Poetry primer

I promised a couple of folks who'd attended this past week's CWPA with me that I'd provide a layperson's guide to the math-based poetry I've begun to write. You'll find just such a guide below, but first I wanted to list the links to posts that have the most to do with math and poetry, for easy access:

1. "I sing the verses eclectic": this is the first post in which I showcase math-themed poetry written by my Calc I students from Fall 2007. This was almost immediately followed by...
2. "Round two," the second such post.
3. "Ars poetica" is the post in which I laid out my idea for (G,φ)-grams, examples of which are explained more fully below.
4. "Six: being a modular G-gram in 14 lines" is the first post containing a G-gram, "Six." (Its structure is explained in layperson's terms below.)
   
5. In "Deep thoughts and more dilettantish dalliances" I printed the first non-modular G-gram. Its structure is a bit harder to describe, but I'll attempt to draw a rudimentary roadmap in a later post (when I'm not so tired!).

So what's, uh, the deal?

Here's the idea: anyone who as a kid played at all with codemaking and codebreaking knows how one assigns numbers to the letters of the alphabet: "A" corresponds to 1, "B" to 2, and so forth, until "Z" corresponds to 26. We'll be making heavy use of this correspondence, so keep it in mind. Mathematicians might denote this correspondence as a certain kind of function (the fancy-schmancy term for it is homomorphism), let's call it φ, that takes each letter over to its corresponding number:

φ(A) = 1, φ(B) = 2, ..., φ(Z) = 26.

And yes, this φ is precisely the φ occurring in the term "(G,φ)-gram."

It might help you to think of the number associated with a particular letter by φ as a the "value" of that letter. Next we're going to compute the values of words by "adding up" the values of the letters the word comprises, but the way we perform addition is going to be a bit wonky.

In what way? We're going to "wrap around" so that our sums never go past 26. For instance, if we add 13 + 4 = 17, since this number is less than or equal to 26, we're golden. 13 + 8 = 21 is still less than or equal to 26, so we're fine. But if we add 13 + 18 = 31, we've gone 5 steps past 26 (27, 28, 29, 30, 31), so the number 31 is really going to be the same thing as 5, by our reckoning. Similarly, 32 = 6, 33 = 7, ..., 52 = 26. And what then, past 52? We start wrapping around again. That is, 53 = 26 + 26 + 1, so 53 should be equal to 1 again. 54 = 2, and 55 = 3, and so on.

By the way, the motivations one might have for counting like this are many. For instance, classical computers do all of their computations in binary arithmetic, a form of counting in which the only digits allowed are 0 and 1; in such arithmetic if you add 1 to itself you have to wrap around to 0, so that 0 + 0 = 1 + 1 = 0, and 1 + 0 = 0 + 1 = 1. We're doing the same thing, only we don't wrap around right away, we wait until we get to 26.

The set of numbers {1,2,3,...26}, along with the "operation" of wrap-around addition, forms an object mathematicians call a group. Groups are often denoted by the letter G, and the "G" in the term "(G,φ)-gram" refers to precisely this sort of object. There are simply oodles of interesting examples of groups. The group G we've described here is called the modular group of order 26 and is a member of one of the most useful families of groups there is.

Now let's apply this weird kind of "wrap-around" arithmetic to words. Take the word "WORD," for example. Since φ(W) = 23, φ(O) = 15, φ(R) = 18, and φ(D) = 4, the value of this word is 23 + 15 + 18 + 4 = 60. But since 60 = 26 + 26 + 4, 60 really should be 4 by our reckoning. The upshot is that φ(WORD) = 4.

So what in the heck does this have to do with the poem "Six"?

If you're blessed with a great deal of patience, apply φ to each of the lines of the poem, one at a time. (Spaces and punctuation don't count for anything, so you've only got to worry about the letters.) If you do all of your addition right you'll find that the "sum" of any given line is the number 6. Kinda cool, huh?

Notice also that there are 14 lines to the poem, and therefore if we want to compute the sum of the entire poem, we only have to add 6 + 6 + ... + 6, 14 times. This sum is

6 x 14 = 84 = 26 + 26 + 26 + 6 = 6,

in our wrap-around arithmetic. Thus the whole poem has the value 6 as well!

Finally, note that the title of the poem has the value φ(SIX) = 19 + 9 + 24 = 52 = 26 + 26 = 26. But what happens if we add 26 to any number between 1 and 26? You get back whatever number you started with! For instance, 17 + 26 = 17, since by taking 26 steps, you wrap around to 17 again. This means that even if we tack the value of the poem's title onto the value of the poem itself, the result doesn't change: 6 + 26 = 6. (26 acts exactly like 0 in this regard, and for that reason mathematicians usually call it "0" and talk about the set {0,1,2,...,25} instead.)

I hope this makes some sense.

As I mentioned above, I'll try to describe "Inverse" in layperson's terms in a later post.

Before I tackle that task, though, I'll be posting a 12-part essay inspired by my conversations with the writing folks at CWPA this past week. In these next several posts I'll be using this space as a proving ground for some of my recent thoughts on writing and writing pedagogy, and likely on college teaching in general. I hope that this window onto my process will prove meaningful to some of my readers, students and faculty alike, and that all will feel free to chime in with thoughts of their own.

I'll try to get the first post in this series out tomorrow, depending on the status both of the first of the presidential debates and of my sobriety, the both of which are sure to be inextricably linked.

For now, however, I'm off to bed, as it's late and I've got another full, fun day tomorrow.

Short short short

This'll likely be one of my shortest posts ever, considering my tiredness it won't be long before I degrade into semi-random stsdofsh kfsd,v...

Yes, I'm most of the way through my first Carolina Writing Program Administrators' Conference, and lemme tell ya...I. Am. Lovin'. It.

I've learned a ton from these folks over the past 24 hours or so, and have had fascinating conversations about just about every aspect of pedagogy you can imagine: communities of learning, discursive practices, applications/pitfalls of technology, issues of resource, economies of scale...you name it. I don't have my notepad in front of me, but I know I've got roughly seven gazillion notes to my self on people to talk to, papers to read, ideas to follow up on...it's going to take several hours just to sort out my scrawl.

The further I go in the academic study of academic writing, the more the parallels between writing/writing instruction and mathematics/mathematics instruction simply leap out at me.

We're not all that different, these folks from me. I'd really like to keep in touch with many of them and follow up on their ideas. (Here's a shout-out to any of you who've tracked me down here!)

Okay, I'm off to bed...it's another early morning tomorrow, and it's all too late tonight.

Achieving self-awareness

I handed back the first exams of the semester in my Precalc class today.

By and large they did quite well. There were handfuls each of As and Fs, slightly smaller handfuls of Bs and Ds, and a passel of Cs. It was a nice trimodal distribution, with the middle mode dominating strongly at a mean of roughly 76.2%. The students struggled most mightily with anything involving absolute values, an assessment with which they wholeheartedly agreed when I made that observation at the beginning of class this afternoon. Long division of polynomials? No sweat: they nailed that. Radicals? Not anyone's favorite, but they came out okay on those. Absolute values? Fuhgedaboutit.

I'm not sure what it is about absolute values that proves so tricky to these beginning mathematicians. Admittedly their manipulation involves a good deal of practice and carefulness...but the same could be said of other concepts over which the students have shown a good deal more mastery.

I'm just not sure.

One thing I'm finding out about myself as I teach this class is just how much about mathematics I typically take for granted. I've blogged elsewhere (strangely enough, in this post about absolute values!) about the mathematical concepts I take as given; I'm only just now, in the midst of our first round of testing, realizing the tenuousness of some of my assumptions about students of mathematics.

I have to remember that though these Precalc students are every bit as intelligent as my Calc I and Calc II students, and though they're often exceptional students who are remarkably dedicated to their studies and who put every measure of effort into their educations, they're simply not as experienced mathematically as the students I find in my calc classes. I'm used to starting off the semester with students who come preprogrammed with knowledge of absolute values, of the algebra of polynomials and rational functions, of radical manipulations. And if a student makes it to my class without such knowledge, I've had the right to turn them around and send them down the hall to...

...precisely the class I'm now teaching.

It's new for me, and I know I'm messing up here and there, simply because of its newness.

I'm probably making assumptions I shouldn't make, about the knowledge you might or might not have, about the skills you do or don't possess. How many of you have ever graphed a function before? Probably most of you, but I probably shouldn't make that assumption. How many of you are bored to tears in having to spend an hour and a half going over simple examples of functions and the many forms they may take? Probably most of you...but I'm guessing not all.

I'm learning, folks. I'm learning what I need to say, what I need to do. And I need your help. If any of my Precalc students are reading this, please feel free to drop me a line, or even comment on this post (anonymously, if you'd prefer), and let me know: am I saying enough, or am I saying too much? Am I making any unsafe assumptions?, making an ass out of...you know the drill.

