments once I see how the first sheet goes over. I may, after all, find that I'm aiming too high, or too low, or expecting too much or too little...the students may want to take the class in a totally different direction. We'll see how it goes.

After tinkering with various ideas for that class's exam structure, I've decided to just play it by ear and ask the students to help me design the exams when the time comes: they'll help in the construction of the tests, and in their organization.

Tidbits: I'll also be asking each of the students to read and present on at least one mathematics research paper, and I'm toying with the idea of giving a small amount of extra credit for learning and using LaTeX in the preparation of written homework problems.

Besides this, nothing much to report, teaching-wise.

Further bulletins as events warrant.

Very busy

That was the "learn Chinese" phrase printed on the fortune cookie fortune I got a few nights back at the Asiana Grand Buffet.

Fitting!

As was the flipside, the fortune itself: "It's tempting to make promises, but can you fulfill them all?"

It's as though the cookie knew to whom it was going.

A few days back the man in charge of planning for next year's Conference on Diversity sent an e-mail to the members of the "Ideals" Subcommittee (on which I am), indicating that the subcommittee is still without a chair; he was fishing for volunteers. My finger hovered over the mouse button, itching to hit "reply," but I thought to myself, "my wife will kill me. This is no exaggeration: she will kill me."

My finger stayed.

She should be proud.

Soon after posting yesterday, I finished grading the Calc exams. They did quite well. The average between the two classes was around 78%, and very few failed. There was one 100% (yay!), one 99%, and one 98%, and a smattering of slightly lower As. Most were Bs and Cs, with a few stray Ds and only a very small handful of Fs. I'm pleased.

I had a chance to finish writing responses to the second section's poems as well. As I did with the first section's, I've saved a number of these, hoping to receive leave to post them.

Yup, the semester's coming to a close. Tomorrow's the due date for the two final exams still out (both for independent study courses) and the final paper (for the third independent study I'm directing), and tomorrow from 11:30 to 2:00 we have the last of the 280 presentations. I'm excited to know that there will be pie, provided by one of the team members deriving a continued radical expression for pi. By tomorrow evening, I should be done.

Then what?

I've had a good semester. I'd give myself a B+/A- for Calc I: high marks for ingenuity, originality, classroom skills, and general derring-do, tempered with a slightly lower grade for just...well...you know, I feel like I never really clicked with a number of the students in my class, especially several in the second section. I think I'm probably overanalyzing and expecting too much of myself, but I feel as though there was a bridgeable disconnect somewhere...we'll see what kind of grade the students give me.

I'm giving myself an A for 280 this semester, I think I outdid last semester's performance. I rather like the decision to use homework committees, I think that came off rather well. I believe my conscious, directed efforts to improve mathematical writing worked wonderfully, and I'm particularly proud of the first day's assignment, as well as the Professional Proof Analysis. We had several truly exciting days of class, on which we simply said Screw it! and dove headlong into relatively unrelated mathematical ideas that struck our fancy.

All in all, the class has been great.

I've enjoyed my independent studies this semester, too. Nicodemus is fantastic at providing himself with direction, so I've hardly had to think about where to take him as he found his own way through our Abstract Algebra II course. And Bethesda's independent study on The History of Math Technology has come along nicely, too, though in retrospect two of the topics on which she's produced papers have dealt more with philosophy than technology, ultimately: her first discussed the advantages of a duodecimal (base-12) system over a decimal one, and her last (of which a final draft is due tomorrow) deals with the validity of computer-based proofs. She's really come along in her writing, I've enjoyed working with her. Felicity's managed to eke out a fine independent study, too, coordinating homework and exams over e-mail, scanning and sending me her completed problems in Linear Algebra II. I'm amazed at the ease with which I've been able to juggle all of them, their hard work has made my job an easy one.

No research students next semester (unless Sieglinde and Trixie make any major breakthroughs in graph labeling over the winter break), but I've got that one reading course in lattice theory, and I'll be working with three of my favorite blasts from the past on their 480 presentations (Fiona's let me know she's already had nightmares about her presentation).

I've got to start putting together my Moore-method notes for graph theory, and figure out a rubric and grading schema. My wonderful friend Griselda and her colleague Natalia have been kind enough to share their respective course materials for discovery-method "proofs" courses they've run at their institution. I'm sure those will serve as handy guides.

