72

The next seventy-two hours are going to suck.

That's all I've got to say.

If they can put a person on the moon...

Embarrassing.

It's embarrassing when you can't assume that the computer technology in your classroom will be error-free enough to allow you to ask your students to (1) download a file from your website, (2) open it using software that's already installed on the computer and easily accessible from the desktop, and (3) use the software to solve a few simple problems...in under half an hour.

My students took Team Quiz 2 in Calc I just now, and there were glitches galore, none foreseeable, all technology related. Of the 18 five-plus-year-old Macs in the room intended for student use, 3 were powered down, 3 were frozen, and 1 was (and has been for as long as I can remember) rendered useless by having no mouse.

That left 11, to be used by 8 groups of students. So far, so good.

Another froze soon after the students managed to download the file they needed from the course website.

Three more performed so...damned...slowly as to render them unusable for the Mathematica computations I was asking the students to perform. One poor team had to hop from one computer to a second, to a third in order to find a machine that would actually carry out the incredibly simple command they'd given it without stalling for several minutes. Since completing the quiz required at least six such computations, there's no way they could have finished on the first two machines.

Keeping score? With the aforementioned 4 of the 11 remaining machines by now out of commission, there were 7 student computers left for 8 teams, and fortunately the instructor's terminal at the front of the room proved functional, so the eighth team could hop over to it.

To my Calc I students: I humbly (and I do mean humbly) apologize for today's technology woes. I'm not sure if it's even fair to grade the quizzes, given the way at least two teams were stalled by recalcitrant computer behavior.

Argh. That really was pathetic.

Overkill

What can intermediate-level math students be expected to get out of a highly technical research seminar?

I'd talked up my out-of-town colleague Seymour's visit quite a bit over the past few days, to both students and colleagues, partly because I knew that if I didn't get the word out, Seymour might end up speaking to an embarrassingly small handful of our strongest students and most devoted faculty members. The last thing I wanted was for a friendly and devoted colleague to drive four and a half hours (both ways, in one day!) to give a talk to a pitiful few. I've given talks to such small groups, and I know how disappointing a small turnout can feel.

As it was eleven of our ever-stalwart students (including five from my Foundations course) came to Seymour's talk this afternoon, and several faculty members rounded the audience out.

As it was I ended up feeling bad not for Seymour, but rather for a few of my students whom I'd strongly encouraged to attend the talk, expecting that though they'd struggle to keep up they'd likely be able to grasp a good bit of the presentation. It turned out that I had a hard time keeping up with the bedazzling details, so I'm certain a number of the students were having a bear of a time.

Nevertheless, by the talk's close I remained convinced of the truth of what I'd told my 280 students in class yesterday: "even if you don't understand every twist and turn of every argument, going to seminar talks early and often prepares you well, as it exposes you to the language, the notation, the terminology, and the conventions of mathematical communication. The earlier you start going to math talks, the quicker they'll start to make sense to you, and the stronger a student you'll be."

To help students get the most out of the talk, I typed up and sent out a "companion piece" indicating to the students who attended what I thought were the important points to keep in mind about the presentation. Leaving aside the technical details of Seymour's highly convoluted argument, I encouraged students to stay focused on the talk's big picture and to try to understand

1. the ways in which Seymour's use of notation for operations in unfamiliar settings suggested analogous, more familiar, mathematical operations;

2. the culture of mathematics evident in Seymour's "humanizing" commentary during his talk, highlighting the importance of collaborative research and the ability to ask good questions as well as answer them; and

3. the mathematician's penchant for ranking, sorting, and ordering that formed the basis for much of Seymour's work.

Intermediate-level students hoping to hone their seminar-watching skills will do well to look for these general "trends" and meanwhile gloss over inconsequential details. Remember that no one should go into a talk expecting to understand every last jot and tittle of the presenter's argument; just because a theorem's proof involves computations you can't possibly understand, not being an expert in a particular field, doesn't mean you're barred from getting a lot out of the talk, provided you know how to take the talk in.

By the way, my thanks go, again, to Seymour, for his unselfish trek up to UNC Asheville this afternoon. I'd also like to give a shout-out to Tomassino, who complained that he hadn't been able to figure out his pseudonym on this here blog. (I've mentioned you before, man, although you'd have to search the blog to find out the context in which your name appeared.)

To be continued, I'm sure!

The seduction of induction

Here's a question I hope some of my 280 folks will be willing to answer in the comments section (other onlookers are encouraged to reply as well): what's so hard about inductive proofs?

