A little conversation

As I noted in my last post, I recently received a very thoughtful e-mail from an old student of mine, Sedgwick, who will soon be studied public policy in the graduate school at the University of Minnesota. His letter concerned a number of points I'd made in earlier Change of Basis posts, regarding everything from my (recent) initial response to Don Tapscott's Grown up digital to my views on the UNC Asheville Integrative Liberal Studies (ILS) program. I'd like to make of this post an open letter in reply to Sedgwick, playing off of his ideas and elaborating further my own, where appropriate.

(Note: He's given me permission to quote his e-mail freely.)

Let's just go in order. Sedgwick begins:

In regards to your Learning Circle book, it's a small world...a few weeks ago I heard Don Tapscott talk for a while on a weekly technology show. Much like you, I found a few of his observations on leveraging technology in the classroom useful, but not particularly distinct from the suggestions that any 'wired' person could make about improving academia. Given how conservative educational institutions can be in many respects, it is not a particularly challenging feat to poke fun at, say, the traditional lecture format.

We're in complete agreement: I've yet to find anything truly revolutionary in Tapscott's book. Of course, I'm not reading it in order to find therein a recipe for a 21st-century classroom, but certainly I expect various prescriptions and proscriptions to appear. Let's continue with Sedgwick's letter:

That said, as the conversation progressed, Tapscott proposed some solutions which I found...interesting. For example, UNCA is rather keen on the need for interdisciplinary thinking and the liberal arts, but Tapscott's solution to an increasingly complex world is to dump the liberal arts altogether and start giving college students specialized training at the undegraduate level (e.g. law school and med school training from freshman year onwards). However, these suggestions are just for professions requiring some specific training, as Tapscott suggests that instead of 'wasting' four years at university, high school students should instead focus on some marketable technical skill or 'passion' and go 'do their own thing,' for lack of a better term. As Tapscott's hypothesis goes, once kids catch on to the fact that anyone can be as successful as Bill Gates or Steve Jobs by striking out on their own, universities will fade away by mid-century. While I can see how Tapscott is attempting to speak to people who perhaps did not get much out of college, proposing that the solution lies in obviating academia altogether just seems too flippant a suggestion to take seriously. Anyway, not to prejudge your book discussion, but did I want to concur that Tapscott's analyses seemed rather aloof from reality, and that he
did get me a bit miffed.

I find it very hard to reply to this portion of Sedgwick's letter, as I can't be sure of the sort of universities Tapscott has in mind. In some ways, the institutions Tapscott's speaking of already exist: technical and trade schools, and a number of community colleges, for instance, already place their emphasis on technical skills and specialized training, asking students to complete only a bare minimum of coursework in "irrelevant" areas. Likewise, students at more research-oriented universities (including "institutes of technology) often pursue courses of study designed to prepare them for graduate study in one or two particular areas, to the detriment of their engagement with other fields. Furthest from the form suggested above are the liberal arts institutions, in which students are required to engage a broad course of study that incorporates classes from highly divergent disciplines, and to synthesize what they've learned into a coherent whole.

I can't know to which sort of institution Tapscott (through Sedgwick) is referring, but in its broadest sense the claim that universities will "fade away by mid-century" seems prideful and cocky. Of course, to insist that the university system as we know it is timeless and eternal and will persist unchangingly as long as the human race lives upon the surface of the Earth is an equally hubristic claim, but I would be very surprised if the evolution of the university follows any course other than gradual change, morphing smoothly from one sort of institution into another, different but only subtly distinguishable from the last, morphing subtly again, and again, and again, its eventual manifestation starkly different from its present form, but only noticeably so by direct juxtaposition.

Let me offer a defense of the liberal arts institution in particular.

What seems to be missing from Tapscott's presumed argument (not having seen the program to which Sedgwick refers, I can't claim to place this argument squarely in Tapscott's mouth) is a certain measure of humanity; for the proponent of the above argument education seems to be cast as little more than a means to an end, and that end is a purely chthonic one, concerned only with the attainment of a job and the wealth that attends to that job. (Come to think of it...I've seen very little mention of art and literature so far in Tapscott's book...) Little regard is given to other learning goals achieved through a liberal education, including an appreciation for art both for the sake of art and for the unique knowledge it can offer us, an understanding of the interaction between all areas of intellectual inquiry (whether they're catalogued under the "natural sciences," the "social sciences," or the "humanities"), and an understanding of the humanist thread that binds together all of our race's endeavors.

Must a medical doctor be able to quote Chaucer in order to perform a successful heart transplant? Assuredly, no. But should we insist that she know a bit about the history of medicine, and the place medicine occupied in that history alongside the various other disciplines from which it grew and with which it intertwined?

Will the learning goals above be met in the universities of 2050 as they're met in universities now? Almost certainly not: just as our universities today in no way look like Harvard of 1900, Oxford of 1500, or Plato's Academy over two thousand years ago, the schools in which the adolescents of today instruct the adolescents of tomorrow will be profoundly different institutions. But I doubt they'll look anything like the schools proposed in the argument put forth in Sedgwick's letter.

Speaking of which, let's get back to that letter. At this point, Sedgwick offers a few bullet points in order to summarize his response to an earlier post I wrote on the ILS program at UNC Asheville, in which I was quite critical of the cluster component of ILS. Here's the first:

Get data on the real-world benefits of UNCA's ILS program. Over the years, I saw both ILS and the Humanities requirements take a lot of flack from my peers, but given the (admittedly few) conversations I've had with people after the fact, it seems they ended up enjoying the breadth offered by a cluster. Now that graduates from the ILS program are beginning to enter the real world, perhaps some alumni surveys are in order about how that breadth helped them. I think that by combining the existing data about the comparatively better performance of liberal arts graduates with data specific to UNCA, one could make a compelling argument for attracting serious students to campus.

I won't say so much about this suggestion, besides pointing out that I'd be more interested in getting data on the long-term perception of the other components of the ILS program (the intensives, for instance). The idea has its merits, and honestly I can't be sure that UNC Asheville's not already performing or planning to perform surveys as described above; it wouldn't be a difficult matter to ask a few questions about the ILS program on an alumni survey (much like the ones I recently filled out for Vanderbilt).

Sedgwick's next bullet point:

Better integrate the cluster system into the curriculum itself. While I appreciate the ILS program's relative boldness amongst generic college curricula, I think part of the criticism by students is that it appears 'bolted on' to the normal college requirements. Although I was a GenEd student, it seemed from the outside that many cluster classes appeared to be just 'normal' departmental classes. Perhaps cluster coordinators could work on 'harmonizing' some of the classes to lessen the appearance of patchwork? Actually, I bet this is already happening to a good extent, and that given the budget cutbacks, additional tailoring would be be difficult if not impossible.