I'm going to bring this post to an end.

Coming this week (I promise promise promise): Newton v. Leibniz, in the students' words, and a bit of mathematical verse.

A few days late: what's up?

I promised another entry last Saturday, now that Saturday's come and gone, and beyond it another week, come and gone. Promises, promises...

Last Saturday brought my first Super Saturday of the season, in which I and three of my stalwart students spent an hour and a half (thanks Deidre, Belladonna, Sieglinde!) with seven little 9ish-year-old munchkins, teaching them the ins and outs of binary arithmetic, and how it can be used to make hard-to-break codes.

This lesson I used last semester with the Spring installment of Super Saturday. The kids then were on average a year or two older and a bit quicker out of the chute: I remember that it took only a few minutes to run through binary arithmetic, almost all of them had seen it before and knew all about it already. In this new group only one of the seven had seen it before, and he was willing enough to let the others catch up. We spent about a half hour learning to count by twos, and then another forty minutes or so running through an example of the code. Time ran out while they were working through their own examples, but I think most of them were getting the hang of it in the end.

Later that day...I finished grading for the weekend sometime in the middle of Saturday afternoon, and left teaching alone until Sunday night, when Griselda and I talked on the phone for about four hours, the bulk of our conversation, as usual, about our respective classes, colleagues, and institutions. She gave me an encouraging word regarding my plans for a Moore-method graph theory course next semester (verbatim: "Do it! Do it!"), as I'd asked her to do several times by e-mail.

Then came Monday, ushering in a hellish week of work, work, work. I was able to get ahead in my class prep on Monday morning, whipping together several note sets, worksheets, and handouts for both of my courses. I got everything set through to this coming Monday in 280 and through Wednesday in Calc I, freeing me to work on other things, like grading 280 homework sets and papers (oh, by the way, here's a link to a compilation I made of what I felt were the most insightful comments in their Professional Proof Analysis papers), helping out with Who Wants to be a Math Major?, putting on Part II of my Research Seminar talk, meeting with several advisees, putting some finishing touches on the grant proposal I'd be bringing with me to Fayetteville for the NSF Day held there on Friday, getting exams ready for my Calc kids' Thursday thrill, touching base with all three of my independent study students, and hauling ass eastward for a less-than-24-hour whirlwind tour of the drizzly and depressingly drab city of Fayetteville. Tons of ideas took shape there, and on the way my highly experienced colleague Quimby giving me plenty of food for thought. On the road there and back, he was able to help me sort out and solidify several promising ideas for future initiatives in both teaching and research. I was also able to meet up with folks at nearby colleges whom I'd not met before. Here's a big fat CoB shout out to Winslow and Queshia, and to Bonnie!

Last night I staggered, overdressed, into Robinson Hall at the ungodly hour of 10:00 p.m., having spent little more than 24 hours away and not having been home since the morning before. Strangely enough, the third floor hall lights were on, the Math Lab door was open. My 280 students Quincy and Olivia were there, working away on their homework from my class. (It's no wonder they're doing so well...) After calling Maggie to bring the chariot round and pick me up, I unlocked the installation blocker on one of the Math Lab computers and downloaded a workable LaTeX editor for Olivia, something I'd promised to do for her long ago. She was, to understate the matter, happy.

Super Saturday returned this morning, six kids, four student helpers, all from the second section of my Calc class. For an hour and a half we worked away on fractals. After an easy definition and a few simple examples, I brought out the by-now-beloved L-tiles. The elementary-school kids were ecstatic when they beat the college kids in creating the L₂ from four L₁s, and it took little more time before both teams worked together to build first several separate L₄s, and from these an L₈. Next came a music video built upon the Mandelbrot set and a round of free-form fractal building, finished off with a level-2 Sierpinski pyramid. I think we all had fun this morning, but I was most heartened by the turnaround shown by the class's youngest child: Boudica began the morning too terrified to even sit next to the class's other young girl, and she spent much of the period working by herself. While working on her own fractal, she sat next to my Calc I student Henrietta, a warm and friendly young woman (whom I've been lobbying heavily regarding a math major...she's clearly passionate about the subject) who was able to put Boudica at ease. By the time her parents came to pick her up, she was hell-bent on putting together a few more components for our communal pyramid, and Boudica didn't want to leave. "I want to come back next week," she said. "This is awesome!"