I've noticed that every time I've taught a proofs class, three times here now, and once at the University of Illinois, the homework set the students found most challenging was the one dealing with induction. Fellow teachers, have you seen this phenomenon at work as well?

Students simply seem to find induction tricky, intellectually irksome.

Here's one reason this proof method might prove (no pun intended...okay, kinda intended) more difficult for most than others: the method relies on the proof of a conditional and not an absolute statement, and aside from the base case, therefore, one is never proving a statement directly. The very reason the proof method succeeds is the "if it's true for n then it's true for n+1" clause. Understanding the sufficiency of this clause (together with the base case) is itself a major leap, and therefore most beginning provers are uncomfortable with the very validity of an inductive proof, and therefore are less motivated to complete all of its components.

Maybe that's it.

Ideas, anyone? Throw me a frickin' bone here.

As it is, though, this semester's crop did pretty solidly on their induction homework. Of the twenty-five or so homework sets I graded over five hours this morning, although there were only two As, everyone who managed to turn in a completed homework set did halfway decently, and the class average couldn't have been much below 80%. I'm pleased. Though this term's students are a bit shyer in class than their Fall 2007 counterparts, their dogged pursuit of homework perfection is easily comparable to that strong semester's work ethic. Kudos!

Moreover, their LaTeX is coming along beautifully. A handful have really gotten the hang of it (Trixie, Tish, and Siegfried, y'all rock!) already and are executing nearly flawless TeX masterpieces, and several others (Omar's comin' atcha!) are using it regularly and their proficiency isn't far behind. More importantly, three or four people have already commented on how having to type the math up makes them think much more carefully about what they're writing, which is ultimately the primary goal of the class.

Keep it up!

Right wrong turns

Sometimes it just all comes together, y'know? It helps having a class full of kids (and a few older folks) who are willing and able to put the responsibility for their learning on their own shoulders and run off with it, leaving me in the dust.

Several times during the past week I've found myself extemporizing in Abstract II as the students have asked questions the answers to which I had no idea. That's the satisfying/scary thing about teaching these high-end courses: on the one hand, the students are smart and savvy enough enough to ask really deep and interesting questions on the fly; on the other hand, the students are smart and savvy enough to ask really deep and interesting questions on the fly. It ain't Calc I in there: although I will never tire of the thrill of teaching calculus (Calc I folks: I'm not kidding when I say that I never get over the beauty of the definition of the derivative. That's not feigned excitement: I really am in awe of the limit definition, after all of these years, and I'll never get tired of seeing the hs magically disappear!), and though those first-year courses are often jam-packed with smart students, it's no more than once every few semesters that I find myself perplexed by a student's question in Calc I: there's just not enough that's unfamiliar, and the concepts are second nature to me by now.

But Abstract II? Wow. That class is a kaleidoscope, and whatever view we find on a given day seems to depend as much on the mood of the class as it does on the phase of the moon.

A couple of the questions that have come up lately in that course are pretty routine and standard ones I was able to respond to without skipping a beat: "If f(x) is irreducible in k₁[x] and k₂ is a subfield of k₁, then is f(x) irreducible in k₂[x]?" "What would an irreducible cubic polynomial in ℚ[x] look like?" (This last was in response to the proposition showing that the irreducible quadratic and cubic polynomials over any field k are precisely those with no roots in k.)

The really interesting stuff happened when I was momentarily fertumult.

"Why would you want to represent f(x) in 'base' b(x)?" asked Bertrand in class on Wednesday. He's always good for a truly probing question or two.

"Um..." For a moment I rethought my policy of encouraging students to ask good questions like why?

"Um...well...if we know that k is finite..." I hemmed and hawed for a few minutes and waved my arms around in an unconvincing fashion. A few muttered words seemed to make some sense, and reason returned, and I felt better as my mental wheels gained a bit of purchase. "...then if we've got a homomorphism φ:k[x] → k[x], φ is completely determined by its action on the finitely many polynomials of degree less than deg(b) and its action on b itself."

Whew!

I've not done so much tiptoeing as I've done with these kids since my first semester at the University of Illinois when they gave me a section of Accelerated Honors Calc III for engineers to teach...those kids were smart. It hadn't helped me that at the time I hadn't done vector integration since my second year of undergrad.

This past Monday's class was the funnest so far this week, and it came about because I'd been sloppy in setting up an example in my worksheet for that day. The exercise asked the students to come up with two quadratic polynomials in ℤ₈[x] which each exhibit more than one nontrivial factorization, thus demonstrating twice over that ℤ₈[x] is not a unique factorization domain.