(Note: "GenEd" was the general education system in place before the phasing in of ILS; it differed from ILS in a number of aspects, but its ultimate purpose was more or less the same. Sedgwick was likely in the last class of students to whom the old system applied.) I fully agree with Sedgwick's perception of students' complaints about clusters, and I agree that the way to adjust the clusters in order to effectively respond to those complaints is to do as he suggests, harmonizing the classes in some meaningful fashion.

However, I'm not so sanguine about the extent to which this is being done: to my knowledge only a small percentage of the clusters' faculty meet regularly to coordinate their coursework. Sadly, there's little little incentive for faculty to put much effort into aligning their cluster courses with their fellows in whatever clusters they occupy: there's generally no release time, there's no monetary reimbursement. The only real motivation is intrinsic: the knowledge that you're providing an optimal experience to the students in your cluster. Would that this were enough incentive, but considering that even moderately diligent faculty members already spend dozens of hours a week in teaching, grading, class preparation, committee work, research, work for professional organizations, et cetera...it's unrealistic to expect most faculty to devote any more unreimbursed time on cluster coordination than is absolutely necessary.

And, as Sedgwick suggests, budget cuts don't help.

The last bullet point:

Tweak the cluster system to make its relevance more apparent. As a merely personal example, I feel that, for all the knowledge I acquired at UNCA, I graduated without much in the way of marketable skills. All the jobs I've applied for in the past year were entry-level. Now I may be an outlier in that my degree was relatively specific and there are few environment-related organizations where I live, but I think the point remains that a certain proportion of graduates wish they had the opportunity to develop one or more specific skills as part of their liberal arts curriculum. Perhaps the cluster system could require the development of two distinct skills? In particular, I think of your effort to combine math and writing, but any combination could leave graduates with both a personalized and marketable skillset. I think UNCA, as a small and tightly-knit campus, is well suited toward this type of skill cross-pollination. Given how often people change careers nowadays, I think UNCA could market the 'have your bases covered' need very effectively. Of course, skill meshing isn't just about career resiliency but also the potential development of new approaches that can revolutionize existing careers or create new ones.

Agreed! Ideally.

There's nothing in this last point that I can find fault with, and it could be that UNC Asheville's ILS program is on a trajectory that will eventually take it to this ideal. As yet, however, it is merely an ideal, and will remain so until the faculty who will implement the program are granted release time or remuneration they'll need to motivate them to design the program and put it into action.

Might we move further beyond cluster courses, implementing team-taught courses in which two or more faculty provide truly integrative real-time instruction on a daily basis? This sort of instruction exists elsewhere...hell, if I received this sort of instruction as an undergraduate at a much larger institution nearly fifteen years ago, what's to stop UNC Asheville from moving in that direction? It's hardly lethargy or laziness (see my comments about being overworked above), and though one might try to finger financing for this fault, why not spend the money currently spent on implementing cluster courses on the faculty development and resources that would be needed to support team-taught courses?

It might simply be that at UNC Asheville, as at many other institutions, it's unstated disciplinary territoriality and parochialism that's to blame for the reluctance to implement such innovative course offerings.

There's much more to be said at a later time. For now, I've been typing too long, and my eyes are demanding that I take a break and enjoy the rest of my Sunday evening.

My sincerest thanks go out to Sedgwick for allowing me to share his e-mail on this blog. I hope he's provided my readers with some rich food for thought, and I hope he'll find my responses to his letter to be worthwhile ones.

Three great gifts

Week One has ended!

And we've not lost a one.

This afternoon's session on fractal dimension, led ably by my colleague Nostradamus (my partner in crime this summer), marked the end of the week's action for the REU. While they're still a bit reluctant to speak up in front of one another, they're definitely growing more at ease with working together, as evidenced by their professions to collaboration behind the scenes (from Nils and Ole) and the ease with which they cooperate in class (two pairs worked together to complete this morning's LaTeX exercise).

Speaking of LaTeX (and other mathematical technologies), all eight students now have installed on their computers some sort of LaTeX editor and compiler, and all eight have installed some version of Mathematica.

The students are beginning to show the first signs of focus as they near their initial selection of topics: Billie indicated specific interest in the "use it or lose it" tree construction, as did Daria. Nigel likes the look of the same algorithm, though like Daria he'd like to hear more about Cayley graphs before deciding on what to do. Several students asked more about graceful labelings and generalizations of chromatic polynomials, too.

All in all it's been a good first week. I've certainly learned from it that there's no single snapshot of a successful first week's work: while I've made no small point of this group's relative reticence, in their own way they've been no less successful in their mathematical efforts than last year's bunch, say, a band of brothers and sisters to whom I was often tempted during lectures to say, "shaddup already!"

If any of this year's REU students are reading this, please know that we're only remarking on your quietness because we find it a striking counterpoint to the previous years' groups. There's absolutely nothing wrong with your reservedness: it says nothing about your intelligence, your work ethic, or your eventual success as mathematicians. It's just very different from what we're accustomed to.

As yet I've said nothing in this post about the three gifts to which I've alluded in the post's title. It's time to remedy that.

All three of these gifts promise to expand my both my own understanding of the mathematical world and my ability to convey that understanding to others.

The first gift comes to me from Daria. When I fetched her from the airport on Sunday morning she and I got into a conversation about ethnomathematics, which readers of this blog might know is one of my less minor interests, especially given my rather unorthodox (among the research mathematical community) view that mathematics is not universal but is indeed a cultural artifact, a socially-constructed system that varies from one people to another. Somehow it came up early in one of our first conversations that Daria had recently taken a course in ethnomathematics, and in fact would soon have with her the textbooks she'd used for the course. I asked her if I could borrow them when they arrived, and yesterday she brought them to me. I have no doubt they'll prove a fascinating foundation for my own study of ethnomathematics, and a good basis for the course on the subject that I hope soon to develop for UNC Asheville students.

Both books, Ethnomathematics: a multicultural view of mathematical ideas (CRC Press: Boca Raton, 1998), and Mathematics elsewhere: an exploration of ideas across cultures (Princeton University Press: Princeton, 2002), are by Marcia Ascher of Ithaca College. In a heedless display of randomosity I began reading the second-written one first just a half-hour ago. It promises to be an interesting read. Having read little more than the introduction at this point, I already suspect I'll find a kindred epistemological spirit in Ascher.