Let me know.

Today's one-sentence essay at the class's close asked the students to define "function," in their own words. Additionally, I asked each of them to indicate whether or not he or she liked poetry. Yes, I'm gearing up to assign another poetry exercise. I'm trying to decide how I'd like to modify this iteration of the assignment: ought I ask them to confine their theme to the topic of "function" in some way? Or should I merely let them traipse about the whole of the realm of mathematics, as I did my Calc I students last Fall?

I'm also toying with an ongoing "Pet Function" project, in which each student will be held responsible for the caretaking of a particular function, whose life cycle they'll sketch as we reach various points in the semester.

That's the skinny in Precalc. The thème du jour in Abstract Algebra was a litany of technical lemmata that read like one of the toledoth passages out of Exodus: "and the existence of cycles begat a decomposition into disjoint cycles, and once these cycles were found, verily the other elements they moved not. Existence begat uniqueness, and so was proven the Fundamental Theorem of Arithmetic, symmetric group version."

The first section hung in there, but the weight of the afternoon's weariness and the length of the week's work and the dark depths of the unintuitive computations we'd had to wade through were too much for the yawning and blinking and note-passing second section. Fuck it, we ended a few minutes early and half a proof short.

Hang in there, folks! Next week we'll cap off the FTA for S_n, we'll meet the alternating groups, we'll start exploring properties of groups in their most abstract form. It's all good.

What else?

In addition to about an inch and a half of grading (Precalc projects, abstract homework) I've got to spend a couple of hours this weekend throwing together a couple more handouts for the Carolina Writing Program Administrators conference to which I'm heading off on Monday afternoon. I've already got a handout that breaks down our (my, Lulabelle's, and Casanova's) views on the "future of writing at UNCA." I just need to gather some qualitative and quantitative data on the previous assessment project...and maybe say something about faculty intentionality. I'll figure it out.

Oh, and! I just found out that I will indeed be co-organizing a short course on designing and implementing writing-intensive mathematics courses at this coming spring's MAA Southeastern Sectional Meeting in Nashville! I'm excited.

Okay, my pizza's coming out of the oven in about five minutes, so off I must be. Take care, Gentle Readers. I'll check in again soon.

Deep thoughts and more dilettantish dalliances

Hey, All! It's been a long time since I blogged here about math pedagogy, so I wanted put out a post that lets you to know that I do indeed still reflect on my teaching, and that I am keeping close watch on my classes as they develop this semester.

I thought long and hard about teaching math this morning as I walked into campus. Specifically, I thought about our use of software in Precalc, and I've come to the conclusion that I'm not very happy with it.

Let me start out by saying that I'm glad that I've had the chance to use the software this semester in teaching the course: it's been an opportunity to gain proficiency with a particular pedagogical technology I'd not used much before. Moreover, I recognize the usefulness of some aspects of the software for some students: the software's ability to generate problem after problem of samples fitting particular problem molds is useful for students who learn well by example and iteration. Nevertheless, while before I couldn't say with certainty that I would prefer to not use computer-graded homework in teaching introductory math courses, I feel that having made use of the software in my course I can credibly affirm that statement.

Without going into detail, let me lodge three objections (relatively briefly! I'll flesh these out later once I've had a chance to further reflect on them, most likely once this semester's behind me) to the software:

1. Its use foregrounds the medium at the expense of the message: in asking the students to master the software's often unnatural and indeed often byzantine commands for entering mathematical notation with a Flash-driven interface, the I feel that all too often the computations the students need to perform to generate a correct answer are overshadowed by the mechanical manipulations they must undertake to enter the answer into the computer.

2. The software places the students at a distance from the instructor, to an extent that effective bridging of that distance vitiates the need for the software in the first place. To wit, even with the ability to zoom in on a single student's solution to a single specific homework problem, the opacity of the software's interface does not allow the instructor to penetrate beyond the student's final response. Should this response be wrong, there's generally no way to deduce from it just what it is the student did incorrectly without asking the student to submit her or his handwritten notes. Of course, I am all for students' working out solutions by hand...I continually exhort them to do their work on paper and use the computer only to enter their solutions...yet if ultimately I have to dig up their handwritten notes in order to tell what it is they're doing right and wrong, what's the point in having the software in the first place? I might as well simply ask that they submit their homework directly to me, let me grade it, see it, be in contact with it, and reestablish the missing and much-missed bond between the students' understanding and my own, without the electronic intermediary.

3. Finally, and most fundamentally, I feel that the software system by its nature reinforces the common and erroneous perception of mathematics as a rigid, timeless, universal enterprise. As the computer is trained to expect only a very particular form of answer to each problem it provides, the student may come away with the mistaken notions that math is a field in which there is a single correct answer, in which there is no gray but only black and white, that process is unimportant if the product is ambiguous, that every instance of a certain calculation requires a single form of solution. All of these claims are preposterously wrong and further the view of math as far-removed from ordinary ways of thinking, as something undertaken only by pointy-heads who've mastered arcane rules of mathematical computation and communication. Human-graded homework, on the other hand, far more sensitive to idiosyncratic-but-correct responses, to slight variations in notation and style, to math's true nature as fluid, era-dependent, humanly-crafted enterprise, allows students to succeed by responding in various ways that reflect their own particular learning styles. The unmediated bond between student and teacher facilitates the latter's ability to convey the perception of mathematics as a ground in which critical thinking can be taught, and the former's ability to construct a personal mathematics all her own.

These are deeply-rooted philosophical objections about which I hope to say more later. I'd be interested in hearing others' take on this matter, I don't claim to have the final word!

On a wholly different note, I've finished my first "exotic" (G,φ)-gram, although I must admit that I'm unsatisfied with one of the steps I took in its construction. I'll admit up front that I wrote another Mathematica notebook to help crunch the noncommutative multiplications that go into the poem's analysis.

The poem is a (D₄,φ)-gram, D₄ the dihedral group of order 8. The homomorphism φ that governs the poem is one Mathematica chose at random; this is the part I'm not happy with, as I'd rather choose a φ that's "meaningful" in some way, that relates each letter to an element of D₄ in a "useful" way. But here's the rub: what choice of φ would work best? My thought was to assign to each letter the longest element of its stabilizer subgroup (relative to the presentation of D₄ in which a represents the reflection in the vertical and b the reflection in the SW-NE line)...but it turns out that practically every letter then goes to an element of the abelian subgroup {1,a,bab,abab}. (The only one that doesn't is "Q", with its funky NW-SE symmetry, which goes to aba...and how often is "Q" used?) Thus if one reflects (or in fact in any way permutes!) the letters of a given line, the value of that line under this particular φ is unchanged.

Boring!

Even worse, any choice of φ(α) from Stab(α) will lead to the same problem! Thus my opting for a randomly generated homomorphism.

My hope was to write a poem in which the value of each line is the group-theoretic inverse of the same line written backwards (letter-by-letter, not word-by-word). The poem below achieves this, although it's an admittedly simple poem. However, it was surprisingly easy to construct (owing probably to the smallness of the governing group), so expanding on this theme would likely not be hard.

So here's the poem, with the homomorphism following it:


Inverse

What kind of mirror symmetry
must a piece possess
for its value to invert itself
when we trade east for west?


Let me give the homomorphism by a listing of the preimages:

φ^-1(1) = {C,L,P,Q,X,Z}
φ^-1(a) = { }
φ^-1(b) = {A,F,O,R,S}
φ^-1(ab) = {I,U,V}
φ^-1(ba) = {G,Y}
φ^-1(aba) = {H,M,W}
φ^-1(bab) = {B,D,J,K,T}
φ^-1(abab) = {E}

Thus only 7 elements ended up in the center of the group, and only 4 of these ("C","P","L", and "E") appear in the poem.

Anyway, I'm having fun. My Mathematica code will enable me to work with dihedral groups of any order, so I may try out a more complicated example later if I have time. Now though, I've got to meet with Sylvester in a few; he and I are continuing research on caterpillar labelings this semester, and we're meeting to debrief after his presentation in the Senior Seminar yesterday afternoon.

Let me close with a note to my Abstract students: keep up the good work! Your committee presentations are already at a very high level. You're doing a great job of highlighting common difficulties and errors, and in giving credit to particularly insightful methods your peers apply. I like that you're all making note of the fact that there's generally more than one way to prove a given proposition. You're also demonstrating a good understanding of the "metamathematical" aspects of mathematical writing. In particular, I appreciate the attention you're all paying to the "Four Cs" criteria, and I hope those criteria are helping you to learn to discern good math writing from bad.

And now, adieu! Thank you for reading.

Six: being a modular G-gram in 14 lines

With the unwarranted assistance of modern technology (on-line word lists and a bit of deft Mathematica programming), I've managed to complete my first nontrivial (G,φ)-gram, Six, printed below.