One of my preplanned examples worked, the other was verkochte (I'd forgotten that in ℤ₈, -3 and 5 are identical, so the "distinct" factorizations I'd expected were one and the same). Did the students give up? No, no, no! Instead, after a few minutes of floundering about trying to fix the failed example I'd started with, we just set up the equation that would have to hold in order to yield two nontrivially distinct factorizations, and solved away. We soon had a whole passel (not a half of a passel, or even two-thirds of one) of examples.

That's a great class.

While I'm bragging on them for their self-directed successes, I should give props to Uri for coming up with the basis for a fantastic question for the first exam. Though several of the students, when asked last week as part of their homework to posit potential exam questions (with accompanying solution sketches), came up with appropriately difficult and interesting posers, it was Uri's question involving the advanced ring theoretic properties of the powerset ring that proved the most amenable to inclusion on the exam. It was just right.

Okay, off to read a few Project NExT-Southeast Fellow applications before I hit the hay.

Au revoir!

Jacta alea est, and all that jazz

[This post was written at the Reagan Airport in D.C. on Tuesday night...but there was no way I was gonna shell out $7.95 to use the airport's wifi for the sake of posting it...]

Well, it’s done.

We did two takes, as that’s standard practice, but we were all pretty happy with the first one. In fact, after all is said and done, I’m happier with the first: I feel I was clearer, and that I did a better job of explaining myself than in the second. On the other hand, I believe the producers thought I was more animated and personable in the second one, and they thought it was a toss-up between the two.

We’ll see. I’m confident. I should hear from them within the next four months or so regarding where we go from here, one way or another. I let them know of a couple of courses I’d be willing to put together for them, a couple of which (including graph theory) they were quite excited about.

Me? I’m tired. Fortunately, aside from the inevitable backlog of e-mails to slog through, I’ve got nothing to worry about work-wise once I get back, at least until tomorrow. It thus promises to be a relatively relaxing red-eye run.

29 minutes, give or take

Ugh...I slept like crap last night.

Just a few hours before my date with the Teaching Company. I'll be happy to have this damned thing behind me.

After a few practice runs, I've got it down to about 29 minutes, give or take a minute or two, which is smack dab in the right range, so I'm pleased. I'm going to do one more run-through of the last half (modular arithmetic and the RSA cryptosystem) in a few minutes, and then call it good.

A word about classwork: I'm in the middle of my reading of the Calc I students' final drafts of their derivative "paperlets," and as one might expect there's a broad range of quality. Some are adequate, and others are positively fantastic. The best are distinguished not only by the completeness of their treatment and the correctness of their computations and statements, but also by the composition of their exposition: the ideal narrative offers smooth transitions from topic to topic that highlight rather than downplay the interrelation of those topics.

I hope to finish grading those over breakfast this morning.

Then it's off to the races.

Nearly there, my friends, nearly there!

Breakthrough

Today's the day I'm going to want to remember in the waning weeks of the semester: in the closing classes of the term I nearly always find myself thinking "what could I have done better? Did I really reach them? Did they ever find that spark?"

This morning my Calc I class lit up like a powder keg. Though it's taken a little longer than I would have liked it to, I genuinely feel that at last I've got a sort of simpatico with these students: I get them, they get me, and we're both willing to work our asses off for each other.

(Incidentally, I apologize for the uncharacteristic potty-mouthery in the past couple of posts. It's just been that kind of week.)

What happened this morning? Things just clicked.

I think the restructuring of the homework, basing it on problems of my own devising instead of on the arcane and ethereal exercises offered by the textbook, has helped a lot. It's a no-brainer, really: only I can create the sort of exercises that challenge the students to grapple with the topics as we see them come up in class. Moreover, while the book's exercises are interesting and thoughtful ones, the lessons the exercises purport to teach are wasted if the density of the problem precludes the student's understanding of the lesson's import. For example, the problem that purports to show that the derivative of a quotient is not the quotient of the derivatives (the now-infamous #36 from Section 2.3) would do well to simply say that that's bloody well what it's trying to show, rather than coyly trying to trick students into that understanding. "That's what it's trying to show us?" said several students, one after another, after I'd shown them the point behind the problem. "Why didn't they just say so?" Understatement's all well and good for French cinema, but with first-year mathematics exercises you're often better off being as subtle as an atomic bomb.

Yes, the few extra minutes it takes me in preparing for each day's class are a small price to pay for the benefits that accrue. Several students have indicated that though the exercises I've made up are still challenging and enlightening, they make a hell of a lot more sense than the ones the book had dealt them.