For instance, "we now know that there is no single, universal path -- following set stages -- that cultures or mathematical ideas follow" (p. 2). Take that, proponents of mathematical universalism. As I'm fond of saying (and have said elsewhere in this blog), mathematical language is hardly more universal than the English language, and the mathematics of an alien race would likely be as indescribable and indiscernible to us as their courtship rituals.

Or take this line: "most practitioners of modern mathematics value their ideas because they believe them to be context-free; others value their ideas as inseparable from the cultural milieu that gives them meaning" (p. 4). Indeed, it's a blight on modern mathematics that so many modern mathematicians might laud math's seeming baselessness and independence from any fixed ground. This view could hardly be farther from the truth, as math is a highly predicated belief system, the truths it embodies obtaining only when certain cultural norms about truth and knowability are applied. How is it that a mathematician unwilling to state her or his hypotheses, elements necessary for the application of any reasonable theorem, would be laughed from the lecture hall, while it can be commonly supposed among mathematicians that the very science of mathematics does not rest on similar epistemological hypotheses?

I'll be sure to blog about these books as I make my way through them this summer.

A second gift, one of recognition and promise for future collaboration, comes to me from a heretofore unknown colleague in South Carolina. Lately my work on the intersections between poetry and mathematics has been getting the attention of more and more poets. Io, a poet and teacher from South Carolia, came across a copy of my paper on using poetry to teach mathematics (the one to appear in WAC Journal), and told me of her interest in the subject. She confessed that abstract algebra had been one of her favorite classes in college, and that she had great interest in understanding more fully the similarities between math and poetry.

Already, in just a short exchange of e-mails, I can tell I've found another likely friend and colleague. I hope to continue my correspondence with this woman as I further develop my own understanding of the ways poetry and math interact.

Side note: next year marks the 50th anniversary of the founding of Oulipo. Perhaps some sort of public and poetical and perimathematical celebration is in order? That's something to think about.

The third gift comes to me from an old student, Sedgwick, who graduated about a year ago with a degree in environmental studies. Sedgwick was one of the star students in the second section of my Spring 2008 Calc II course, a close-knit class that was a lot of fun to work with. He's still, a year after graduation, a regular reader of my blog (shout-out, Sedge!), and after reading a relatively recent post (this one, I believe) on the effectiveness of various components of the Integrative Liberal Studies program at UNC Asheville, and an even more recent post on Don Tapscott's Grown up digital, he wanted to offer a former student's perspective on the ILS Program, and did so extensively in an e-mail he wrote to me about a week ago.

His e-mail is, as is all of his work, thorough, well-thought out, and well-organized. This guy's always been a top-notch thinker. He makes many excellent points about various components of the ILS system. I asked Sedgwick's permission to excerpt his e-mail to me and to form a response to it in the form of an open letter consisting of a blog post here. Having been granted his leave to do that, do that I shall, in a post I hope to write this weekend.

For now, though, the dinner bell is readied to ring, and after a long, long week or work with a new crop of talented young researchers, I'd like nothing better than a few hours off. (If only I could get this damned channel assignment problem out of my head!)

To be continued...

Stand and deliver

Day Three has come and gone, and though they're still quite quiet, they've begun to strut their stuff, mathematically speaking.

Yesterday evening I presented them with the most substantial "homework assignment" I'll be giving them during these first couple of weeks of the program (the dreaded list of 34 terms and concepts from graph theory for which they were asked to find definitions and examples they would then take turns presenting to each other in seminar), and they completed it well. They'd divided the work up almost perfectly evenly, each taking about four of five of the terms for her or his own, and working together to produce a single Word document (by Friday it would be LaTeX) containing all of their findings. Impressive! They're the first group we've had who's created, unprompted, their own lexicon at the stage in the game. (Or at any stage, for that matter...)

The first few turns taken were orderly ones, each student presenting on several subjects appearing consecutively on the list.

Demeter started things off with a discussion of the many different sorts of sequences of vertices and edges one can consider: walks, paths, trails, circuits, and cycles. Her presentation was straightforward and confident. It was solid, and left little room for questioning.

Dora's presentation on cliques and blocks and related ideas raised a few more questions, which she handled smoothly. I appreciate how hard it is to think on one's feet in any setting, and I can't imagine how much harder it is when the questions you're being asked to answer (on the spot!) involve high-level math you began learning about just two days earlier.

Next it was Daria's turn, and her introduction to independence and domination led to good many more questions, which she too fielded handily.

Nigel's turn came, and he approached his presentation a bit more lightheartedly then his peers had before him. He was particularly adept at using the board and the colored chalk, and he seems very at ease working in front of his peers, providing clear and correct descriptions for each of the terms he'd been assigned. I hope he can build upon that confidence.

After a break for lunch, the rotation became more scattered. Billie's presentation on coloring and all matters chromatic stood at the center of a scattered maelstrom of turn-taking by Omer, Ole, and Nils, who traded off with one another as they discussed everything from graceful labelings to adjacency matrices.

Billie, who comes to us having taken a graph theory course (the only one of the bunch to have done so, I believe), had no trouble at all searching through her old course notes to root out a good working description of the Deletion-Contraction Algorithm. Like Demeter and Nigel before her, she was confident and clear.

To be honest, the tag-teaming trade-offs the guys made made it hard for me to get a good grip on their presentation styles. Even though they spent as much time at the board, cumulatively, as had their colleages, they weren't up before the class for a long enough chunk to get a sense as to how they'll be in sustained presentation.

Ah, but that will come later!

All around, the students did well. There were several minor slips, but that's to be expected. As I'm fond of saying, truthfully, hardly a day goes by without me making a dozen mistakes at the board. Sure, there were a few misstatements and a few definitions that might have been made clearer, but in the end it was good.

One thing I would like to see more of: the students challenging each other and asking questions that serve to further the work their colleagues have done.

And I'd like for the student who's presenting to not look at me when she or he asks "does this make sense?" or "is that right?" Who am I to say? I don't want to be thought the only expert in the room. I realized earlier this evening that in the last couple of days I've made the mistake of sitting at the center of the room's semicircular arc; tomorrow I'll decenter myself by moving to one side. (Funny: it took precisely one day for the students to fix their places in the classroom. From the near side to the far side, they sit thus: Nils, Ole, Dora, Billie, Daria, Demeter, Nigel, Omer; the last two days I've sat between Daria and Demeter.)

I find myself wondering what it is that's interesting them, mathematically...what sort of projects will they opt to undertake this summer? We've now posed maybe a score of open problems (with another score or so lying in wait), but I've no sense as to which ones they're finding appealing. I'm excited to find out.