Six is a (Z₂₆,φ)-gram, where φ(A)=1, φ(B)=2, ..., φ(Z)=0. Every line l of the poem satisfies φ(l)=6, except for the title, which lies in the kernel of φ: strangely enough, φ(SIX)=0. Thus, as there are 14 lines to the poem P, φ(P)=6 as well, since the order of 6 in G=Z₂₆ is 13. (Since the title lies in the kernel, the φ-value of the poem does not depend on whether or not you include the title!) To summarize: the title, although mathematically in the kernel of the defining homomorphism, literally gives the image under that homomorphism both of each line and of the poem all told, with or without the title.

I couldn't resist retaining some classical rhyming conventions, as you'll see in reading the poem below. And the bulk of the poem is iambic, so there's some regular cadence there as well. I'll leave it to the poets among you to scan it properly.

What's next, mathematically speaking? I'd like to challenge myself with a non-cyclic group, perhaps even a non-abelian one...something small to start with, like D₄. I fear that the Mathematica code I've now written to help me search for fitting words in the abelian case will be useless when faced with even such a simple group!

Enough analysis...read! enjoy! And have a wonderful weekend, all!


Six

Are we so free that we must build
tight cages out of self-wrought bars?,
that, every barrier of old o'erperched,
bold walls we must before us raise
to repress our over-anxious powers?

Dreary now is Spenser's sonnetry,
and empty are Keats's odes:
a crowd of words must measure more
than the essence it encodes.

So vowing, to this priesthood
I, in metered ciphered characters,
make an offer of my novice oath,
though a cheap and clumsy canticle,
in a passionately davened prayer.

Ars Poetica

Let G be a group, and let φ assign to each element of {A,B,...,Z} an element of G. Extend φ homomorphically.

We say that the poem P is a (G,φ)-gram if there exists a fixed g in G such that for every line l in P, φ(l)=g.

Theorem. Every poem is a ({1},ι)-gram, where ι represents the trivial homomorphism.

Exercise. Construct nontrivial (G,φ)-grams. In particular, construct gematriyists' (G,φ)-grams, in which G = Z₂₆ and φ(A) = 0, ..., φ(Z) = 25.

This assignment is due whenever you feel like turning it in. Spelling counts. Pagination is optional.

Post #200

It's been a long and tiring week, but a pretty good one.

We're now about three weeks into the new semester, and I'm feeling good about it so far: my classes are fun, I've set a good pace in both, my students are engaged and willing to work.

The past week or so I've felt on edge about something, though, as though I've not been able to find a groove. It wasn't until yesterday morning that I put my finger on what it was that was keeping me from settling in: homework deadlines.

In every class I've taught since I started teaching here three years ago, all homework, projects, papers, projects of any kind, have always been due at 5:00 p.m. on Friday evenings. This singular deadline meant that I didn't have to think about when a particular assignment was due, it meant I could flip from class to class and be able to say in the wink of an eye when students owed me something. It was easier on the students, too, for they knew the answer they'd get if they asked me when something was due to my desk.

It's the course software for Precalc that's been throwing me off: freed from having ("getting"? I'm missing it!) to hand-grade the students' homework, I've been able to glibly toss about due dates without any restriction imposed by my grading schedule: Monday at midnight? No problem. Wednesday at noon? Fine, again...and somehow that glibness carried over to my assignation of Abstract deadlines, too, so their homework has been due willy-nilly throughout the week.

No more.

The nerve-wracking chaos ended yesterday when I informed all of my classes that I'd be reverting to the Friday at 5:00 deadline. It's a seemingly minor change, but we'll see if it doesn't put me at ease!

Meanwhile, as I mentioned above, all's going quite well. I'm well aware that we're making our way through Precalc a little more slowly than we should be, but I imagine once we get through the algebra review I'll pick up the pace a little bit; I feel that it's important to establish a firm foundation of algebraic skills before moving into a place where we'll have to apply them. I've just passed out the first written assignment, asking the students to minimize the cost of constructing a box with a fixed volume and certain dimensional constraints, and to put all necessary data and computations in the form of a written report to a container manufacturer. I'm eager to see how wel they handle this project. I'm imagining that it'll be a snap for some, and a mountainous challenge for others.

Abstract's chugging along; we've had two sets of committee reports so far. Yesterday's reports were very strong, they've already begun to break away from the "here's the answer" mode of reportage, in which they don't really comment on their peers' work so much as solve the problem for the class. "That's not really helping anyone," I've reminded them. I've exhorted them to provide helpful, respectful, and specific feedback: "good!" is as useless as "wrong!".

I was particularly gratified by one of the students' reports yesterday, in which she indicated the importance of clarity and composition, emphasizing how even if someone had the right answer it was often difficult to discern this if the proof's wording were awkward, if its structure were nonexistent. It was clear that she'd read the "Four Cs" style sheet, if nothing else.

Well, I'm off...the weekend lies before me, and I've got relatively little grading to do (a few factoring problems and a team quiz from my Precalckers...shouldn't take more than a couple of hours during some football game later), so I might actually get some down-time tonight.

Happy 200th post!

Studies in comparative pedagogy

Today's been a bit better. I've gotten a lot done, both research-wise and teaching-wise, I've met several more of my new students, made a Math Club meeting and a meeting with the high schooler I'm mentoring as she helps me run Super Saturday this semester, and I sat in on Lulabelle's sociology class, Social and Cultural Inquiry.

To me the most beneficial aspect of my sitting in on her classes (and her sitting in on mine...tomorrow it's her turn to come to Precalc) is the chance it gives me to observe many of the same pedagogical techniques I use in my classes in an entirely different disciplinary setting. Many of the same tactics are at play: small group work, writing-to-learn, brainstorming, student discussion. And while the groundwork for Lulabelle's course is entirely different from that of my own, the ends met by the tactics she employs are ultimately very similar: both of us want our students to learn how to think critically, how to assess data and other evidence, how to solve problems and make connections between the distinct and discrete bits of information they pick up here and there.

After 75 minutes had passed, I'd punctuated my notepad with no fewer than nine ?s indicating follow-up questions I had for her when we met after class:

"How do you draw the quiet ones into the discussion? Do you?"

"Have you read The experience economy?"

"Do you like people to raise their hands [before speaking]?"

"Might this work well with small 'break-out' groups to cogitate solutions before reconvening?"

"What's your motivation for reading the daily reading out loud? Is this to achieve a multimodal pedagogical style?"

And so forth. I'm finding our exercise in comparative pedagogical practices very fruitful; it's making me much more aware of my own choices as a teacher. I hope Lulabelle's getting as much out of it as I am.

Now, I'm off, into the rain. More soon, I'm sure!

Weak Two

You ever have one of those days when you just can't seem to get traction?

Today began early, and pleasantly enough. I had a nice walk under cover of slate gray skies, the sun's rise obscured by a feathery blanket of cloud. It was muggy, but comforting.

I'm trying to figure out where my Abstract folks are right now: this morning had me meeting with several of them about the committee problems that were due in class today. A few of them seemed a little overwhelmed, but in class most seemed on the ball. My hunch is that at this point what we're doing doesn't seem relevant, since we're developing skills and techniques, not results that are important per se. Moreover, with the scant background we've developed so far, it's difficult to see how what we're doing fits together with the overall thrust of the course. Once we start doing congruence arithmetic (which we're poised to start discussing on Friday), things will start to come together. In particular, I'll be able to point at congruence classes and say "See? See?!!? Groups!"

From this morning's Abstract section (I'm sorry I was a bit sluggish, y'all...I had a too-short weekend, too!) I went back to the office and took care of a number of bureaucratic tasks...those damned things pop up like whack-a-mole moles at a second-rate funplex. I was busy enough to not even notice the clock inch towards 1:45.

I felt out of sorts in Precalc, too. I think what threw me off at first was my continued frustration over the software. I'm sorry, y'all, that registering on Educo is such a pain. I promise you, I promise you, that once you're all in the lane and plugging away on-line, things will go a bit more smoothly.

My stutter-step start to that class continued, and I felt a half-beat behind for the whole hour. I think part of it's due to my sub-par preparation...I realized about halfway into one of the exercises I'd prepared (describing the interval defined by |x+6|<3) that I hadn't completely prepared a description of the algebraic solution to the problem. That oversight's due in part to my unfamiliarity with teaching this material (again, and again and again: first time doing this, folks!), but part is also due to my abhorrence of "plug 'n' chug" explanations.

Me: "You'll wanna do it this way because if you remember, what the absolute value |a-b| represents is the distance between a and b..."

Not me: "Why do this computation? Because it works, that's why."

Ouch.

I need to find a middle ground.