Today's in-class activities also proved exciting. We went on a limit hunt along the lines of an exciting game of Battleships: Having shown that the ratio (a^h - 1)/h tends to roughly 0.693 when a = 2 as h tends to 0, and to something slightly more than 1.098 when a = 3, the students set about trying to find the value that would make the limiting value 1 on the dot, asking Mathematica to graph the ratio for increasingly precise values of a. Several times Hera literally jumped from her seat in excitement as the race to estimate e tightened (granted, she's an exciteable soul in the first place, but still...). The excitement was hardly hers alone: several others were visibly intrigued, and by the time it was revealed that the number we were seeking could never be numerically known, there were warm smiles of understanding on faces scattered throughout the room.

To the Calc I folks who may be reading this: thank you. Thank you for your feedback in leading me to the change in course on the homework, and thank you for the warm and supportive reception of the change once we made it. Thank you, too, for the willingness to work with me in class and to take an active role on in-class exercises. I got more out of class today than I have from any class in a long time, and that success is based largely on the cooperation you've shown in creating a healthy learning environment.

Meanwhile the students were a bit more subdued in Foundations, a class that mercilessly meets just after the lunch hour, in the valley of biorhythmic cycles that tend to pull people bedward. "I'm sorry I was so quiet in class today," Trixie told me later. "I'm quiet in all my classes, but I feel bad that you were getting mad at us for being so quiet."

"I wasn't mad," I said, "I was just trying to wake people up!"

I will credit 280's Nighthawk with the line that brought me the greatest joy today, though: "it's [this course's emphasis on clarity in writing is] bleeding over into my other courses." It would shock me not if Nighthawk's colleagues found him guilty of felonious brownnosery and bullshitting in the first degree, but the fact remains I was tickled by the comment.

Meanwhile, my twelve-member Abstract II class was cozy and familiar. Indeed, I remarked at one point, as I paused to find my place on our worksheet, on how they chatted with one another warmly like old friends. "You are old friends," I said. "You've been in many classes together by this point, and most of you know each other really well." Then I waxed a bit pre-nostalgic. "I've had many of you in several classes, too, and for some of you this will be the last class I'll ever see you in! Many of you will be graduating in a few months, and it's going to be a very different school without you."

"Oh no, I'm going to cry!" Nadia said, hiding her face behind her notebook.

Fortunately Euclid's Algorithm intervened and saved her from her tears.

All told, it was a good day. A mite busy (two hours of unceasing student deluge after Abstract II...can y'all make an effort to finish the homework with a liiiiiittle bit more of a margin than a few minutes before five?) as the day neared dusk, but a good day in the end.

And now it's time to end this day (14 minutes to go!). Tomorrow brings a batch of grading and a little bit more class prep for which I won't have time as I'm winging it to Washington next week...and I hope too tomorrow brings a happy Valentine's Day to one and all.

From a hopeless romantic to his faithful readers, my thanks for your words of wisdom, your help, and your support!

Why can't it be Wednesday?

I estimate that by this point in my career I've given nearly a hundred talks of some length or another in seminars, colloquia, symposia, or conferences. A conservative estimate puts me in the driver's seat during more than 1500 class periods (to say nothing of exam reviews, study sessions, and problem group meetings), and I've stood in front of audiences ranging in size from five or six to five or six hundred. Although I admit to perennial pre-semester jitters and no first day of class would be complete without a few abdominal lepidoptera, it's been years since I've felt more than a slight twinge of anxiety in preparing a live presentation.

So what's the deal, huh?

I've got a date with a learn-at-home organization, The Teaching Company, in a few days; they're flying me up to their Washington area offices to record a demo lecture they'll use to test my suitability for future work with their video lecture program. Heads: I "win," and they retain me to record a series or two with them; tails: I "lose," and I return to hearth and home and an already rich and utterly fulfilling career.

No big deal at all, really.

Why am I so anxious?

I've run through my half-hour talk god knows how many times in the past few weeks. From start to finish for the first time it came out as I walked home from campus three weeks back. Before an empty classroom silenced by a snow day I did two walk-throughs with the slides I've by now memorized. The tougher parts have tumbled from my lips over and over as I've tried to loosen their verbal knots with my tongue, searching for a way to ease one end over another and unwrap the mystery tied tightly within.

Still there are parts that seem alien and abstract, even to me.

Tuesday's the day (it's Thursday night as I write this).

