Chapter 2. The parallel postulate

For two years now Umberto's school has been contemplating scaling its two-semester first-year composition sequence to a single semester. His position was not an unfamiliar one to most of last week's conference-goers: many schools have made similar moves in the last so many years, including UNC Asheville, at which first-year seminars have been granted writing-intensive status in order to, in part, take away the sting of divesting ourselves of a second semester of introduction to academic writing. (Whether the sting is fully unstung will help make up a later chapter of this series.)

Having made a cursory assessment of the pros and cons, and having heard his administration hem and haw for the better part of two years now, Umberto's head was spinning. He came to the conference looking for answers.

Much of his presentation last Tuesday afternoon could be summed up in three words: "at what price?" What do the students lose in not having a second semester of focused, intentional writing instruction? Will a single semester only short-change them in terms of content? Will halving the time they spend in practicing academic writing significantly affect their proficiency? Will it do irreparable damage to eliminate one of the few courses in which students at a large comprehensive university might receive personal, one-on-one assistance with their instructors?

On the other hand, there are potential benefits to the move. Umberto's handout mentions several: "the promise of more resources," "the promise of a WAC [writing across the curriculum] program and a WAC director," and "the promise of fewer part-time dependence (staffing and space)."

What to do, y'all?

Judging from the tone of the follow-up discussion a half-hour later, the collective mind of the room had already been made up. Nearly everyone seemed to believe that the more attention students paid to writing early in their college careers, the better off they were in the long run. It was taken as gospel that more is better, that students require two full semesters of meaningful, intensive instruction in academic writing in order to gain the confidence and proficiency they need in order to succeed as writers.

After several minutes of heated exchange on the issue, Nora played a daring devil's advocate and suggested "why not get rid of first-year writing instruction altogether? What proof have we got that it works?" As an adjunct to this insinuation my own colleague Euterpe, having with me weathered the yearlong storm of assessment of our writing intensive courses, suggested that Umberto might consider designing a similar assessment for the first-year program at his own school. There was further discussion on these ideas.

After a few minutes Leona ripped a gauntlet from her hand and flung it at my face: "I don't want to put Patrick on spot," she said, placing me on the spot, "but one could make the same argument about first-year math programs! How can it be that we're sitting here talking quite seriously about cutting first-year composition programs altogether, while the same case isn't ever made about corresponding courses in other fields?"

After several more minutes of heated debate on this and related points, Euterpe, ever the mediator, stepped in to smooth down ruffled feathers and suggest adjournment for pre-dinner refreshments. As the room emptied I approached Leona.

"Touché," I said. "You're absolutely right: the same case can, and has, been made. At our own school we've watched Precalculus shrink from its onetime incarnation as a two-semester course to its current single-semester format."

I told her of the difficulty I'm facing now: I'm finding myself pressed for time as I attempt to navigate an unfamiliar content-driven course with clumsily-designed student-centered methods meant for deeper yet more leisurely engagement of the course's concepts. ("Coverage!" bellows the beast.)

Leona patiently let me unburden myself to her, though surely she hoped to sidestep me and make for the pinot grigio that was chilling in the canteen in the basement of the South Lodge, not a hundred feet away. "I'm just not familiar with the course, never having taught it before," I confessed. "I'm not sure at what pace to teach, I'm not sure of the effectiveness of the techniques I'm using, and I find myself traveling entirely too slowly. I'm already a week 'behind' where I'm 'supposed to be' by now."

(By the way, here's a tidbit for my students, any of my students, in whatever class I'm now teaching or ever have taught before: I'm a veeeeeeeeeeeery slow teacher. It's just the way I am, I like to spend much time on few concepts, discussing them more deeply and thoroughly than I would were I to whip on through without so much as a by-your-leave. Occasionally students will tell me the pace of the class seems fast. This might be because I write fast, for which I compensate by making my typewritten notes available on-line. In any case, if you think we're moving fast, my friends, you ain't never seen fast before. We're moseying, ambling, out for a collegiate constitutional. Don't worry, I'm planning on speeding up. I like the slowness. I won't let our pace worry me if you don't let it worry you. It's been working well for me for a decade now, and I don't intend to change it at any time soon.)