Happily, I've noted a couple of students who are very attentive, diligent, and responsive to my efforts...they're clearly engaged in class and give great non-verbal feedback as I'm teaching. By watching them I can tell if I seem to be getting through. My thanks to you!

I have to say that I enjoyed my second section of Abstract today, overly large as the section may be. Although there were a few people who I sensed were having difficulties with Euclid's Algorithm, I think most were picking it up, judging from the alacrity with which they offered to share their computations with the class at large.

My grades for the day: Abstract: B+; Precalc: B-; Overall: B.

Bleh.

I'll be ready for Wednesday, folks. You can count on it. Thanks for your patience with me.

Now, off to bowl!

When must it be that |x+1| - |x-3| = -4?

Math works.

We've built it that way.

Having been doing math "professionally" for quite some time now (long enough that I'm not going to bother counting the years) I tend to forget just how well-built the mathematical machine really is.

This semester's already forcing me to reflect on the inner workings of some simple math concepts that I've sadly learned to take for granted.

Like absolute values: how marvelously they work! You really can depend on |a-b| = a-b to hold as long as a > b. Who'd'a thought? From the textbook I pulled the following absolute value problem with which the students are soon (today or Monday) to grapple: "find a formula for |x+1|-|x-3|, for any x."

Just now I was sitting here at my desk working through the problem so that I know we won't encounter any roadblocks during class...and I was marveling at how well one fares if one forgets everything else one knows about absolute values beyond the its basic defining formula. It's all just adding and subtracting, after all: if x < -1, you get |x+1| = |x-(-1)| = -1-x, and |x-3| = 3-x, so the difference you seek is -1-x-(3-x) = -4. It's marvelous!

This is what I'd hoped teaching this precalculus course would do for me: it's bringing me back into touch with the bits and bobbins at the base of mathematics that I've too long taken for granted. It's making me rethink why one does the things one does, and reflect on what one's really doing when one thinks one is doing something one's not really doing at all.

It's marvelous.

Day Two: Return of Dr. Happy

I'm lovin' this semester, lemme tell ya. I'm still pretty much on top of the world (a friend called me "Happyman" today).

I spent the day getting ahead in my Precalc notes and further revising my paper with Tip...and meeting with my new students! About my job: I come for the glitz and the glamor, but I stay for the students. These people are great. They're animated, dedicated, smart. It's just one day into the student interviews and I've already scoped out another potential undergraduate research partner and another maybe math major.

167 was a lot more fun today, we actually got to talk about math! If only there were no software glitches. It's proving a pain to get everyone enrolled in the Educo website.

Okay, I'm sorry for cutting this so short again, but I must away to repast.

More later!

In the bag

I am tired.

But happy.

Going straight from 167 to 461 with only a five-minute breather was a pain in the butt. I guess technically my "velocity" was 294/5 = 58.8 course numbering points per minute. Shoot me now.

I thought my first day of Precalc went well...to the extent that I can tell that it really was a Precalc class. I was a little frustrated that explaining the course software and demonstrating its basic functionality to the students takes so damned much time. Fully half of the period today was devoted to pulling the software up, showing students how to log in and pull up the homework assignments, and showing them how to register the software. (It would help matters if there were one way and only one way to register. Grrrrr...)

Tomorrow, folks, tomorrow: I promise we'll start doing math tomorrow. If any of my students from MATH 167 are reading this, another thanks goes to you for slogging through today's bureaucratic rat's nest. I appreciate your attention, and I look forward to working with you.

On the plus side, I believe I managed to nail all but one of the students' names on the first try...at least among those whose names I should have known. 22 out of 23 ain't bad.

From 167 it was a 30-second walk down the hall to the bigger, rowdier section of Abstract. I'm going to need duct tape and thorazine for this crowd, something tells me. They're a lively bunch, but very dedicated to their work, too, and we had some good brainstorms. (The first day's activities asked the students to abstract from the set of integers the essential properties that make this set a group under the operation of addition.) I hope now that everyone's got some sense of a what a group is, so we can promptly push the definition to the back of our minds for a few weeks and work through some preliminary hoo-hah.

I am bushed.

But elated.

It's been a great day. What will tomorrow bring? How could it possibly get better than this? I love my job!

One down, two to go!

MATH 461 is underway. Today's class went splendidly. The students worked well together, showed only the slightest trepidation about hitting the board first thing out of the gate, and came together to craft a great classroom experience. I'm happy.

God, I love my job!

Okay, now to prepare for the next class...Precalc, here I come!

28 minutes to go!

28 minutes to my first class of the semester!

I feel ready to face today (it's a beautiful day, by the way), having had the help and support of many wonderful colleagues and friends. In particular, many thanks go to my colleagues Fyodor and Ignatz for helping me to get on the ball with the Educo software for MATH 167, to Lulabelle and Thalia for their perpetual moral support, and to Maggie for everything!

Now, to get set for the first (and smaller) section of MATH 461...

...further bulletins as events warrant!

Jitters

Yup, it's that time of year again...Fall's nearly here, classes begin in less than a week, and I'm freaking out that I'm nowhere nearly as well-prepared as I ought to be.

Likely that's not true, but hey.

What's eating at me most this time around is my preparation for Precalc. I've never taught Precalc before (not here, not anywhere...the closest thing I've ever taught was Vandy's one-semester "Survey of Calculus" that took the highlights from a two-semester course of calculus and condensed it into a single semester for the benefit of folks like business majors who didn't want to bother with proofs), for one thing, and for another the course as we teach it involves a substantial on-line homework system with which I have next to no familiarity. It's easy enough to learn, but I'm still puzzling through the details myself and I know it's going to be a few weeks before I'm proficient enough to help guide my students through it.

I'm definitely feeling underprepared. With a week to go, I've got the first couple of days of both of my classes ready: Precalc will start off with a review of the structure of the real numbers, before vaulting into absolute values. Abstract begins with an examination of the Division Algorithm and related results. I'd like to be a week further ahead than this...I've still got next Monday and Tuesday to prepare, though, and with classes starting on Tuesday the hum and thrum of surrounding activity should invigorate me.

Just a couple of days back I was chatting with one of the REU students, and we talked about how we both feed off of the energy of the semester, we get high off of our work and the work that's demanded of us. I'm looking forward to getting that high again.

On a positive note, I've come up with several good ideas for relevant, meaningful Precalc projects that should not only prove challenging and enlightening to the students, but should also give them a leg up when they get to certain cognate problems in their future calculus classes.

For instance, a standard optimization problem in first-year calculus courses is to maximize the volume of a box that can be built from a piece of material of fixed dimensions, according to fixed parameters. The only step that actually involves calculus is the differentiation of the formula the students obtain for volume as a function of one of the linear metrics for the box; deriving that formula in the first place, plotting it and interpreting the graph, and determining maxima and minima visually are all actions that require the skills taught in Precalculus, and students who accomplish such tasks now will be more ready to handle the cognate problems when they recur in the calculus curriculum.

Another plus: while running into campus this morning I finally figured out the general form for the function that governs the work Thalia and I have both kept working on since parting a little over a week ago. I don't know why I didn't see it before, but now with the general equation in mind we might be able to make real progress on actual proofs.

Ah, research.

Here's hoping this semester goes well! Once things get underway, I'm sure I'll slip into the right frame of mind. I always have, for nearly three decades now. I'll be fine. Besides, I've already got pledges of moral support from two dear friends should Precalc prove to be the back-breaking straw.

Much as I know I need to prepare a bit more, I'm having difficulty focusing. I'm feeling very sluggish today. And why should I not be? The weather took a much-needed turn for the rainy overnight, so the sky is a shroud of white. Campus (where I sit as I type this) is dead: my department's a tomb, the building's air is still and warm (the cooler's on the fritz...again), all is quiet.

All right, I'm going to head home...no sense in cocooning myself in this scholarly sarcophagus any longer today. I'm sorry this post has been so scattered and erratic.

To be continued...

Halfway in

I've had a week now to recover from the departure of my summer's colleagues, the wonderful students that made up my REU team for the past eight weeks. The last of them took flight almost exactly a week ago this moment, winging her way back to New York last Sunday afternoon.

I miss them. They're great students, smart colleagues, and close friends. I look forward to seeing them at conferences, working with them on papers, growing up and learning by their sides.

For now they're gone, but the work continues: with two of them (those with whom I worked most closely with) I've already exchanged write-ups, graphs, ideas...yes, work continues.

But the cycle's begun anew: I arrived at my office at 7:15 this morning to find a note left on my door by two of my favorite students from last year, informing me that they're back (for RA duties, no doubt) and were surprised that they'd missed me. They were operating under the (not-all-that-erroneous) assumption that I'm never more than a hundred yards or so away from my office.