Between now and then I've got two meetings each of my three classes, and another weekend offering a pile (considerably lighter than the last, mercifully) of grading. I've got three tests to write and a homework set or two. I've got a few more meetings with my students, a couple of writing projects that demand my attention, and at least three colleagues' visits to plan for. There are several other things I'd rather be writing about on this very page than this one impending one-day trip (fascinating conversations with several different students today come to mind, conversations about the nature of proof, the nature of math, the nature of knowledge itself), but I feel blocked, unable to say anything about anything unrelated to that goddamned talk.

It'll be fine, I tell myself (and others close to me say).

I am right, of course, and they are right: no matter what, it'll be fine. I always give a decent talk. It'll be fine.

But I want it to be more than fine. So much more.

I want to wow their socks off and blow them away. I want them to say "this goes to eleven." I want to be fucking awesome.

We'll see.

Meh. Whatever.

Life goes on, right?

Life goes on.

Yeah.

Why can't it be Wednesday?

Committee report report

I've just picked a perilous path through the eye of the perfect storm of grading, fourteen and a half hours of toiling and troubling over computational exercises and low-stakes writing on derivatives from my Calc I kids, first-order logic problems and truth tables (not to mention first-time LaTeX exercises!) from my Foundations students, and a passel of problems on ring theory from my Abstract II class.

Fortunately the cold I came down with at the end of this last week made sure I didn't want to be anywhere else but home this weekend.

I've been meaning for some time now to say a bit about how the homework committees are going this semester, since so far I've been proud of their smooth functioning. (For the people reading this on my first-ever cross-post on the Young Mathematicians' Network, you can read a number of my older posts about using homework committees here. Briefly, it's a peer-review technique I use to encourage students to (1) begin their homework more promptly, (2) engage in self-authorship, (3) grow accustomed with the frequent multiplicity of correct solutions, and (4) develop their teamwork skills. Students volunteer to serve on committees tasked with reviewing and offering feedback on their peers' drafts of solutions to particular homework problems. After reviewing all submitted drafts they lead a brief class discussion on the problem they considered and return the drafts to their respective authors, who then have time to revise their work before submitting a final draft one class period later.)

There have been four reports in each of Foundations and Abstract II, and despite their relative inexperience with the genre, the former students' committee reports have been stronger than those of their Abstract II counterparts: they've skillfully avoided simply answering the problem placed before them (this was a major problem the first semester I asked students to serve on homework committees), they've intentionally made use of the course's writing stylesheet (the now-infamous Four Cs), and they've done a marvelous job of indicating common pitfalls, clever solutions, and helpful hints.

When the first homework set was handed in there was a little misunderstanding concerning exactly who received which draft of which problem, and as a consequence I was given a glimpse of the feedback the students were offering to one another on the drafts submitted to the committees. It was heartening: the few comments I saw were meaningful, respectful, and helpful without being too much so.

The quality of these students' committee work greatly exceeds that of the students in the first Foundations course in which I assigned committee problems. Back in "the day" the kids'd frequently just get up in front of the class and solve the given problem for their friends, resulting in a couple dozen nearly identical eventual submissions in which the students would faithfully render every jot and tittle (whether correct or not) of their colleagues' proffered solution. During the past couple of semesters I've very deliberately pointed out that it is not the job of the committee to perform this disservice.

Word's gotten out.

So, props.

In other news, I'm happy to see that I've received a few (not even anonymous!) replies from my students regarding Gil Strang's free text, Calculus. I might be misreading the message here, but it seems that though people are not yet ready to chuck the text through the nearest open window, it might not hurt to mix a few of my own exercises in with the relatively recondite ones offered at the end of Strang's sections?

I believe this is exactly what I shall begin doing.

Thank you all for your feedback, I appreciate your attention to improving our classroom environment! Together we'll make it to the semester's end.

Strang strangeness

So...we're about four weeks into this semester, and the Calc I students are showing signs of stress.

What gives?

As far as I can tell from class, things are going swimmingly. They're enjoying the class, they're actively participating, they're doing very well on the quizzes, and though there were a few folks who didn't turn in last week's homework set (the first set featuring problems from the textbook), those who submitted solutions managed to puzzle through those problems as well.

So what's the big deal?

It's the textbook.

In the fall semester our department decided to go with Gilbert Strang's Calculus as the backbone for our Calc I and Calc II courses, making the switch from the eminently mediocre Stewart (now in its gazillionth edition, an ever so slight modification of the gazillion-minus-first). Our reason? Like many of our colleagues in other departments and at other schools, we were concerned with the rising cost of textbooks, and we wanted to do something meaningful about it: Strang's textbook is available on-line at no cost to the student.