A two-semester Precalculus course makes much more sense to me, but there are practical reasons why this dream is not likely to come true. For one thing, we just don't have the person-power to pull it off: we're scrambling as it is to schedule all of the Nature of Mathematics sections we need, to say nothing of Calc I and Calc II. (The latest round of budget setbacks likely means we'll have another lean year or two, since it's dollars to donuts that the faculty search we'd planned to run this year isn't going to make it.)

This chorus is common to both math and writing: my fellow conference-goers intoned a long and dirge-like litany of budget cuts, over-enrollments, and staffing shortages. For instance, East Carolina University was faced with a surplus of 1500 freshpersons this past year, fodder for several dozen unplanned composition courses taught by twenty-some hastily-hired instructors. (All this and a $6 million budget cut!) North Carolina State University saw a dip in funding yet its writing program administrators have to cope with ever-rising enrollment at a school committed to growth.

Assessment aside, I suspect that every one of us present at last week's conference would agree: in math and writing both, surely two is better than one, and one better than none at all...to claim otherwise is to stake a daring claim indeed. More than most any other topic discussed at CWPA, the trial of Two v. One made me feel close to my colleagues on the other side of our metaphorical quad.

Anyone who's read this blog for more than a month will surely know that I believe that math and writing are not so different after all. "Lost in translation: demystifying mathematical writing" is the title of one of my talks at this past May's International Writing Across the Curriculum Conference. In this talk I aimed to show that what makes good math writing good is more or less the same as that which makes good writing good in general. (Repeat after me the Four Cs: correctness, completeness, clarity, and composition!) My hope was to show writing folks that we're not so different, they and I:

1. We all work in a rich and robust linguistic medium. (Mathematics, though hardly a universal language, as many wrongly claim it is, is a language nonetheless.)

2. We all conduct our business in an economy based on metaphor and imagery.

3. We all value clear and critical argumentation, and strong composition and communication.

4. We all deal with colleagues who often have little understanding of the ways in which meaning is constructed in our disciplines.

5. We all deal with a studentry largely underprepared for the complex tasks we charge them with, and we all therefore exert Herculean efforts in remediation.

6. We all perform tremendous "service" to every other department on campus, at levels unparalleled by any of our colleagues in the college. After all, students may need to take a single course in a laboratory science, and to satisfy this requirement they might select a course in physics, or chemistry, or biology. Students may need to take a single semester, or even a full year, of a foreign language, and they will generally have several languages from which to choose (there are eight at UNC Asheville). Most liberal arts universities have "core competency" or "intensive" requirements, but the student can often meet most of these requirements by taking courses entirely within her major.

On the other hand, every student must take at least a semester of composition, and every student must take at least a semester of mathematics: writing programs and mathematics departments are thereby burdened with boatloads of "service" courses whose delivery requires that departments either retain a large number of adjuncts and lecturers or direct the efforts of its full-time ranked faculty towards teaching introductory courses.

This last point came up over and over and over at CWPA. While I am certainly not opposed to teaching the odd introductory course (I despise the term "service"; thus the ever-present quotation marks), many tenure-track faculty are opposed, and vocally so. At CWPA Leona related a story about a colleague of hers at her previous institution: predictably, this colleague had expressed great interest in teaching first-year composition courses during her interview. Once hired, her interests "changed," and she lamented that she hadn't been given a chance to teach a literature course. Clearly she felt her creative talents were being wasted.

While such an attitude is condescending and does little justice to the intelligence of first-year students and others who enroll in introductory courses, the following question is a fair one, and was asked more than once last week: should a school entirely eliminate its first-year composition requirement, whither the expertise of those faculty formally trained in composition and rhetoric? Few schools offer undergraduate degrees in these fields, and if first-year programs are scrapped, these folks' services may be rendered redundant.

This last question and its offshoots (In what way does the teaching of composition dovetail with the teaching of literature? Can one direct meaningful undergraduate research in composition theory? Are there parallel issues in mathematics or other "hard" sciences?) and possible responses will be the topic of the next essay in this series.