For now, speak up, my young colleagues! Let us know what's caught your eye.

Credo

I may be a bit of a Pollyanna (given how treacly, or even declassé, the following might seem to some), but...

Credo of a Humanistic Professor

I believe in cooperation.
I believe in friend-making and in bond-building.
I believe in mutual respect and in shared responsibility.
I believe in achievement of understanding, and in always-open lines of communication.
I believe in controlled, consensual risk-taking.
I believe in honest intellectual inquiry.
I believe that little is more important in an academic relationship, up to and including mastery of the material being studied, than ensuring the conditions and goals indicated above.

Day One, Part Deux

It's been a good day, all in all. First days are always a bit awkward, simply because (a) no one really knows everyone else yet, (b) no one's really got a full sense of what's about to go on, (c) everyone's testing each other out and figuring out what to expect from each other, and (d) there's a lot of tangentially-related bureaucratic crap to cut through before you can get to the fun stuff.

The period from 9:00 to 11:30 this morning was spent almost entirely in filling out forms, getting pictures taken, coding in identification numbers, and so forth. The upshot is that the kids now fully exist, according to most of the file systems on campus. Their existence so verified and redundicated (I know that's not a word, but I felt like using it anyway...at this point my brain is more or less tapioca, so I hope you'll cut me some slack), we were finally free to start working on some math.

We got through a couple of pages of set theory and notation (in about half an hour) and two pages of graph theory (in another hour) before it was time for lunch, after which we returned to discuss some high-level open problems in fractal geometry (yet another hour) and some more graph theory (the last hour of the day).

So far? They're a bit shy about presenting in front of one another...but who isn't at first? It's definitely too early to tell how well they'll come together as a team, but as friendly as they all are (I've had lovely conversations with them all as individuals) I can't imagine they won't coalesce into a terrific theorem-proving team.

From the "Oh, and" Department: today was also the first day of my Learning Circle for the summer, on the book Grown up digital: how the Net Generation is changing your world (McGraw Hill: New York, 2009), by Don Tapscott. I'm yet to be impressed with the book (so far, though it makes some insightful and worthwhile claims, it's a rather uncritical gathering of anecdotes, only marginally relevant data, and personal observations), I very much enjoyed the conversation I shared with my colleagues in the Circle, and I had several thoughts I might blog about later...especially regarding the construction of collaborative syllabi and other course documents. I'm looking forward to next week's discussion on the text.

So much to learn, so much to live for!

Day One

Ugh.

No sleep last night, too excited.

Much to do today, but much of it boring (sorting out housing snafus, filling out paperwork, getting folks into the library system, etc.).

We'll make it through all right.

But if I don't bowl well tonight, Wes'll kill me.

First impressions

My first impressions of this year's REU group:

1. They're a bit more timid than last summer's bunch. This isn't a good thing or a bad thing, it's just a thing. It's all good, it just might mean it'll take them a little longer to come out of their shells. (Nothing a bit of bonding over a long-ass list of graph theory definitions can't cure...)

2. They seem highly dedicated. They all talked about how much they love math, and several have mentioned how they become absorbed by problems on which they're working, and several of talked about their futures in math (grad school, teaching, etc.)

3. They strike me as talented, but modest. I already have a feeling they've got a great deal of math smarts collectively, but I don't think we've got any hotshots: almost every one of them mentioned to me within minutes after meeting me that she/he was excited to be here, and they seem to understand their participation in the program as a privilege and as a responsibility rather than as a right.

Tomorrow, we begin. Excitement!

Mission: mathematician

This summer's goal: turn eight math majors into mathematicians.

Any minute now

The 2009 Summer REU is set to start.

Although I usually count the Sunday night potluck at our place as the "official" kick-off, the first of this summer's students will be arriving any minute now, if her estimate is anywhere near correct (and I think it is).

Monday should be fun. I've adapted several of the handouts from last summer (they all worked out pretty well) to reflect changes in the problems I'll be pitching to the kiddoes by Week 2. There was little change beyond removing a few graph-theoretic definitions that don't seem all that crucial anymore and adding a few missing ones that do. If this year's crew is as outgoing and eager as last year's, the first week's going to be a fun one.

I guessed I'd better get busy with some new pseudonyms, huh? How about these: Demeter, Billie, Daria, Dora, Ole, Omer, Nils, and Nigel?

Those oughta do.

Is everything in place?

Housing? Check.

Checks? Check.

Plans for managing their arrivals? Check.

First week's work? Check.

Whew.

We'll see how this goes.

I'm excited.

What comes after "four"?

Deep not-specifically-mathematical thought for the day: it's been a long time since I've been at the same school for more than four years without some major milestone (e.g., receiving a degree) marking a significant turn in the road. I was at UIUC for three years only, and at Vandy for four years before that...there were five years in Denver, but I earned the B.S. at the end of the third year and at that point entered the wide, wonderful world of graduate school, so I may as well have been on a different planet.

I'm about to enter my fifth year here (and my fourth year of blogging about it). I wonder how it's going to feel? Steady? Stable? Institutionalizing?

We'll see.

Everything's more or less in place for the REU to start in a little over a week. (A week from right now is when I expect the first of the students to arrive.) Aside from the personnel, little is changing from last year's program: if things go well (and they did), why bother messing around? I've managed to put together a list of 39 at least moderately tractable questions to start with, we'll see where we go from there. Obviously the REU is going to take up much of my summer, and what little work time I've got outside of that will go toward prepping for the fall semester's courses (Calc I, with a new text, and 280 again).

In other news, my colleague Euterpe agreed to help read through last year's students' weekly reports, in order to assess the development of the students' writing skills along various axes. Thanks, Euterpe!

Plans for the weekend? Finish up the revisions on the paper on mathematical poetry that I wrote for Math Horizons. The editors gave me some great suggestions, and I hope that after I've hammered out the last remaining details the article will be more engaging and enlightening still.

Independence

What shape of mountain might you be,
what manner the curve
about your single upward-arched umbilicus,
what steepness the slope of your coefficients' climb,
that after convolving
and contorting
and clever recursion,
crushing you
and crashing you
against your brother's craggy scarp,
a more vertiginous mount remains,
a peak more sharp, more stately still,
looms loftily above your head?

A half-dozen deep(ish) thoughts

I never cease to be amazed by the simultaneous simplicity and utility of guided free-writing.