Last semester several sections of Calc I were run using Strang's book as their basis, with mixed success. Four faculty used the text, and while one was very happy with it, the remaining three expressed significant reservations. A couple of these supplemented the textbook heavily with problem set, examples, and other materials on Educo (my feelings on which I've blogged extensively elsewhere), while all three faculty members with reservations admitted that they'd had more success with the book when they "distanced" themselves from it, almost treating the class though it were text-free.

After a good deal of debate (several hours of several faculty members' time) we decided to give it another go. I figured it wouldn't be such a big deal because I tend to teach in a rather text-free fashion generally. As it is, I usually end up writing the equivalent of a supplementary (or at least "companion") text to the course's regular reader anyway. So I thought, what the hell, I'll give it a shot.

Now, as I said above, we're four weeks in, and...well...the students aren't lovin' it.

"I totally understand everything that's going on in class, and the quizzes make sense, and I enjoy all of the activities we're doing...but when I open up the textbook to do the homework, I can't figure out what's going on."

"The wording of the problems is really...weird. Like, it'll ask you to do the same things we've done in class, but in the middle of all of these words. Why can't they just say what they want you to do?"

"When I look at the chapter to see if it reinforces what we've been doing in class, I can't read it. It's just impossible to read!"

"I can't get started on any of the homework problems, and then when I ask one of the Math Lab people for help, they'll show me that it's really just something we did in class, I'll be, like, 'why didn't they just ask that in the first place?' "

I'm getting these sorts of comments from some students I know to be among the smartest, hardest working, most dedicated in the class. If they're having a rough time of it, I can't imagine how the folks who find math a struggle are feeling.

How are you doing?

Students, I'd like to know how you all feel about this text. Write to me about this. I'm asking you all to offer a comment to this post and let me know how it's gone down for you. Vent away. Or gush, if you'd like. Comment anonymously if you'd like to, I'm really just trying to get an overall sense of the class's view. If we can figure out where we can stand, we can troubleshoot the problem and we'll find a solution. I've got some ideas in mind, but I want to know the depth of the situation first.

And to my colleagues elsewhere: have you used this text before? Would you like to share your experience with us? Maybe you've had a different experience than we've had. I think it may just be that what works well for Strang's MIT students just doesn't work well for the folks at UNC Asheville.

That's all for now, folks. Lemme have it!

Problem solving, a.k.a., we'll see where this goes...

...in light of the conversation I had with my colleague last Friday (and the subsequent blog post last Sunday) on student perceptions of our university, I've just sent an e-mail to our new chief admissions officer, requesting a meeting with her at which we might discuss outreach activities involving area high school guidance counselors, whether that outreach takes place at the level of the university or at the level of the department.

To be continued, I'm sure...

Can we talk?

With all the thought I've put into designing writing assignments and writing-intensive courses over the past few years, it's nice to know that I still have a lot to learn. Every now and then I'll trip over one of my own assignments and realize that there's always room for self-improvement.

The past few iterations of my MATH 280 course have included a number of "dialogue" homework exercises, in which I ask each student to construct a dialogue between herself and a putative member of the class who's having trouble understanding whatever key concept that particular homework set is meant to address. I feel that the dialogue format offers students a number of challenges and opportunities, including

1. the chance (and challenge!) to explore high-level logical or mathematical concepts using everyday language,

2. the opportunity to assume the role of the expert by taking the lead in the fictitious dialogue, and

3. the chance to break free of conventional mathematical writing and experiment (in a relatively low-stakes setting) with a novel discourse that meshes well with a number of students' learning styles.

I don't intend to do away with the dialogue exercises any time soon (students often comment that these exercises are among their favorite and I feel that they are truly effective activities for the above reasons), but in light of my students' performance on the most recent run (graded yesterday), I realize that I'm going to have to be a bit more intentional about this sort of assignment.

What went on?

Several students' submissions were superb (I'll say more about this in a bit): not only did they construct a dialogue (and not a monologue, as some of their colleagues submitted), but they made effective use of the dialogue format to truly engage their befuddled interlocutor. While the "friend" in the weaker dialogues would say little more than "oh, I see" or "I don't really get this," the best students' conversation partners actively pushed the dialogue forward, asking probing questions that highlighted subtle difficulties in the concept and offered a window onto gradually improving understanding.

On the other hand, the weakest student responses didn't even offer dialogues at all, but merely gave a brief paraphrasing of the relevant concepts (universal and existential quantifiers). Ultimately I don't think these students understood exactly what a "dialogue" was supposed to look like.

And whose fault is this? How often have they been asked to construct "dialogues" in the past, and how often in math or science classes?