I spent this morning helping first-year students move in. For three hours I shuttled luggage carts back and forth between parents' cars and dorm rooms, schlepping all manner of stuff into Mills Hall. Fully laden one way, emptied out on return, several trips for several students. One's a budding music major, another's contemplating environmental science. A few are undecided: biology? History? "I think I want to teach."

"Math?"

"No! I'm not really good at math."

"That's what everyone says until they find out it's not true," I assured her.

It was refreshing, revitalizing. Its sense of newness and rebirth stood in contrast to the moving out my REU students did the weekend before.

My move into Johnson-MacFarlane Hall at the University of Denver 15 years (!) ago was nothing like this: my dad parked his truck in the middle of the phalanx of vehicles that lines that lined High Street for a block in either direction, and as soon as the dorm's outside doors were unlocked students and parents descended on the dorm like sharks in a frenzy. It was chaotic. There were no faculty or staff helping people to move in, there wasn't even anyone to direct activities or answer simple questions (where's the bathroom? Is the dining hall open? Where's the nearest ATM?)...that day I saw my RA cry for the first of what would be many times that year. (She was a gentle soul and didn't deserve some of the folks who lived on our floor.)

This coming week will see me getting the first few days of my two classes ready. I've read through the first few chapters of the text I'll be using for Abstract Algebra, and I've brainstormed a few opening exercises for that course, but I've yet to commit anything to paper. Tomorrow I'm meeting with one of my colleagues who's taught Precalc a number of times, and I'm hoping he can then give me a rundown on the topics I should be sure to address during the next fifteen weeks.

I've also got to get some more work done on the presentation on the writing program I'll be making at the Carolinas Writing Program Administrators conference in a month or so. Lulabelle and I met late last week to discuss that issue, but we spent most of our meeting chatting and going over the draft of my math poetry paper and syllabi.

Much work to be done, I just don't feel like doing it right now.

Time's running out, though...not long before classes start, and then where will I be?

Argh.

Okay, I'm off...I feel like I've got a lot to say, but so few words with which to say it. I've got to gather them together and compose my self before I can make any more sense.

Halfway gone

As you may surmise from my four-week blogospheric absence (an absence that has elicited chagrin from at least a small handful of dedicated readers...most of them connected to the primary reason for my absence in the first place), things have been a mite busy in my neck of the woods.

The program's nearly complete: four have now left, four remain.

Yesterday was the last "official" day of the 2008 REU, on which the students gave their final presentations and partook of a summative ceremonial repast at the West Asheville Asiana Grand Buffet on Smoky Park Highway.

Huzzah.

The presentations were wonderful, truly of a quality greater than that of most faculty conference talks. Although a few of the kids were obviously nervous, they did marvelously.

I'm stunned by the caliber of research they've produced, as well. The new knowledge they've brought into the world should make fodder for at least four or five papers, and all of the projects they've begun are worth following up. Some have barely started! I'll probably spend a good deal of time in the next few weeks beginning my share of the papers that should come from the program.

The better part of my time, however, will be dedicated to getting ready to teach this coming term's courses: Precalc and Abstract Algebra. I've not taught Abstract here before (I did a semester in Illinois), so that'll be something new...and I've never taught Precalc...nowhere, no how. I'm looking forward to that experience. I've already met a few of my students (they're friends of past students), and they seem a good lot. It's going to be a departure for me content-wise, but a worthy challenge pedagogically.

Elsewhere in my professional career, I've now written about four or five pages of my paper on math poetry. I've yet to really pick apart many of my students' responses to the experience of writing math-themed poetry, aside from identifying a "sense-making" trend in their poems (about which I believe I may have blogged before). My reflections on the students' reflections should form a core part of the paper, and I hope to get to that soon.

I've also got two papers to referee and one to review...reading material for the beach when Maggie and I head down to Savannah for a few days at the end of the week after next.

Ah, well.

I'm going to go now and have a bite of breakfast. I plan to return more soon than four weeks hence, so stay tuned.

Let the world wait

Last night
a weary worldly friend
crept in and sat down by my bed.

"What's happening?"

"What's not?"

She settled in.
I sighed.

Staring at the ceiling fan,
I told her of a summer long before,
of when sixth grade had ended
and for the last time until September
I'd bounded home from school.

"My pack slid from my shoulders,
my shoes fell on the floor.
Down the stairs I flew, twelve of them in seven steps,
my brother a blur behind me.
On the concrete floor we crouched,
a pair of academic acolytes.
Gleefully we lit a fire:
in the wood stove went
papers,
notes,
diagrams.
More still:
maps I'd drawn,
games I'd made,
worksheets,
outlines,
quizzes,
tests,
an unsent love letter or two?
page after page after page after page after page after page of
all that made
a year of school
vanished,
a pile of learned lifeless ash."

Over.
So easily ended.
As though a heavy blade had sliced the summer from the spring,
I was cut free
to waste those sunny days in whatever way I could.

The ceiling fan spun on.

"And now?"

"And now it's hard to extricate
tomorrow from today.
So much undone
makes what is done
remain unsung.
I can't rest on my laurels
if my laurels aren't yet won!
If I flew westward fast enough,
might I outrace the sun?"

"Now it's time to get some sleep,"
she said.

Nodding, I agreed.
I let her close my eyes.

"Close the door.
Shut me in.
Make me a cup of chamomile tea.
As I sit sipping,
let the world wait.
For a day,
let the world wait."

Week Four progress report

Halfway.

Damn.

This summer has been flying by.

Our program evaluator, Ophelia, is currently in town (she's been down in Jacksonville, FL, preparing for her upcoming year of teaching there). She arranged for the mass comm people to videotape yesterday's student presentations, which she herself attended as well. (Not to mention one student's mother and another student I'm doing research with who is not taking part in the program...) The talks are improving steadily: they're all getting better at pacing themselves to say what they need to in fifteen minutes and no more, and they're getting better at making use of board space and multimedia tools. I was particularly impressed with Ethelred(not the Unready!)'s presentation, in which (I only noticed in reflecting on it, while composing a feedback e-mail to him) he didn't write a single alphanumeric character on the board. His fifteen minutes were focused on a series of well-chosen diagrams that illustrated his technique, a complicated generalization of a method I developed last year for navigating internally disjoint edge paths in a certain kind of graph. His method seems to work, but he's yet to make it one hundred percent rigorous. I eagerly await the finished algorithm.

Others made solid presentations, too: Desdemona seemed hurried, but she too made great use of appropriate pictures in explaining how to find homomorphisms between graphs; Camilla and Dione worked together to weave a multimedia presentation on growing networks that was bursting with differential equations; Norton couldn't resist replaying his PowerPoint presentation illustrating Bach's unintentional use of the theory of cyclic subgroups in his Violin Concerto, BWV 1041; Uwe gave the first public presentation of his algorithm for finding optimal L(2,1)-labelings of internally three-regular trees; Hector talked about his progress through the research program that should allow him to show that all algebraically realizable finite hyperbolic tilings are indeed constructible; and Thalia put forth her new technique for hunting down gracefully labeled paths...her work this past week has been really interesting stuff for me, and I've had a blast trying to tell Mathematica how to help us out in our search.

From a pedagogical point of view, it's clear that the students are helping each other out as much as we (the faculty) are helping them. Though we see four or five of them regularly during our common morning office hours, and we see a few more somewhat regularly in the afternoons, in between those sessions they're all progressing in great loping strides. That progress is often attributable to one another: "yeah, Ethelred helped me code this in Mathematica...", "Thalia can help me with the differential equations...", "Uwe figured out how to do that..."

The quality of their written work is improving substantially, too: several of them are already producing work that's beginning to look publication-ready, and we've got four weeks to go yet!

Craziness.

I really need to start thinking about my courses for the Fall semester. Ouch.

Not to mention the two grants that I promised myself I'd be working on for the fall.

And the review I've got to be writing.

And the papers I'd like to finish off and send on in.

Why is it that the summer is busier than the school year?

Anyone who says academics get a three-month paid vacation is full of shit.

Pardon my French.

Of course, I've been spending my free time reading up on the pedagogical uses of poetry as I start to write my paper on poetry in mathematics, so I can't complain overmuch about not having time to complete the above tasks. I've got a stack of papers, mostly from educational theory journals, some from general qualitative research publications, dealing with the roles poetry can play in research and education. The literature is somewhat sparse, judging from what I've found, but I have found some relevant material. Not surprisingly I've seen nothing about poetry in the "hard" sciences apart from the offshoots of Clemson's Art Young's Poetry Across the Curriculum movement.

I have managed to get some great responses from my student poets from last Fall. I'm going to send out a reminder to those from whom I've yet to hear, and maybe include a couple more students in the process before I begin writing.

For now, I've got to go and clean up a bit for the company we've got coming over for our inaugural Thanksgiving in July party.

Arrivederci!