Here's the skinny:

1. Choose a topic on which to write, or let someone else choose a topic for you.

2. For five minutes (set a timer for yourself), write without stop on the topic you've been given. If you get stuck and can't think of anything more to say on the matter, just write "I'm stuck I'm stuck I'm stuck" or "what in the hell am I thinking right now?" or whatever you'd like to, over and over again until you become unstuck and refocus on the chosen topic. Don't stop writing, and don't correct yourself, grammatically, orthographically, or otherwise. And don't hurry. You don't have to write quickly, but be sure to write continuously.

3. When your time is up, stop writing.

4. Now review what you've written and select a few words or phrases you find startling, surprising, or important in some way or another.

5. Choose one of those words or phrases, copy it at the head of a new piece of paper, and...

6. ...begin anew, writing for five more minutes, without stop, on the key word or phrase you've selected from your first piece of writing.

7. If desired, repeat.

I helped put together another writing workshop for my colleagues today, and my colleague Euterpe, director of our First-Year Writing program, all-around wonderful teacher, and majorly cool individual, led the workshop participants in a guided free-writing exercise centered on the topic of writing assignments. Through the exercise, she hoped, we'd be able to more fully develop our vision of a writing assignment we hope to pitch to students in one of our writing-related courses.

I don't think I was very successful in that effort, but the discoveries I made more than offset my lack of progress towards constructing a meaningful writing assignment. I made no fewer than six major revelations in the course of my writing, four of which I realized right away, even as I was writing, and two of which I realized only later, as I was transcribing my handwritten work onto the digital page below.

Where'd it all come from? With the MATH 280 "equivalence class" exercise about which I blogged the other day fresh in mind, I began thinking about how I could ask students to further interrogate the idea of "equivalence class" through writing, and my output from the free-write is as follows (underlined handwritten text has been replaced by italicized text; everything else is verbatim):

***

Given a set of objects, how is it that we can make some sort of mathematical sense of them? How can we group like objects together, and what does it mean to be "like"? How can we put objects in order, from smallest to largest, and what does it mean to be small or large? This activity will help you to accomplish this task. By presenting you with a seemingly chaotic pile of objects you'll be asked to provide structure where structure is not immediately apparent, and in so doing will learn to recognize what it is that defines structure in the abstract: what properties does structure encompass, and how can you recognize these properties?

You'll be asked to come up with a short list of characteristics that define what it means for the sorting of a set of objects to be an "equivalence relation": that is, the way you sort the objects should be in such a way that two objects are sorted together if and only if they're "equivalent" in some meaningful way. What properties much such a means of sorting have? That is, if I asked you to say, "if x and y are paired together, and y and z are paired together," what can you say about y and x? about x and z? What about x and x?

Let's consider the example of the random objects sorted by color...

Provide structure where structure is not immediately apparent

What does it mean to provide structure? What is structure? Maybe it's a way of organizing things so that they make "objective" sense to someone other than yourself: you of course understand what you mean by an assortment you've made, but how can you help others to see your thought process? "Structure" provides a "user-independent" means of organization: you agree with others to establish a set of rules or properties that define what you'll mean when you declare a certain kind of structure exists. For instance, in the case of an equivalence relation, we speak of the following structural characteristics: reflexivity, transitivity, and symmetry. These are the defining characteristics of this particular structure. Thus if you tell someone, "oh, this relation is reflexive, symmetric, and transitive," they know that whatever structure stemming from that relation will "look like" an equivalence: every object will be equivalent to itself and so on.

How to best get students to recognize these "atomic" properties on their own? They'll be asked to sort, but can they understand their own method?

How to best get students to recognize these "atomic" properties on their own?

In a sense we have to first move students from intuition to mechanics before we can get them to go in the opposite direction! Students inherently recognize that a structure is present when they're faced with it; they just have a hard time articulating what it is that that structure encompasses. That is, how can we bridge the gap between "oh, I see it!" and "Ah! Here's what I see!"? It seems like the same problem we just discussed regarding good writing: students know good writing when they see it, but can they explain why it's good? Brainstorming about what makes good writing might help students with that recognition task, so maybe a similar brainstorm about equivalence relations and other structures is a good starting off point? From the fruits of a brainstorm session, the students can be asked to reflect and decide which are the ripest, the sweetest, the most delicious and worthy of keeping? Can students then make mathematically precise what these fruits are? I use a first day exercise...

***

Ready for my revelations? In order, they were as follows:

1. "...provide structure where structure is not immediately apparent..." Isn't this, at the end of the day, what math is all about? Isn't this all I'm really doing when I'm going about the business I've selected for myself? Is this what I'm asking my students to learn to do, ultimately? If it's really that simple, can I convey the basic notions of mathematics to my students more successfully if I pitch it to them in those terms?

2. "What is structure? Maybe it's a way of organizing things so that they make "objective" sense to someone other than yourself..." Isn't this, at the end of the day, what is meant by "mathematical structure"? In this case, isn't mathematics really little more than an elaborate metaphor, a linguistic convention, a highly human and humanistic mode of communication used to convey often abstruse and technical ideas from one human individual to another or to others? This is hardly the first time these things have been thought (hell, it's not even the first time I've thought these things), but I feel as though the free-writing exercise helped me to think these things more clearly: I was successful at writing to learn, and writing to discover.

3. "In a sense we have to first move students from intuition to mechanics before we can get them to go in the opposite direction!" At the college level (good) math teachers are always trying to get their students to transcend mere mechanical computation and instead develop good mathematical intuition: it's far better to understand precisely where a formula comes from than to merely memorize its concomitant parts. (If nothing else, with true understanding of a formula's provenance you can rederive it from scratch.)

How funny, then, that I realized through this exercise that in order to develop the most basic building blocks of mathematics (relations, functions, sets, orders, et cetera), one really does have to begin with an intuitive concept and work backwards from there, axiomatizing our intuition with rigorously defined concepts like "reflexivity" and "transitivity." Moreover, every time one adds new ideas to the existing mathematical corpus, one must develop new axioms and new definitions: new mathematical discoveries almost always come about through intuition, which is then succeeded by the axiomatization of the newfound ideas.

I find it ironic that I made this revelation today in particular, as just this morning I found myself facing the unpleasant task of writing to a colleague to disrecommend a student who shows profound inability to make the jump from mechanical computation to intuitive understanding.

4. "From the fruits of a brainstorm session, the students can be asked to reflect and decide which are the ripest, the sweetest, the most delicious and worthy of keeping? Can students then make mathematically precise what these fruits are? I use a first day exercise..." As I was about to point out to myself, my current first-day exercise in 280 challenges students to develop a theorem from scratch: beginning with raw "data" concerning the sums of certain pairs of numbers, students first posit a couple of definitions (of "odd" and "even"), then make observations about numbers having the properties of "evenness" and "oddness," then make a claim based on their observations (there's the theorem), and finally prove their claim carefully.