I spent the next hour doing a bit of grading, and after lunch with Maggie at Noi's Thai Kitchen (mmmmm...spring rolls...), I got a couple more hours in. And (you'll be amazed by this!) after dinner came...more grading! I'm finally done grading all of the Calc kids' exams, and about a third of the way through the homework.

Whew.

Damn.

I'm tired.

The last couple weeks have both been 90-hour weeks.

And you know what?

I'm pissed.

Weeeeeeell...not pissed. Just disappointed.

Why's that? Let me answer with a question.

What's up, Calcsters? What happened?

Let's just look at the numbers: the course average (both sections) on Thursday's exam was just a hair under 68%. While 9 out of 54 people taking the exam got As (including a 99 and, yes, a single 100) and another 8 got Bs, there were also nearly as many Fs. There weren't many low Fs, but a few too many Fs overall to let me sleep easily tonight. The performance was perceptibly bad: on Thursday afternoon, Neville, easily one of my first section's best students, offered me a humble apology as he handed in his homework. I nearly melted.

"I wanted to apologize for my performance on this morning's test."

"Neville, it's okay. We all have our off days."

"I just know I did really bad. I've never done that bad on a math test before."

"Did you feel that it was too hard of an exam?"

"No! That's the thing! I knew how to do all of the problems, I just looked at it and went blank."

"Even so, there's no need to apologize."

"I just didn't want you to think any less of me."

"If there's one thing a good teacher can't afford to do, it's let his students' performance affect the way he feels about them personally."

So I ask again, what happened? I agree with Neville: I don't think the exam was too hard. It was hard, to be sure, but not too hard. Moreover, it was very similar to the practice exam, and some of the exam problems people did worst on are ones dealing with topics through which we spent a great deal of time working (like related rates).

As I started grading the homework just a few hours ago, I got a clue as to what might have gone wrong: it's clear that nearly none of the students did any of this most recent homework set before taking the exam. Though I assigned the related rates problem set early last week, early enough so that I was able to offer those who wanted it the opportunity to turn in the homework a week early and get feedback on it for study purposes, most of the students didn't get started on it and the two accompanying homework sets (exponential and logarithmic models, and linear density) until the wee hours in the lead-up to the exam, or later still, after the exam was over.

I really hate to use this word this term, y'all, but there's one word for that sort of academic behavior, and that word is stupid.

Settle down, settle down: I'm not calling you stupid, I'm just saying you made a stupid move. There's just no other way to put it.

You simply cannot expect to well on a difficult exam if you've not done the homework for the sections with which the exam deals by the time you've taken the exam. (One or two of the brighter students might be able to get away with this kind of crap, but they're living on the edge. Don't follow their reckless example.)

I'm feeling disappointed right now. I'm feeling disappointed, helpless, and, to be frank, a little hurt.

Disappointed: you're smart kids. You wouldn't be here if you weren't. UNCA's a tough school to get in to. You're the cream of the crop, in many ways. You're smart cookies. Every last one of you has the potential to do well in my class, and I'm disappointed that many of you are clearly not doing what's needed to wrestle down and master some of the concepts we're dealing with in this class. You can do it, that's what frustrates me: you can do it, but you're choosing not to.

Helpless: what else can I do? I sometimes feel like I've done all that I can. I work my rear end off each week, forcing me to ask myself whether I need to invoke the tired old aphorism and work smarter, not harder...but how so? I pride myself in my ability to blend traditional teaching methods with more innovative ones; what better way to make most effective use of the time given to me? I troll the Math Lab, helping out any students I come across in my travels. I offer up nearly endless office hours, my open-door policy allows students to drop in and chat just about whenever they want to. I put together practice exams and solutions, similar, as I always say, to the actual exams in length, content, and format. I give review sessions, in some of which are solved problems that will appear verbatim on the next day's test. I do a lot, and moreover, a bit immodestly, I must say that I think I do it well. Some might say very well.

Mother of Claude, what more can I do?

Thus, helpless.