"So they remember you when you're gone"

Life seems shorter after dark:
you can barely see your fingertips.

The concrete path ahead ends
in the umbra of the streetlight's edge.

What's out there?
No one knows.

Acre after acre
of unplowed, unfurrowed, virgin land.

Mortality is closer
when the sun has gone to bed.

The stubborn say "I'll beat this,"
while the tired say "I'm dead."

Week Three progress report

Doin' good.

At least, I think so.

After gentle admonishment about the lengthiness of a few of the student presentations last Friday, yesterday's talks all came within a minute or so of the universal 15-minute maximum we'd prescribed in advance. ("15 minutes, MAX!") The talks were solid, with more eye-popping graphics, slick LaTeX work, and tons and TONS of data. They're beginning to come up theorems of their own, they're making conjectures, their laying out formal proofs. It's coming together. I've yet to get a look at their written work for the week yet, though I've gotten sneak peeks through reading their drafts (thanks for bringing that to me, Uwe!) and seeing their presentations. Their level of sophistication appears to be rising. I have to admit that I'm looking forward (sick scholar of scholarship that I am) to the point at the summer's end when I can lay their weekly assignments side-by-side-by-side-by-side-by-side-by-side-by-side in an attempt to track the development of their technical writing proficiency throughout the program.

One more thing I'll say about these students: they (and one or two of them in particular) ain't shy about asking for what they need from us: more ideas for things to do around town, modifications to the presentation schedule, caffeinated tea...At their behest we've scheduled a "grad school forum" in a couple of weeks, at which time we'll give them the chance to ask anything and everything they'd like to about selecting, applying to, transitioning to, and surviving grad school.

I hope that their forthrightness is indicative of a sense of belonging and collegiality, and not just out-and-out boldness. Probably a little of both.

I really wish that I'd had a chance to take part in an REU when I was a student. I know I'd have been up to the challenge. I realize now in reflecting back on my undergraduate experience that though I had solid professors who were active in their fields and who did a great job in the classroom, their mentorship didn't extend far beyond the classroom door. I fear that I was simply a student in a Ph.D.-granting department whose Ph.D. program was undergoing a period of latency, and that the faculty didn't really know how to properly mentor talented undergraduate students...I really should have been told about journals geared towards undergraduates, I should have been encouraged to go to undergraduate-appropriate conferences, I should have been pushed to apply for REUs (there weren't as many back then as there are now, but there were a good number by then, and I'm sure I would have landed a spot in one had I applied to a few).

Ah, c'est la vie. I'm pretty happy with the way my life has turned out, and now I get a chance to live the REU experience vicariously through my own program. Woo hoo!

In other pedagogical news...well...there's not much more right now. I've started to plan my syllabi for the Fall semester (two new preps! I've not taught Abstract Algebra here before, and I've never taught Precalc). I met with Lulabelle and Casanova over lunch this past Thursday, and we planned our next move in the new writing assessment study. Casanova's going to focus on analyzing the syllabi as faculty products, and I'm going to try to better articulate my hypothesis that the rubric we constructed at the conclusion of the last study was (perhaps unintentionally) skewed towards assessing more "traditional" research theses. "We might just have to report that it's not possible to create a universal rubric capable of assessing the writing of any given course," said Lulabelle.

Well, for now the waterhole calls...I must away.

I'll check in again soon...next weekend, at the latest!

Week Two progress report, v.1.0

Yowza, I'm beat.

I'm not going to write much right now, as I'm tired as hell and the needle's pegged to the far-right end of the Incoherometer.

I will say simply that the students made their first end-of-week presentations today, and they were magnificent. Aside from the so-so time management displayed by a few overeager presenters who might have taken a taaaaaaaaaad too long in singing their songs, the talks were marvelous. All eight made great use of visual aids (including Mathematica notebooks, LaTeX files, PowerPoint, good ol' fashioned boardwork), all were careful to provide their peers with history, background, and context for their respective projects, all were able to make clear some very high-level and abstract mathematics, and all held one anothers' feet to the fire through careful and clever questioning...Camilla and Thalia were particularly good at this.

To those of the students who may be reading this: I promise personalized feedback on individual presentations, no later than Monday!

There are some great projects afoot. At this point they're working on color-critical graphs, channel assignment utilities, group theory as applied to musical theory, the limiting connectivity of field automorphism graphs, the structure of growing networks of churches, statistical measures of preferential attachment models for random tree growth, variations on previous definitions of finite hyperbolic tilings, and an algebraic/geometric characterization of graceful labelings on graphs.

This last one, Thalia's pet project, has really got me intrigued right now. Thalia's taken as a starting point a "classical" observation (that a graceful labeling of a graph must place the highest- and lowest-labeled vertices next to one another, and in order to obtain the next-highest label exactly one of two configurations must obtain, and so forth), but her logic pushes her along a path I've never seen taken before, on which she's found a way to view the collection of graceful labelings on all graphs of a given size as a collection of lists of states, on which an algebraic structure can be placed. Her insight's led to some interesting ideas, and I'm eager to see where they go. We met for a couple hours this afternoon to hammer out some Mathematica code and to organize our collective thoughts on her construction. She's a fast thinker, and is a lot of fun to work with.

In other news, I met with Lulabelle and Casanova yesterday to begin our analysis of the materials submitted by our colleagues at the close of this past year's writing assessment study. The pickin's are slim, and we've decided to start our analysis by reading through those syllabi we've so far collected from our colleagues who were directing discipline-specific writing-intensive (doubly-hyphenated-adjectival-phrasedly-modified) courses last fall. Our task at this point is merely to reflect upon our critical reading of these texts: what is there in the syllabi that indicates the instructor's attention to writing pedagogy? What implicit elements might the students (or faculty) infer from a reading of the syllabi? What elements are common to the syllabi? Which are present in some but not others?

And so it goes.

Judging from the penultimate parenthetical comment in the previous paragraph (oh, the alliteration!), it's time for beddie-bye.

Until tomorrow, then, I am always yours...

Week One progress report

How we doin'?

If I had to say off the bat, I'd have to come back with a "not too shabby."

The students are fast, sharp, and diligent. They excel not only at the formal mathematics but at the auxiliary activities: LaTeX, Mathematica, you name it. They're kickin' butt and takin' reservations. I've already begun to think of them as able colleagues, and I and my fellow faculty have encouraged them to adopt the same viewpoint. Hell, no doubt in less than a month they'll be slowing down to let us catch up to them.

I don't know how much of their apparent progress this past week can be chalked up to the design of our program (a design which I feel is much tighter, clearer, more well organized...well, just so darned much more effective than last year's) and how much to the strengths of the students themselves. I'll be eager to know their take on the matter.

Meanwhile, I'd like to take a moment to look back over the first week and hold it up against the learning goals I'd put forth for the program's participants...how well do they match up?

1. Have a grasp on the fundamental concepts of graph theory, group theory, metric geometry, and dynamical systems; and understand cutting-edge open problems in these fields. We're there. I figure that in the first week we've given them the equivalent of one half to two thirds of a semester each of graph theory, abstract algebra, and dynamical systems. (Not to mention a year's worth of seminars, as Tip has said.) They've soaked it up, and they've been far more than passive note-takers. We've given them some pretty challenging exercises, all of which they've navigated with aplomb. Their mastery shows best in the brilliant questions some of them have asked (Norton's got a thing for topology, judging by his constant queries about that area) and in the frequent and astute observations they've made (late yesterday afternoon Thalia noted an omission in a project Tip and I have been working on for over a year; it was easily fixed, but her insight led us to find and fill the gap in our method...summer research has begun in earnest!). I think they've got the "fundamental concepts" down and are ready to hammer away at the open problems. A couple have already started to apply the concepts we've covered to original ideas. Uwe's application of graph theory to analyze the efficiency of saxophone fingerings is absolutely delightful!

2. Be able to make effective use of research databases, including MathSciNet and arXiv. We spent an hour or so on Wednesday talking about these databases. Our conversation, directed largely by Camilla's questions about how one becomes a referee, how one becomes an editor, how much publication is expected of faculty, and so forth, took us, on this third day of the program, deeper into the issues surrounding professional development than we ventured all summer last year. I don't think the conversation's tangents were distracting; the students did admirably on their first research assignment. On Friday each submitted the results of at least two literature searches, each including at least three academically rigorous references. The annotated search results proved the students' comfort in working with the relevant databases. I think they're ready to ply their skills to authentic searches.

3. Feel comfortable in generating new questions and topics for research, either by modifying and generalizing existing statements, or by branching off into uncharted territory. The verdict's still out on this hard-to-assess issue, but judging from the number of questions several of the students have put forth during the past week, I don't think we're going to have to worry about this one. A few of the students are more reticent than the others, and it's hard to say whether this is out of shyness or lack of skepticism. Time only will tell.