Why on Earth have I not thought to pattern further 280 exercises on this model? Why can't this model serve not only to develop the ideas of "equivalence" and "order," but also "function" and "set" and "combination" and "universal" and "existential" and...

...now for the two revelations that struck me later:

5. "How to best get students to recognize these 'atomic' properties on their own?" I tell my students in 280 over and over and over again: "whenever you get stuck, whenever you don't know what else to do, go back to the definition."

Why? First of all, often the definition is all you've got: if you're asked to prove something about continuous functions, then you'd by god better know what a continuous function is.

Second, definitions are generally atomic, or at least molecular. A definition concerns first principles, and is free from unnecessary clutter. At the 280 level, at least, if the definition doesn't offer an entirely self-contained description of the object or idea being defined, then unraveling the definition's meaning generally involves no more than tracking back to one or two slightly more basic definitions, the atoms in the molecule.

In a similar fashion, theorems are broken into propositions, and propositions into lemmas. You can't possibly come to a proof of a complicated statement like "the expected diameter of a use-it-or-lose-it tree grows linearly as a function of time" without breaking it down further, into simpler statements about the center of the tree, about its diameter and vertex eccentricities, about the way in which those quantities are likely to change as the tree grows according to the defining process, and so forth.

The upshot of all of this is that I realized why it was I'd been hung up on my own research (into "use-it-or-lose-it trees," in fact) for the past few days: I'd forgotten my own mantra and had been attempting to prove too much at once. I needed to step back and break things down into simple lemmas, the mathematical equivalent of Bob Wiley's baby steps.

I've done that now, and I've made more progress in a couple of hours than I'd made in a week or two before I realized my misstep. (280 students, take note! It works. It really works!)

6. I use a hell of a lot of colons when I write.

Seriously. Go back and count 'em. Nearly every other sentence I write has a colon in it.

I wonder why this is? Is this trademark quirk a function of the way my mind processes what I'm writing about? A colon generally precedes elaboration or clarification: what comes after it is meant to provide an illustration of what's come before it. (See?!)

Maybe teachers, prone to using examples to illustrate their ideas to their pupils, are more apt to use colons than people in other lines of work.

Ya think?

I don't know. I just find it fascinating that I so often use that particular piece of punctuation.

I'm going to end this post in just a moment, as it's been a long one, full of fun things to think about. But I'd like to leave you with an exercise, those of you who actually read this thing (I know you're out there!). Given the great deal I learned about myself today through free-writing, I thought I'd assign you, the reader, a brief free-writing task.

For those who'd like to try it out, please respond in the comments section to this post with the fruits of your labor on the following activity. I really do think it will prove a meaningful and enlightening activity, and I hope that you'll consider trying it out. (Former students: how 'bout it, huh? I know you miss my classes! It'll take you a half hour, tops, and I promise it'll be harmless and fun.)

1. We begin with the following question: "What is mathematics?" Now we follow each of the steps below.

2. Give yourself five minutes (set an alarm on your watch or cell phone), and write, continuously, on the topic above. Don't correct yourself, don't change anything you've written, just keep it intact, word-for-word. If you get stuck, write some sort of nonsense until you get unstuck and refocused on the topic above. You can type or write, whichever you prefer.

3. When five minutes are up, take a few minutes to look over what you've written, and select a word or phrase that strikes you in some way. Copy it to a clean sheet of paper (or a clean file in MS Word) and begin anew, writing continuously for another five minutes, starting from the word or phrase you've selected.

4. At the end of these five minutes, once more select a word or phrase that stands out, and copy it to a clean sheet of paper. Write continuously for another five minutes, starting from the new word or phrase you've selected.

5. What results? I'd be delighted if you could share your personal revelations (even anonymously) in the comments section to this post. You could even share your entire free-write, if you'd like to, but you certainly don't have to. Think of this as a semi-public performance of mathematics, a project undertaken in the spirit of Algebra al Fresco. I wouldn't ask you to do this exercise if I didn't think that in doing it you'd make some meaningful observations about yourself.

Please do give it a shot. (It's also a great activity for overcoming writer's block.)

In closing, let me say to my colleague Euterpe: many, many, many thanks for once again proving yourself an exceptional teacher!

Farewell

As I've said many times to many people in the last few weeks, the class of students who yesterday were graduated from UNC Asheville is in many ways "my" class. Having arrived at this school in Fall 2005, were I a student with a traditional sense of timing, I would have joined them in their sunlit march across the stage in front of the Ramsey library yesterday morning.

My last teaching gig, my first one post-graduate school, was a three-year research postdoc at the University of Illinois's main campus in Urbana-Champaign. I only taught one course per semester at that school with roughly 25,000 undergraduates, and, being given "juicier" teaching assignments like Accelerated Honors Calc III for Engineers, Abstract Algebra II, and a special topics graduate seminar in Coxeter groups, I never once taught one of the "core" courses like Calculus I or II, and thereby my chances of having the same students for more than one semester were further diminished. All told I taught 150 or so students during my stay at UIUC, and not a one of them more than once. The same holds true for my teaching career as a graduate student at Vanderbilt: I worked with maybe 200 students over three and a half years there, with no repeaters (well, one exception...but that's a [not altogether pleasant] story for another post).

So it's been radically and refreshingly different seeing the same students in class after class, some taking as many as 19 credit hours with me, watching them grow from timid (or not-so-timid) first-year students, many still clinging to their high school memories like a tattered spit-soaked security blanket, to mature, clear-thinking, sophisticated scholars capable of creating their own original mathematics.

It's been wonderful.

My class has come of age, and it's time now for them to leave. They leave to undertake new adventures, to seek out new experiences, to learn more, to do more, to contribute more to this world.

I say farewell now, fondly, to Bertrand, Cassio, and Davina; to Deidre, DeWayne, and Farina. I say my goodbyes to Farrah, Katya, Kaytlynne, Leonardo, Nicolette, Nidra, Oswald and Stanley; to Ulrich, and, last and certainly not least to Sylvester, Tatiana, and Nadia, the last three of whom shared 16, 17, and 19 credit hours of classes with me over the course of the last four years.

Though I know that I now have and will continue to have in the coming years many more wonderful students, those to whom I now bid adieu will always be dear to me, they'll always be my class.

I love you guys, and I'm proud of you all.

Good luck in all that you do. Keep learning, keep loving, keep living. Keep doing marvelous and incredible things with your time on this Earth, and with the talents you've gained in the years in which I've known you.

And for Pete's sake, keep in touch!

What's in a word?