Hurt: why? As I told Neville, it's unwise for a teacher to let his students' performance affect the way he feels about them personally...moreover, I shouldn't let my students' performance affect the way I feel about myself. But I do, if only a little bit. I feel hurt because I fear a disheartening answer to the question, "do they just not care?" And I want so much for you to care. I want you to care, as much or more than I want you to succeed. I don't feel that I've succeeded in guiding you through my course unless you not only understand, but you also share at least some of the excitement that I feel for what I do. See, I very truly believe that mathematics is a beautiful thing, I care deeply about it, and I bring my enthusiasm about it with me to the classroom. My excitement isn't an act, it's genuine, and I hope that excitement rubs off on you, if only a little.

Where's that leave me? Where's that leave us?

Look, y'all: I know the score. As a rule, you're young, you're full of life. You're free, many of you for the first time in your lives. Most of you are being pulled in a zillion and a half different directions, and you're trying to figure your lives out. I understand that it's hard to manage your time, your space, your resources, in some fashion that allows you to finish everything you need to do in your day in time for you to plop your head restfully down on your pillow before the next morning comes.

Yeah, you've got a long list of things to do.

And guess what? My class, and the work it comes with, are on that list.

You signed up for it, you've fought through the semester this far, we've got about five weeks left. Let's stick with it, all right?

So where do we go from here? Here's a short to-do list:

1. Do the exam revisions. Seriously. Do them. Even for those of you who scored in the 40s, it shouldn't take you more than a couple of hours. Sit down, get out your book, find yourself a quiet hour or two in the library or your dorm room or your apartment, and do the revisions. Do them neatly, fully, correctly. If you all can manage half credit back, you can bring the class average on the exam back up to a more-than-respectable-in-fact-pretty-damned-good 84%. As you do the revisions, try to understand how you messed up when you did. Keep in mind that I allow revisions not because I want you to get a high score on the exam (that's a happy bonus), but because I feel that your understanding of the ideas we've talked about comes before everything else. That being the case, I want you to take this chance to grapple with the concepts that may have eluded you before.

2. I've said it before, I'll now say it again, and I hope that maybe with those pretty sad-lookin' exam grades standing right behind me, you'll all take me seriously this time: do the homework. Seriously. Do it. Get started on it early. "Early" does not mean "on the evening before the homework is due." Rather, "early" means "right when the homework is assigned." That night. Set aside a half hour or an hour to get started. Don't feel the need to do it all at once, unless you find yourself having fun and cruising right through it. Do a few problems, enough to get the hang of what you're doing. If you get stuck, get as far as you can, write down whatever ideas you've got for solving the problem, and move on. Make a note of your difficulties so you can catch me after class with them, or bring them to me or to the Math Lab folks later on. Work on the homework a little at a time. Don't get frustrated. If it's just no fun anymore, put it down for a few hours and come back to it later. I don't assign all that much of it; almost without exception the work I give you in any week should be doable within three or four hours all told, and if you spread that out over the week you shouldn't have more than a half hour or an hour a day.

Hey, if you want to be where the action is, go to the Math Lab, hang out there, do your work there. You'll find a lot of your friends in the Math Lab: a lot of you already come and spend a good deal of time there, so you won't be alone. There's absolutely no stigma attached to hanging out there. It's a free service, they've got cheap coffee, cheap tea, and often a ready supply of free food. Smart people roam the Math Lab. They're paid to help you. It's a good place to be.

All that I ask is that if you make use of some of the Math Lab's resources, like the solutions manuals, don't abuse them. That is, don't allow yourself to become fully reliant on them, don't use them as crutches. A few of you, I can tell, "complete" your homework by copying the solutions from this book. You might not mean to, you may have every intention of doing the work yourself. But you come on in, you camp out in front of the manual, and you start to work. Here's how it might go:

"Okay, let's get started! Problem number 17...all right, here we go. Okay, I know how to get this started...there's f(x)=sin(x)...okay, now let's see...hmmm...what's next?...[15 seconds later]...I'm stuck...let me just peek real quick here...oh, yeah! Duh! Okay...now...yeah. Hmmm...I know this...[10 seconds later]...just another peek. Zomg! I knew that! I'm almost done now, I can do this in my head...write it down...and...there! Okay, let me just check my answer...[peeks]...wait, how'd they get that?...oh, yeah, I see...[erases]...that's what I meant. Duh. Okay, next one, number 22..."