4. Appreciate the qualities that make for a friendly and effective research community. Progress towards this cognitive goal will be exceedingly hard to measure without hearing from the students themselves. We've got our first meeting (by conference call) with Ophelia (our evaluator) on Monday, and by then I hope she's had a chance to glean some feedback from the students.

5. Understand how to make high-level use of a computer algebra system such as Mathematica. On Thursday morning we spent about two hours plowing through several Mathematica notebooks put together by my currently-honeymooning colleague Nostradamus. About half of the students have had previous experience with Mathematica, and a couple others had worked with its main rival among computer algebra systems, Maple. As far as I could tell, there were no major snags. The most heartening indicators of student success are the applications the students have already found for the software. Uwe's made extensive use of its high-level functioning as he's undertaken his analysis of saxophone key structures. During the last few days he's developed incredible proficiency in using Mathematica's Combinatorica package. Meanwhile a few of the others put the software to work in crafting a T-shirt design featuring a complete graph on 11 vertices, each of which holds the name of a program participant. (Ain't they sweet?)

6. Have facility in working within the LaTeX typesetting environment. They made mincemeat of yesterday morning's typesetting exercise, in which I asked them to mimic as closely as they could a one-page sample document I gave them. Though I could have made it substantially harder, that one page featured a number of minor obstacles (double quotation marks, positive horizontal spaces, etc.) as well as some tricky mathematical typesetting conundra (arrays, commutative diagrams, variably-sized delimiters). Three or four of the students have used LaTeX before, and they made short work of the exercise (Norton finished in about 50 minutes, over a half hour faster than the final finisher). Even the least experienced of the bunch managed to put together a beautiful document, and in a totally reasonable amount of time. Just a week in, and they've all got text editors and compilers installed, ready to be pressed into service. I'm eager to see how their first "papers" turn out; they're due on Monday.

And speaking of Monday's assignment...

7. Be familiar with the structure of a mathematical research paper, and be able to construct such a paper. Monday's assignment calls for them to produce a brief report mimicking the structure of a formal mathematical research paper. At this point the content is unimportant; I merely asked them to write on a topic we've talked about at some time during the first week. What's important is that they begin to understand how a mathematics paper is structured. Though I'm by no means expecting perfection from the students on Monday, I've already come to expect excellence.

8. Have confidence when communicating mathematics orally, to an audience of peers or to an audience consisting of research professionals. At this point we've had no structured oral presentations, but all three of us faculty facilitating the program have asked the students to perform impromptu exercises at the board. To a one they've performed with skill and grace. As I mentioned in Tuesday's post, on that day the students ran the show for roughly four and a half hours, during which time only infrequently were I and my colleagues at the board. Surely their first oral progress reports, to be given next Friday morning, will be strong.

9. Understand the dynamics of, and feel comfortable working in, a collegial research group. As somewhat of an outsider, it's difficult for me to say how the students are functioning as a unit outside of the seminar setting. However, it seems to me that the students have already formed a tight-knit social group. They're supportive, they stick up for one another, they're good at reading and meeting one anothers' needs. Camilla's fantastic at sensing the mood of her friends and conveying it to the faculty if there's any doubt we don't grok what's going on. With her confidence, intelligence, and outspoken character, she's a natural leader. Meanwhile, as I'm sure you can judge from the comments to the above learning goals, the students' interactions with faculty bespeak a high level of comfort in working with us. I think we've already made good progress towards meeting this particular goal.

10. Possess a healthy skepticism and authority to challenge as yet unproven results. This is a tough one to measure as yet. Once the students have begun to put together coherent research agendas, I'll be able to say more about the strength of their skepticism. Nevertheless, if confident questioning is any indicator of the students' sense of authority (and I believe it is), these students are well on their way to developing the ownership of mathematical knowledge it will take to make them exceptional scholars.

11. Be ready to present their findings to an appropriate audience at a regional or national conference. This one too is difficult to assess, and might remain difficult for some time yet. The students have hardly begun to present to one another, so it's understandable that they might be nowhere near ready to present to a broader and perhaps (though as likely perhaps not!) more sophisticated audience. I expect great strides will be made in this direction before August 1st comes around.

12. Have a sense as to the structure of the mathematical community at large, and understand their own place in it. This goal too is a bit nebulous, and evidence of its achievement will really only become evident by the end of the program, at which point it might be witnessed by maturity, poise, and overall mastery of the tasks demanded of a professional research mathematician. However, as I said above, we've already had a number of serious conversations about what it means to be an academic mathematicians, and I delightfully anticipate many more before the summer's through.

Keep it up, my friends! You're doing wonderfully, and you've made the first week a joy. I look forward to the next seven weeks, and beyond, in which time many of you will no doubt become my colleagues, coauthors, and coworkers in the mathematical community. What a wonderful time it will be!

Well, by Wednesday we'll be done with our intensive "seminars," at which point the students will be cut loose to seek out research problems and begin their work in earnest.

Before I close this entry out, I'd like to mention that I've now heard back from one of the Calc I students whom I approached regarding her poetry. I'm happy to say that her responses to my "interview" questions showed honesty and depth of thought, and they showed that she really did give a good deal of consideration to the exercise last Fall semester. I also heard from Magdalena, another of those students I contacted: she indicated that she's "still working on" her responses, implying that she's taking time to compose robust and meaningful replies. I can't wait to read what she's written.

I'll check in again in a few days. Until then, your homework: think back on the most meaningful academic experience you've ever had, and explain in 250 words or less why it left its mark on you. Spelling counts.

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!

All in

We're set.

In ten and a half hours we'll begin officially the 2008 installment of our REU.

I'm excited.

And nervous.

These are some smart people I'll be working with; it's evident in their conversations, their interests, in the way they carry themselves and express themselves. They're sharp.

What'll they be like as colleagues, tomorrow and beyond? As I was telling Thalia last night, I hope that this summer we can succeed in forming a community of learners in which no one is deemed as The Expert™, in which we all feel free to question and learn from one another as equals. The only difference between me and her is that I've been doing it for a few more years than has she; time will erase that advantage too, and if she keeps at it ten years down the line we'll be sitting in a side room at some conference, yammering away and remembering our conversation ten years back.

Last year's group: where are they now?

I've been in fairly close contact with five of last year's eight.

A couple of days ago I invited Kieran to come down at some point in the next eight weeks and give a talk about his current research program. He may be able to make it, he's not sure. He's got a lot to get ready for in the fall, when he begins graduate work at the University of Kentucky. He's got his name on a paper with me and my next-door neighbor in the department, on one that's been submitted to Complex Systems.

As I mentioned a few days back, I've been spending some time during the last couple of days retooling a draft of a paper I'll be submitting with my name alongside Wilhelmina's and Mirabel's. I may shoot for Communications in Algebra again; we'll see. Mirabel will be headed southward in the fall; she begins grad work at Wake Forest in August. I hope I'll get a chance to do some more work with her.

I'll be incorporating work by Nestor into the sequel to my paper that will soon appear in Discrete Mathematics; I don't think this sequel will sit for long before getting snapped up by a pretty decent journal.

And twice this past half-year I've seen Opal at conferences. I don't know if she intends to keep on keepin' on once she finishes her undergraduate degree, but I'm sure she's bound to do good things.

Where will this year's eight find themselves in the coming years?

Six of the eight are rising seniors (three last year), one is a rising junior (five last year), and one a rising sophomore (none last year), so this year's crowd is slightly older on average. As a whole they seem a little more sure about themselves and their mathematical futures than last year's group did. Grad school's a likely feature in all of these kid's futures.

I'm sure I'm setting myself up for a heavy round of rec-letter-writing in the fall.

That thought in mind, I'm off to bed; tomorrow promises an early start and a long, long day, crowned by (havah nagilah!) bowling night. I'll so be ready for that by the time 9:00 p.m. rolls around.

Who's who

Six in, two to go, both of whom will arrive in the next seven hours or so.

Loads of meetin' 'n' greetin' and gettin' to know one another goin' on. Everyone seems to be getting along so far (some more well than others...I'm going to have to separate certain people during the introductory seminars), I don't foresee any obstacles to this crowd's forming a hard-core tight-knit learning community within the first few days!

Further bulletins as events warrant; for now, everything appears to be copacetic.

Twenty-five minutes to go...

...At least that's what it feels like.

Desdemona, our REU student who's scheduled to arrive first, should be here within the next hour and a half.

Am I ready?

I've spent the last couple of weeks putting together material for this year's program, and now I've got a list of learning goals, a rough timeline, a list of about 45 possible research questions, five handouts involving set theory, graph theory, and metric geometry (and several more on the way), five handouts on Mathematica, four handouts on LaTeX and the structure of mathematical research papers, an exercise asking students to uses research databases to track down primary source material, and a map of North and Central Asheville, complete with markers indicating locations of yummy eateries and other useful establishments. (All of these are available on the REU website, here.) Not to mention a couple dozen research and survey articles on various topics of interest to me.