Would one be more apt to call the academic community afforded by our department "healthy" or "vibrant"...or both?

A different class

I've always found it compelling to think that our ancestors from thousands of years ago were no less clever, no less smart, than we are today, and that they merely had a bit less experience, had had a few fewer millennia in which to sort things out by trial and error and intentional experimentation, than have we. Given several dozen more centuries in which to try their hands at various critical and computational maneuvers, certainly they'd have come to many of the same conclusions as we have by now. (You must admit that we've been given a distinct advantage by the astute application of printing technology and modern methods of data storage, data recovery, and data transmission.)

One day at some point during my third year of undergraduate study at the University of Denver I was idly toying with some polygons that I'd circumscribed with a unit circle and I noticed it wasn't hard to recover an inductive formula for the lengths of the polygonal segments that made of a circumscribed 2ⁿ-gon a circumscribed 2ⁿ⁺¹-gon instead. With a little basic trigonometry (it turns out that the Law of Cosines works best) you can arrive at an iterated radical formula for the number π.

I was flabbergasted, thrilled by my discovery, and the next day I told my adviser, excitedly, about what I'd found.

His response was something along the lines of "oh, Euler's formula!" I'd recovered a formula first noticed by the great Swiss mathematician Leonhard Euler (the 300th anniversary of whose birth was recently celebrated in the math community), akin to an even earlier formula, the first successful arbitrary approximation of π, due to the French mathematician and astronomer François Viète.

If you're going to get scooped by someone, Euler, one of the most prolific mathematicians in history, is not a bad one by whom to be scooped. Still, that discovery that your discovery is not a discovery at all, or at least not a new one, can be unsettling. Certainly it's happened to us all, and it happens more frequently when you make it your business to ask tough questions. How often do even the biggest names in math research get one-upped by slightly cleverer colleagues?

Asking tough questions is the job of the mathematician, so it's imperative that young math-minded minds get used to tackling tough questions in a controlled environment, one in which the answers are already known to be known, and in which tough but tractable questions can be set up for what they are: challenges and tests of skill, yes, but not traps meant to lure the student into a sense of hubristic invention.

Put another way, if you know from the get-go that the discovery you're about to make is not a new one you can take your attention from the statement of the theorem on the page in front of you and place it where it really belongs, on the path you're about to trace out that will lead you to the theorem at its end. That same path, you'll know as you walk along it, is the same as or similar to the one taken by hundreds of highly intelligent human beings who came before you...but like they did before, you'll make your way along the path yourself, and the fact that the land at which you'll find yourself at the end has already been mapped out and explored doesn't make that land any less beautiful or wondrous.

Discovery is like that.

While running this morning I thought of a discovery activity I can use in MATH 280 this coming fall when it comes time to rap about equivalence relations, a topic that proofs dauntingly difficult to a large number of students.

I'll gather several dozen small objects of various kinds and bring them to class in a big ol' bag and empty the bag onto the classroom floor.

"Sort 'em out," will be the order of the day.

"How?" I can imagine students asking.

"You tell me." They'll pick through the pile of stuff scattered before them, and after a bit of trial and error patterns will emerge: the Tonka truck matches up with the lemon-shaped lemon juice bottle (for obvious reasons), and by the same logic the wingnut and the nickel get tossed in the same subpile, and the magnolia leaf meets up with the mango. Without realizing it, the students have constructed an equivalence relation, creating classes whose elements exhibit demonstrably reflexive, symmetric, and transitive properties.

"Can you do it another way?" The next iteration takes a bit more thought, and perhaps now inorganic objects are grouped together while once-living things share a different class. Or perhaps size proves to be the most distinguishing characteristic. Somehow a new partition emerges, and another equivalence relation is born.

A similar exercise may well work to demonstrate order relations. Confronted with a disorderly mess of objects, can the students impose some kind of order on them? What properties does this "order" satisfy? What properties does it not satisfy? Does the order need to be a total one?

Surely the students, without formal knowledge of the definition of the phrase equivalence relation will be able to build several such relations of their own, and having done so will be far likelier to recognize such relations when they encounter them in more mathematical contexts. Moreover, they'll have a greater appreciation for the technical definition of equivalence relations when it's given to them.

That's the power of discovery: you're much likelier to remember and understand something you discovered yourself than something someone else discovered for you and merely told you about.

Why in the hell don't we teach like this more often?

I know an answer to that question already (and my colleagues and students should feel free to supply many more in the comments section): because it's difficult to do so. Setting the stage for incipient discovery is far more difficult than describing what discovery looks like.

I admit that, though I hope that my classes set students up for discovery more often than those of less ambitious instructors, I make use of discovery-based pedagogical methods more rarely than I should. I'm trying, my friends, I'm trying to address that. I hope to devote a good deal of time this summer both to my own discovery (during the hours I spend with my REU students and the other students with whom I'll be doing original research) and to developing means by which I can facilitate others' discoveries on their own.

What discoveries, new and old, await us? I'm tremendously excited to set out on this summer's journey.

I am not Euler, and you are not me. Yet we're all human, we're all clever and intelligent, we're all naturally inquisitive, and we're all equally capable of discovery should we put ourselves in positions from which discovery is easily possible. In this regard no one of us is in a different class.

Making sense of making sense

As you've surely surmised, given my incessant ballyhooing about my Newton v. Leibniz analysis coupled with the continued absence of said analysis from this blog, it's proving to be a helluva chore to extract the essence of my students' reflections on the project and distill it into anything simultaneously meaningful and manageable.

I really do hope to have a nice post on the project up before the REU starts (June 8th!) and my life becomes decidedly busier once again.

For now, I've started to catalogue my notes on the students' reflections according to several rough headings, as follows:

The nature of discovery. Reflections on this topic center on the ideas of discovery and invention: is there a difference between the two, and how can one tell one from the other? Who is entitled to discover or invent, and how does one go about doing these things?

The nature of mathematics as a discipline. Reflections in this category include comments on the way in which math is made and the way in which it's communicated. The deepest such comments were purely epistemological, striking at the very heart of knowledge itself.

Personal reflections. What did the students gain from the project personally? What did they learn about themselves as people, and as learners?

Suggestions. How might the project be modified in the future in order to make it more effective? The suggestions I received ranged from minor comments on the process to major structural overhauls.

Given the depth of these ideas (and the prolixity with which my students produced them), I hope you'll agree it's no surprise I've been prevented as yet from digesting them and writing on them.

This week, an abstract; next week, a post. I promise!