If any of my writing friends are still blundering through this blatherskite, I hope that they'll free to chime in by posting a comment or two. I have a feeling I'm going to need their help in understanding my own next post. Stay tuned!

Chapter 1. From either/or to both/and; from trespass to transcendence

"You're really brave to come here," was a comment I got from a few different people on a few different occasions during this past week's Carolina Writing Program Administrators conference at the Wildacres conference center just off the Blue Ridge Parkway. As a mathematician at a writing conference, I was a bit of a black sheep.

However, the camaraderie and conversations I shared with my Carolina colleagues on Monday, Tuesday, and Wednesday showed that without doubt we all face the same challenges in helping our students meet their academic goals. The further I go in my study of writing and writing pedagogy, the more similar teaching writing and teaching math begin to seem. As frequent readers of this blog can attest, I'm often struck by the parallels between the disciplines themselves.

Last week's conference was a particularly enriching one for me. I've returned more than typically energized by all that I've learned from my new-found colleagues. I'd like to share some of my insights on this conference in this forum, my blog, my personal reflective space.

This is the first of twelve posts I've planned in which I hope to lay out some of the ideas I've brought back with me from Wildacres. I make absolutely no claim about the originality of these ideas, and since I aim merely to trace outlines of some very complex notions I have no doubt that my exposition will often be skeletal and wan. I'll leave it to my readers to fill in details as they feel fit, and as always I welcome input from my colleagues and my students, my fellow-travelers in academia: please feel free to post comments!

The theme of this first post is false dichotomy and overcoming it: how does one achieve transcendence through the act of trespassing from one academic territory into another? How does one craft a "both/and" from an "either/or"? How does one efface the line between research and teaching, between control and collaboration, between "coverage" and meaningful student-centered learning? I'll address each of these apparent dichotomies in turn, and a final false dichotomy (between mathematics and writing) will lead us to the next chapter, in which I'll take a look at the parallels between first-year composition courses and entry-level mathematics courses like UNC Asheville's Nature of Mathematics or Precalculus.

Let me begin with a dichotomy that doubles itself by laying bare (one of my writing colleagues recently claimed) distinctions between academic disciplines whose modi operandi bear witness to profoundly different means of interrogating ideas.

(Incidentally, the sentence I've just written conveys a subtle such difference: just this past week I had a mildly heated discussion with Lulabelle [a sociologist, recall] over whether or not an inanimate object is capable of "witnessing" anything. In mathematics it's common to say that a computation or an instantiation of some variable "witnesses" the truth of some statement or another. Lulabelle adamantly resisted my attempt to put such language into a table she was creating for my handouts at Wildacres.)

"Research" and "teaching" generally constitute two of academe's three pillars of support (the third is "service"). One, two, or all three of these vessels into which is poured copious amounts of faculty time may be construed broadly or narrowly, depending on the whims of a particular institution's leadership. Even when granted leave to grow with utmost manifest destiny these lands, though they frequently border, don't often overlap. This is made most evident when it comes time to account for a faculty member's expenditures of effort. Even at UNC Asheville (where holism undoes parochialism on a regular basis) the faculty record, to be completed annually by every faculty member, asks faculty to categorize their achievements under the headings "Teaching," Scholarship, Creative, and Professional Activity," and "Service."

I spurred a fervent academic discussion on the way back from last week's conference when I pooh-poohed hard-and-fast distinctions between research and teaching, suggesting a dialectic between the two that synergizes my activity in both.

My teaching informs my research: in my students I find not only collaborators and future colleagues but also sources of novel insights and new perspectives. In mathematics (as in other fields) the solution to a puzzling problem may be reaped not from years of toilsome groundwork in a heretofore fallow field but rather from a few days' reworking, in some novel fashion, of a field already furrowed, sown, and freshly watered. With fresh ideas students are frequently able to do the work that's needed.