Yeah, I shouldn't have assumed that they knew what to do. It's on me to lay out a bit more clearly the expectations I have for that particular exercise. Especially as it's not a genre frequently encountered in a mathematics course, it was incumbent upon me to explain more clearly the powers and the limitations with which the genre comes.

Oops.

What to do? Recovery begins tomorrow: I've TeXed up the two dialogues I felt were exceptionally good (there were about five or six more who earned 10 points out of 10; I felt these two warranted a couple of points extra). Besides distributing the TeXed copies of these dialogues as models to the other students, I figure we'll spend about ten minutes in class discussing what it is about these dialogues that makes them so strong.

For now, the Super Bowl calls. More later, I'm sure. I'll try to check in tomorrow and let you know how the in-class discussion went.

Identity crisis

One of my colleagues swung by my office Friday afternoon in order to let me in on the latest tidbit from the university's Branding Committee.

It seems that according to a recent survey of North Carolina high school juniors and seniors who achieved the highest scores on various metrics (SAT, ACT, GPA, et cetera), our school, the state's public liberal arts school, scored dead last out of the sixteen-school state system when it came to student perception of the amount of individual attention faculty provide to their students.

"What in the hell?" was my reaction.

My colleague nodded his vigorous assent.

"We should be first!"

More excited nodding. "It is frightening that students believe they're more likely to get individualized attention from their professors at Chapel Hill than they'll get here," said my colleague.

"Do the students know what 'liberal arts' means?"

The nodding turned to a solemn side-to-side shake of his head. "No," he said. "That's the thing. High school students do not understand what a liberal arts college is."

"Why is that?" I mused out loud. "Is it because they associate the phrase 'liberal arts' [however erroneously!] with 'unemployability' and fear that if they attend a liberal arts college they'll spend four or more years learning useless skills that will ill, if at all, prepare them for the job they'll need to recoup the money they put into the program in the first place?"

"I don't know."

"Remember the exit survey data that Beauregard sent around a few months ago, indicating seniors' perceptions of the education they'd received here? I seem to recall that we did pretty well on that survey, as regards perception of faculty attention."

"Yeah," my colleague agreed. "The seniors get it, they realize what we've got. But the students we're trying to recruit...the very best of the students we're trying to recruit, don't understand."

"Whose fault is this?" I asked rhetorically.

"Oh, it's our fault."

"But what are we supposed to do about it? What can we do better?"

"We need to get out the message that we offer small classes, that we offer individualized attention. Whenever we're advertised as a 'liberal arts college,' it turns students off. Whenever we emphasize the small class sizes students will find here, they're attracted to us."

"Craziness."

My colleague had another meeting, and I had another class, so the conversation ended there. Busied with grading for the past day and a half, I've only now had a chance to reflect on the conversation and give more thought to its topic.

I'm not sure I understand any better now what I didn't get then.

Let's say that students simply don't understand the phrase "liberal arts," and perhaps instead of merely passively misunderstanding it and chalking it up as a phrase they don't know and are unlikely ever to (like "diffeomorphic" or "Weltanschauung"), they actively and horrifically misunderstand it to entail instruction irrelevant to any likely future employment. (I should note that historically this is the most frequent misinterpretation of the phrase. That it truly is a misinterpretation is shown by data I was made aware of this past Thursday at a department liaison luncheon with the university's Career Center staff: the National Association of Colleges and Employers, a professional organization for college staff devoted to career counseling, human resources, and so forth, recently released the results of a 2008 survey showing that all of the top ten skills employers desire in the people they hire are frequently and explicitly listed as learning goals in liberal arts courses.) Even given this misunderstanding, how on Earth do the students misunderstand "liberal arts" to entail large, impersonal lectures or standoffish professors who have no time for personal interaction with their students? Surely the next most common stereotype of a liberal arts education (after the erroneous one detailed above) involves a hirsute and betweeded tenured professor gathering several of his favorite students to climb up to the rooftop of their classroom building and there sit, overlooking the quad, while they smoke weed and contemplate the relevance of Derrida to modern-day indigenous dairy farmers. What aspect of that (also, perhaps more sadly!, erroneous) stereotype implies impersonality or aloofness?

I've just now pulled up the survey results that I'd mentioned in my conversation with my colleague, to see if I'd misremembered anything. I'm happy to say that the data are even richer than I'd remembered: we have midcourse data, too, from the sophomores. Here are the relevant figures.