Still I don't feel ready.

I'm just anxious, giddy as a schoolboy on the first day of class.

I met with Lulabelle and Casanova yesterday to talk about the direction our new year-long writing assessment study is going to take. Suffice it to say we've got a mountain of data (both quantitative and qualitative), making the selection of research questions a leviathan task. For now we're focusing on collecting the final round of data from the previous year's participants, including syllabi and course assignments, as well as statements regarding these folks' experience in constructing and applying the writing rubric we all worked so hard to put together.

Lulabelle's visibly more relaxed since we've finished up with Austin. I hope she's able to get some good nights' sleep in these days.

After our meeting yesterday I stayed on to help her out with reading math papers (a scary proposition for a non-math-type person!), and then we talked about Kevin Rathunde's work on family environments that are most conducive to optimal experience in children. (I came across a reference to Rathunde's work in reading Csikszentmihalyi's Flow for our upcoming summer learning circle, and tracked down the original work, appearing as a chapter in Csikszentmihalyi and Csikszentmihalyi (ed.) Optimal experience: psychological studies of flow in consciousness (Cambridge, NY: Cambridge University Press, 1988).

Central to his study are five traits (all starting with 'C'; a fitting companion piece to my "Four Cs" rubric!): clarity (of rules and purpose, providing a stable set of experience parameters), choice (giving children the opportunity to direct their own experience within clearly defined parameters), centering (of activity in the present, making it an "autotelic" activity worthy of pursuit for its own sake), commitment (or trust, giving children a feeling of security that will enable them to boldly strike out on their activities), and challenge (ensuring that children are given tasks that are adequately interesting).

A question to ponder: in what ways are these traits best developed and nurtured in the classroom, and to what extent do they there serve the same roles as they do in a "well-flowing family" context? (I can already see how these traits mesh well with what I've always thought of as "good teaching practice.")

That's a question for later, since in the middle of my typing the above paragraph, Desdemona arrived with her family in tow. She's now safely moved into her dorm room, alone for the night but not for much longer.

I'm so excited!

And I just can't hide it!

I'll attempt to keep control.

But I'll still like it.

Until next time, ponder the following: just how long did it take Dr. Csikszentmihalyi to learn to spell his name? And how long does it take him to type it?

Channel assignment

Channel assignment

Were you a step away from me,
and I a step from you,
then you could hear
were I to speak.
Yet another step between us
would make two,
and to speak would mean to scream.
Beyond two steps
no sound could penetrate:
we’d sit alone
on silent isles
until some friends between us stood
to mediate,
one a step from you, one a step from me,
with a single step between them, too.

Sleepless in SoTL

One of the questions that arose during yesterday's midmorning meeting with my colleagues Ophelia (official evaluator for my REU) and Fidelis (world-renowned expert in science education assessment who just happens to have retired to the Asheville area) is: "How does student technical writing improve over the course of a summer REU, given that this learning goal is given attention?" Fidelis assures us that next to nothing is known about process when it comes to REUs, especially regarding the attainment of specific learning goals surrounding professional development: writing proficiency, oral presentation, acculturation to the career of a professional academician. Any one of these would make an excellent target towards which to aim the arrows of study.

I wonder why it is that I had writing on my mind?

This summer's REU will feature much more intentional instruction in writing, the use of research databases, and the use of Mathematica. Fortunately I've got access to a good store of material relating to the second two activities: I can recycle a lot of the handouts on research I used in Fall 2006 for the now-infamous MATH 365 course, and I can borrow liberally from (or just simply reuse) the Mathematica files my colleague Nostradamus made up for the REU last summer.

As far as writing is concerned, it's always a pleasure to make up some new material. I've already got a few handouts that I gave to the students in Week 6 last summer, introducing them to the rudiments of LaTeX. Instead of holding off for six weeks, they'll be given those handouts right away, and they'll get the ball rolling in Week 1:

Week 1, Wednesday: students will be asked to download freeware LaTeX compilers and text editors (OSTeX, MiKTeX, TexnicCenter, WinEdT, etc.).

Week 1, Friday: by the end of the day, students will be asked to submit their first assignment in LaTeX, a rudimentary paperlet discussing one concept with which we've dealt during the first week's seminars. The paperlet will have the form of a research paper in miniature, structured roughly as follows:

1. Introduction:what exactly is the concept you've chosen to write about? (Give a simple definition.) What is its history? What is known about it?
2. Elaboration and examples: say a bit more about the concept by elaborating on your definition from above. In what context does your concept live? What concepts are related to it, and how? Give a few examples of your concept.
3. Conclusion and discussion of future work: Bring your work together, summarizing the findings, and indicate open questions surrounding the concept you've chosen to talk about.

I hope that by asking them to so structure their work right away they'll grow accustomed to this genre.

Week 2, Friday: students will be asked to submit a research prospectus indicating at least three topics into which they would like to look during the coming weeks, including an abbreviated (and unannotated) literature review for at least one of them.

Weeks 3 through 7, Friday: students will be asked to submit "progress reports" taking the form of mini-research papers. These reports will be structured much as the paperlets described above and will continually update the reader on the students' active research. A student may choose to report on only one week at a time, or to maintain an ever-up-to-date comprehensive report summarizing all work performed until the present.

Week 8, Friday: the "final version" of a paper will be submitted. If all goes well, this paper should be a fairly close approximation to something publishworthy.

Last year only Wilhelmina's and Mirabel's paper approached publication quality by the end of the summer (speaking of which, I've got Wilhelmina's most up-to-date version sitting in my in-box, I hope to get to that today...), though with the intentional iteration highlighted above, I have hopes that many more of the students will leave with a strong piece of work in hand.

In other writing news, I'll be meeting with Lulabelle and our colleague Casanova on Thursday to discuss the directions our new writing study might take. We've got a ton of data to sift through, and any number of ways we could conceivably slice it. Between faculty journals and conversations, student pre- and post-tests, student assignments, the faculty-developed rubric and the results of its application to the students' work, there's a lot to work off of.

As I indicated towards the end of this previous assessment project, I have a hunch that this study is going to say more about faculty writing practice than it is student writing practice; this is why I'm interested in tracking the elusive "instructional intentionality": how is it that faculty's attention to writing instruction manifests itself in the classroom, and to what extent does that attention pay off in (at least perceived) significant student proficiency gains?

Beh. Mumbo jumbo mumbo jumbo...

...finally, I should say that I've already heard back from most of the former Calc I students I'd written regarding their poetry from last Fall semester. I need to make up an "interview" form to send them so that I can help them reflect on their writing process.

Well, I'm going to get going and take a look at my graph theory notes from this past semester's seminar course; I have a hunch I'll be able to modify the problem sheets to serve as a basis for next week's graph theory instruction. We may end up playing a lot of it by ear and making it up as we go along.

Avanti!

Back

Damn.

That was productive.

It's going to take me a while to unpack all that I've carried back from Austin.

It's been a long time since I've had a trip that fruitful; I've come away with so many ideas, insights, opportunities for collaboration, connections to new colleagues...what a wonderful crowd these WAC folks make!

It's safe to say that there's a great deal of interest in writing in the math curriculum.

Lucky me.

People were enamored with the work Lulabelle's put together, and with my piece of that project. If anything, though, the writing folks were more interested in hearing about my math poetry project from the Fall 2007 semester. I've already contacted several of the students from that Calc I class and asked if they'd like to work with me by providing me with reflections on their creative process. I'd like to know: what did they get out of the exercise? Did it help them to engage mathematics, and if so, how? I've spent the last hour or so looking over Arthur Young's Poetry Across the Curriculum and other related projects. This is such wonderful work!

As I said, it's going to be a long time before I'm done unpacking the mental baggage I've brought home from that conference. Already, though, I've begun thinking about how I can assess this summer's REU students' writing as they proceed from novice article authors to more seasoned technical mathematical writers, and I've begun puzzling through the process of assessing just how it is that a writing instructor's intentionality leads to students' gains in writing proficiency. And I'm coming up with ways I can use blogs in Abstract Algebra and wikis in Precalc this coming fall. And Lulabelle and I are meeting with Casanova on Thursday to discuss the direction(s) we'll be taking in our follow-on to the last assessment project.

And...

...and...

...and...

...too...much...to...think...about...

...and with only a week to go before the REU begins.

Until I return with a longer and more meaningful post, here's a question for all of you out there who happen upon this blog: would you rather be bored shitless or busy as hell? Discuss. You will be graded on ramblingness, incoherence, and lack of a thesis sentence.

Think 'pon it carefully. Surely your response says more about your personality than a thousand Cosmo quizzes.