Moreover, as I've discovered again and again, my research informs my teaching: in countless classes I've been able to enliven the exposition of various topics from theoretical mathematics by bringing a particular course's techniques to bear on the research problems with which I'm grappling at the time. The Fundamental Theorem of Calculus played a crucial role in much of my recent work on random graphs, so I trot it into my Calc classes. My analysis of asymptotic connectivity required mastery of Markov processes, and I put my Linear Algebra students to work on those. Right now I'm merely waiting for my Abstract students to learn about group homomorphisms so I can challenge them to write their own (G,φ)-grams.

To me, research and teaching are inseparable, and I credit my successes in both of these areas to my willingness to let them intermingle.

I told this to a colleague who teaches here in Literature and Languages, and she responded by indicating that the situation is different in her field. There, she claims, it's more difficult to involve students in undergraduate research efforts since it's no easy matter to carve out bits of one's own research program to make fodder for undergraduate inquiry. Whereas many of my projects can be dissected into chewable chunks, the ways in which knowledge is interrogated in her corner of the humanities frequently make overseeing undergraduate research difficult for her colleagues.

I must admit my ignorance regarding undergraduate research in literature, or in composition theory, or in rhetoric. Is it so difficult? Why can it not be, as in mathematics, seamlessly grafted onto typical classroom experiences? What forms does research take in these fields? Are these forms of inquiry inaccessible to most undergraduate students? Does meaningful inquiry require the student to long immerse herself in a body of literature so deep as to preclude its plumbing by an undergraduate with a typical courseload? Are there no ways to effectively circumvent such arduous preparation?

I don't ask these questions to be obtuse or snarky, I ask because I don't know the answers.

So I drone Ben Steinianly into the blogosphere: "Anyone? Anyone?"

I cannot imagine teaching without challenging my students to engage themselves in their own original inquiry. There's no more authentic learning experience, no better means of encouraging ownership of a discipline's core concepts, than to involve students in real research. It's the ultimate form of student-centered learning.

Speaking of which brings us to another dichotomy that was addressed only occasionally at the CWPA last week: "coverage" versus student-centered classroom methods. It's understandable that this topic didn't get so much attention at this conference, as entry-level mathematics courses like Precalc are likely much more content-driven than their correspondents in the writing curriculum. In introductory writing courses, I imagine little is lost in taking time out of class to ask students to write in groups, to perform peer review, to engage in ungraded freewriting exercises: these projects all serve the course's learning goals in direct and obvious fashions.

Meanwhile, traditionalist math teachers kvetch: "I can't use group work in my class, we've got so much stuff to get through." "It'll take the students three hours to uncover this concept that I'd explain more clearly in three minutes." "If they get to Calc I without having seen the Law of Cosines, they'll be in a world of hurt, I can't slow down."

I can sympathize. I've been there. I know how it goes. Even now I catch myself about to mutter or moan a comment like one of the above, as I'm struggling to keep up with where I "should be" in my Precalculus class. On the verge of sacrificing a group project or some other active-learning exercise (one that at the very least gets them taking turns coming up to the board to jot down a number or two, a graph, or even just a point in the plane), I stop myself and say, "whooooooa, now. The hell, Patrick? What are you thinking?"

"What are millennials [students born after 1982] like?" asks Nora, one of my colleagues at CWPA, in a handout she passed around during her presentation. In partial answer Nora offers that they "gravitate toward group activity...are fascinated by new technologies." Millennials' learning styles include "teamwork, experiential activities, structure, and use of technology...collaborative style." What's to stop us from giving them every opportunity to engage in these effective practices?

The false dichotomy rears its hairy head, bellowing an eight-letter four-letter word: "coverage!"

To which I bellow back: "bullshit." Indeed, the astute reader of this blog will note that there are a few words I never use, because it pains me to do so; "cover" is one of them.

Quick quiz: I've got fifteen minutes left of class, and we're working on cleaning up some trigonometric minutiae. Are my students better served by watching me babble about the Law of Cosines or creating and solving their own problem that makes use of more fundamental computations involving sines and cosines themselves?