How can you effectively make use of the manual? Use it only to check your final answer. If you didn't get the right answer, walk away from the manual and see if you can find your mistake yourself: look back over your work. Does the problem present itself? It might. Many errors are easy to catch yourself. If you can't find anything wrong, flag down the Math Lab assistant, elbow your friend working next to you, or come across the hall to my office. Ask someone else to take a look; generally another set of eyes, even if it belongs to someone less mathematically adept than you are, will do a good job of spotting something amiss. If you find a mistake, revise. Give it another go. Only go back to the solutions manual once you've got a revised answer.

By the way, I can tell if you're relying too heavily on the solutions manual when you complete your homework. (I'm not a fool, however well I play one in the classroom.) If you're addicted to the manual, you're only hurting yourself, since it'll nearly always kick you in the pants come exam-time. Here's a tip: a high percentage (over half) of the people who are nailing the exams to the wall ("nailing" being defined as "getting As"), including the one person who's received the highest scores on both of the exams so far, are doing sub-optimally on the homework. They're not doing horribly, but they're not doing perfectly either.

You wanna know why that is? Because they're actually doing the work. They're actually (horrors!) making mistakes. Because even though they're really smart, they're also really human. They're making honest mistakes, and they're giving themselves the chance to learn from their mistakes. Their homework isn't perfect because they're legitimately trying, and not just copying the answers from a book, however innocently that act of copying might be.

And by the way, if I sound cliché in offering up all of this tired old advice, I apologize. I'm only shoveling all of this shit because it happens to be true. Believe me, I don't wanna sound like an old-fart fogey, any more than you want to hear me sitting here lecturing you. I wouldn't say any of this if it weren't true, and if I didn't care.

3. Come and see me if you're having difficulties. I'm not a scary guy, I'm probably one of the most approachable professors on campus. (I'm a bit of a nerd, but I can't really help that. At least I recognize that fact, and I revel in my nerdiness.) Don't worry, I don't hate you because you're doing poorly. I don't hate you at all. As I'm fond of saying, I've had two students out of the thousand or so I've taught over the past decade whom I really just couldn't stand. And you are neither of them.

I don't hate you, I'm not mad at you. I might feel bad for you, that you've gotten a rough start, that you've slipped so far behind. Whatever position you're in, whether you're one of the best students in the class who's having a momentary lapse of reason or one of the weakest students in the class who's struggling mightily to just keep up, I'm ready, willing, and I hope able to help you. It's my job. It's what I get paid to do. More than that, it's what I'm most passionate about in life. Funny, isn't it? It's funny that you've got someone sitting there nearly 60 hours a week who gets all fired up about the possibility that you'll traipse into his office and ask him for help? Funny.

4. In class, take notes (please don't think I don't notice you when you're not doing so, and don't think I don't see the correlation between not taking notes and doing poorly on the exams...as I said before, I'm not a fool, and I'm probably one of the most observant people you've ever met), ask questions, and if I ask you to take part in a group activity, please take part in that activity. Some of the group activities seem corny, I know. (Just wait for the Mean Value Theorem exercises coming up this week!) I'm a bit of a cornball, it comes with the nerdiness. But cornballery can be fun if you just let yourself be taken away by the cornballitude. What's more: I've spent thousands of hours designing my classroom activities, and I don't do anything in the classroom unless I think it serves a useful purpose, so nothing's ever corny for the sake of corniness. Please keep in mind that my primary goal is to help you understand what we're talking about on any given day, and I'll do anything I damned well can to meet that goal. I'd really appreciate your cooperation in this endeavor.

You know what? I'm feeling less disappointed than I was, less hurt, and less helpless. I feel like I've actually done something, I've managed to unburden myself. Writing this entry, though it's cost me another three hours, has really helped me work through this issue.

I'll finish grading your homework tomorrow, folks, and I promise that I'll be in a better mood. I'm there already, in fact, as the end of this leviathan post draws into sight.

We've got four or five weeks left in the class, and a fair chunk of hard work ahead of us. I won't promise that it'll be easy, but if you stay with me and you keep on top of the work, I promise you'll make it through alive.

So how 'bout it, huh? Are you ready?

I'm ready.

Let's do it.