From sophomores:

1. In response to the prompt "Please evaluate how well faculty members at this campus encourage student-faculty interaction in and out of the classroom," 83.0% of Asheville sophomores in 2008 indicated either "good" (3 on a 4-point Likert scale) or "excellent" (4 on the scale). This is 6 percentage points higher than the state university system as a whole (77.0%).

2. The prompt "Please evaluate how well faculty members at this campus care about your academic success and welfare" elicited even better results: Asheville sophomores responded with a 3 or a 4 87.8% of the time, compared with a score of 78.6% across the system (a 9.2-point differential).

3. While the prompt "How would you evaluate your access to your adviser?" garnered almost identical numbers from Asheville sophomores (77.1% 3s and 4s) and from UNC sophomores in general (77.0%), the prompt "How well would you evaluate [obtaining] sufficient time with your adviser?" received 3s and 4s from 77.7% of Asheville sophomores and only 64.7% of all UNC sophomores.

4. In response to the question "How many of your classes, if any, do you feel have been too large for you to learn effectively?" 64.8% of Asheville sophomores responded "none," while only 34.7% (!) of all UNC sophomores were able to make the same boast.

All of this goes to show that students catch on quickly once they've gotten here: while they might not know what they're in for before they get to campus, they soon realize that when it comes to attention from their faculty, they've got it better here than they would elsewhere in the system.

What about the seniors?

1. and 2. The same prompts given to the sophomores elicited similar results from 2008's crop of graduating seniors: 89.2% indicated that student-faculty interaction was either "good" or "excellent" (this mark is 6.2 higher than the Asheville sophomores and 5.1 points higher than the 84.1% mark earned by all UNC seniors in 2008) and 92.2% (4.4 points higher than the sophomores' mark and 5.6 points above the 86.6% given by all seniors) gave the same marks to faculty concern for students' academic success and welfare. If anything, the last two years of their studies further convince students that they've got a good thing going at a smaller school.

3. When it comes to their advising experience, 85.8% of Asheville seniors (as opposed to 77.0% of the sophomores) indicated that they had good or excellent access to their advisers, and 81.4% (as opposed to 77.7% of sophomores) indicated that they'd found it easy to obtain sufficient time with them. These figures compare even more favorably with their system-wide counterparts than the corresponding sophomore survey results: the "access" grade exceeds the system-wide mark of 77.2% by 8.6 points and the "time" grade exceeds the system-wide of 73.2% by 8.2 points.

Strangely enough the seniors were not asked about class size, as were the sophomores. I would hypothesize, however, that the bigger schools are likely to close some of the distance between themselves and UNC Asheville during those last couple of years, as juniors and seniors are much more likely to take a large number of seminar-style classes in their respective majors and therefore are likelier than sophomores to have experienced smaller classes, even at schools like UNC Chapel Hill and NC State.

Lest you think I'm cherry-picking the data, I should mention that the above items are all of those on either survey that deal directly with student-faculty interaction: I'm not trying to sweep anything under the rug.

In the middle of compiling the data above I summarized my conversation with my colleague to Maggie, and she posed an interesting hypothesis I'd not thought of: "maybe the kids who took the survey you were talking about are the ones who will get the attention of the faculty at the big research schools."

She's got a really good point: the superstars are the ones who get the lion's share of the bigger schools' top-tier researchers' attention, even in the early, pre-college, years of their academic careers. (I remember being courted by the University of Denver, my eventual alma mater, as the Engineering school attempted to get me to take part in its summer program for talented high school students. I believe this wooing was, whether or not I was consciously aware of it, one of the reasons I ended up going to DU.) Therefore it could be that the students who were surveyed already suppose that bigger schools' faculty are likelier to give them attention than are the faculty at a smaller school like UNC Asheville. This is even more surely the case of students at schools like the North Carolina School of Science and Mathematics and other hotbeds of adolescent excellence: which of NCSSM's students is going to think twice about heading for the mountains in the far-flung western corners of their state when world-famous scholars at Chapel Hill and State are already sending them personal invitations to join their research groups?

Nevertheless, we've got to get them to think twice.

So I ask the questions again: What are we supposed to do about this? What can we do better? Perhaps, to paraphrase one of my favorite Calvin and Hobbes cartoons, we don't need to do a better job, we just need better PR for the job we're already doing.

I'd love to hear from my readers, students and faculty alike: are you at a liberal arts school? Faculty, how is your school perceived by prospective students? Students, what's your perception of your school? What are the causes of that perception? What's your take on the conversation I had with my colleague? Do you have any answers or suggestions?

Please discuss. Five pages minimum, double-spaced, one-inch margins. Due Friday.