The former concept they'll use maybe once in each of Calc I and Calc II, and a few times Physics, assuming they go that route. And if they need the law at those times, it'll be presented to them anew and re-explained anyway. (Such "just-in-time" explanations are often the most effective, especially if the topic is one that's unintuitive or difficult to motivate: by deferring the concept's introduction until the time at which it's needed, students aren't asked to remember pointless facts and figures any longer than they have to.) Meanwhile, the latter concepts are foundational and come up continually in later coursework, and through direct engagement with them the students better understand their workings at a practical, as well as theoretical, level.

No sacrifice is made here. While I might remove a few of the ornamental baubles and bijoux that typically adorn the face of an introductory course, all that needs to be "covered" is "covered," and the face is left intact. Students then engaged the rudiments of my discipline more directly, more meaningfully, than they would have otherwise. All we need to do, as practitioners of content-rich academic fields laden with centuries and sometimes millennia of knowledge, is divorce ourselves from the rather arrogant notion that our students need to know every last fact about our fields of study.

Guess what, folks? They don't. They need to master a few core concepts, and they need to become adept problem-solvers and critical thinkers. Everything else? Icing on the cake. Accepting this fact is difficult for some of us. After several years of graduate training in highly specialized and technical fields, it's hard to remember that our students don't need to know everything there is to know about CAT(0) spaces, or about Sterne's use of the subjunctive in Tristram Shandy. We need to let go.

And we need to let go of the reins and let the students take the helm every now and then. They must be allowed to be our collaborators, not our convicts, and although guiding them in collaboration requires that our students be given a bit more control over their studies, it by no means implies a loss of control on our parts. The most meaningful learning experiences I've ever shared with my students have come at moments when everyone in the room agrees (often tacitly) on a common course of action and each person present assumes control over her or his own role in the action taking place. I can think of three such relatively recent moments off the top of my head:

1. The day, roughly two-thirds of the way through the Fall 2007 semester, on which my 280 class and I agreed to go off-script and just blundered our way about through the wilds of set theory for over a half-hour. That was the day on which Quincy suggested something along the lines of a "fishbowl seminar" in which all participants spend each day's hour discussing a topic drawn at random from a fishbowl placed at the room's center. (I'm still not sure that wouldn't work...) I blogged about that class in this post.

2. My second section of Calc I's rendition of the classic Newton v. Leibniz debate last fall. (Blog posts related to this class include this one, in which the students shared their reflections on the experience, and this one, in which I provide my own initial reflections.) It was clear that everyone had prepared carefully, and after a few minutes most people seemed to have forgotten their real identities as they let the experience take hold of them.

3. The second day of "class" during this past summer's REU, on which the students took turns presenting their findings on the roughly 40 terms and topics from graph theory I'd given them the day before. With one exception (Dione was a bit shy at first) they strode boldly to the board, one after another, and by lunchtime we'd ticked off nearly every item on the list.

In each of these experiences, every individual involved took control and directed her own actions to create a meaningful part in a collaborative project. None of these projects required abrogation of authority. None required the teacher to play the part of dictator, nor the students the part of the peasantry. Each was more than a monologue, more than a dialogue: each was a polylogue. The result in each case was what L. Dee Fink would surely call a "significant learning experience."

Control versus collaboration, coverage versus deep conceptual mastery, research versus teaching: each of these tugs-of-war is played out every day in classes across the college curriculum. With several steps forward, one combatant pulls the other across the line dividing one territory from the other...but if we're wiling to admit that the line between the two is arbitrary, then trespass becomes transcendence.

Of my new-found friends in the writing community who may be reading this, I ask: how do these false dichotomies, these plays for power between imaginary opposing forces, manifest themselves in your classrooms?

I'm curious to know the extent to which our disciplines run parallel to one another. I'm eager to learn more about the form your instruction takes, the ways in which you engage your students in your courses and programs. I'm hoping you will take the time to share your thoughts.

In my next CWPA-inspired post I'll continue by discussing parallels I've noticed between the pedagogy of writing and the pedagogy of mathematics, parallels which no doubt witness (there I go again!) more profound similarities between the disciplines themselves.

Until then, have a pleasant evening, I wish you all well.

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!