Good advice

I've just finished typing up my 280 students' responses to the final question on their exam: "Please compose a brief statement detailing any advice you would like to give to someone taking 280, to be given to the student on the first day of class. You can be as specific or as general as you like, but please be sure that your response offers honest and thoughtful advice to a new 280 student." The advice will indeed by posted on the website for the Fall 2009 MATH 280 class, and it will be the first reading required of the new 280 students.

The advice ranges from comical ("I recommend keeping a running total of how many times Professor Bahls yells, throws a chair, drops a table, ...or does anything else that startles or amuses you") to metaphysical ("I also found that meditation helps out a lot"), hitting nearly every point in between. Some of the advice is, shall we say, idiosyncratic enough that it might not prove so useful to anyone but its purveyor; other advice is solid.

One thing I noticed about the advice this group has offered to future generations of 280 students is its general practicality: it's very focused on the basic "mechanics" of homework completion and class survival. For instance, the several students emphasize things like starting the homework early, unfailingly attending class, and asking questions of the professor as the keys to academic success; surely this is good advice, but it's hardly different from that one would offer to a student about to begin any other class.

What is it about 280 in particular that makes it such a difficult class? And what specific advice might one offer to a new 280 student, as opposed to one ready to begin Language 120 (our first-year composition course) or Humanities 124?

The students had a few words to say about this. I'll be posting the full text of the students' advice on the website for the Fall 2009 280 course in due time.

For now, below, I've compiled a laundry list of things I want to be sure to tell next semester's students, early and often; admittedly the list betrays my own personal bias, but I think every item is a worthwhile bit of wisdom:

1. Write. Write to communicate, write to learn. Write knowing that every jot and tittle of every bit of notation means something precise, something definite. Write clearly, and keep an eye on your composition. A proof, even a correct proof, is meaningless if it's not also clear and well-composed. But don't forget that writing is an iterative process, and that rough drafts are meant to be rough. In all likelihood, your first draft will be shit, but it's important to get it out in front of you so that you can work on cleaning it up.

2. When faced with a new problem, write down what you know and what you need. Often 50% or more of the solution of the problem consists of formulating a clear statement of the problem's hypothesis and a clear statement of its conclusion. Once those statements are on paper (literally) in front of you, often the path you must take to connect the two becomes evident.

3. Trace out that path in baby steps, applying a single definition at a time, a single logical inference at a time. Don't combine steps or take more than one at a time until you are fully confident that you've not misstepped. Most of the mistakes you make will be made when you attempt to take big steps or to skip them altogether; the smaller the steps you take, the likelier you'll be to stay on course. If you're not sure about something you've written, read it out loud.

4. When in doubt return to the definition. As noted above, every symbol, every term, every penstroke means something, and something definite: if you're not sure what that something is, look it up.

5. Work together. Very few are the problems in mathematics for which a single solution is known. Moreover, there are manifold viewpoints on any single solution to a given problem, and it's likely your friends' viewpoints will differ dramatically from your own. In working together you can more easily combine your viewpoints to create a richer picture of the problem with which you're all faced.

I'll leave it at that, as the best lists of advice are short lists of advice. For now, it's bedtime.

Coming soon: the follow-up to Newton v. Leibniz, further reflections on the current graduating class. and more!

Thank yous all around

It's 9:00 on the "morning after," and I've just finished grading the Calc I exams. Only one person failed! I don't think anyone will be failing the class, a success by any measure.

I wanted to check in briefly and send a quick thank you to the hundreds of wonderful students I've had at UNC Asheville, and to dozens of my wonderful colleagues, without whose hard work and support I would not have been able to have earned the honor bestowed on me yesterday. Any award for teaching excellence rightly belongs as much to the students whose dedication drives them to academic success day after day with intervening sleepless work-filled nights, and as much to the faculty members who reach out to their fellow teachers with new ideas for class activities, assessment techniques, and innovative learning experiences, as it does to the award's recipient himself, a single person who is the product of the environment in which he does his job.

Once again, thank you, all of you, my fellow-travelers on this neverending intellectual journey.

More soon, on what it's like saying farewell to "my class" of students as they prep themselves for graduation, and on this summer's coming wave of researchers from near and far, and on the dawning of Algebra al Fresco...all in due time, once I've got my head above water.

For now, it's on to 280!

Reading day

One course down, two to go.

Abstract Algebra II is now wrapped up, all grades successfully submitted by noon today. Overall the students did very well, which is hardly a surprise since the class had a number of our best and brightest and most senior students, five of whom are leaving us after commencement in a few weeks.

So far the questions from students in the other two courses (both sets of students are dealing with take-home exams) have come in a steady trickle.

Yup, two take-home exams.

I was a bit reluctant to make the Calc I exam a take-home exam, after the debacle from Fall 2007, but I decided that what I perceive to be the benefits of granting the students a take-home far outweigh the negative aspects, including the risk that one or more students might again decide to cheat.

1. By affording the students time for meaningful reflection on the concepts learned in class, take-home exams offer a further formative learning opportunity rather than simply a summative assessment of a student's ultimate performance.

2. By eliminating the largely artificial "high-pressure, high-stakes" environment of an in-class test, take-home exams are more apt to measure more clearly students' understanding of (or at least ability to recover and synthesize deep ideas involving) the course's subject matter than simply aptitude in test-taking.

3. As hinted in the previous point, take-home exams more closely approximate "real-world" settings in which students will eventually find their skills tested, in which they will generally have access to resources (books, notes, mentors, and, yes, the outlawed-even-on-take-home-exams colleagues!) in order to better to meet the challenges with which they're faced.

We'll see how it turns out. I did give a little bit of a lecture when I distributed the exam sheets a few days ago, an uncharacteristically stern admonition at the outset, hoping to instill in the students the gravity of the trust I'm placing in them.

I've received one student's exam so far, but I've not had a chance to look over it. We'll see.

For now, I'm working away, one day at a time.

Tomorrow: donuts and derivatives, and a meeting about writing assessment. Oh boy!

One down...

...two to go. Somehow this end-of-semester has lacked the spark with which most terms terminate. Today's last session of Calc I seemed anticlimactic.

I dunno. Maybe it's just me.

Not ones to disappoint

The 462 presentations were solid.

Though they too have yet to get so far in their careers as to overcome the butterflies that bubble up when one's at the front of a darkened room, all six students whose task it was to speak today did very well. One of the presentations was the tiniest bit rigid, another could have stood slightly stronger preparation, and another seemed a little hurried as the speakers tried to fit in all that they had to say, but the slips were only of the slightest kind, and overall the talks were of the quality you'd expect from this set of mostly seniors. It'll probably be a bunch of As, all around.

For my part, I feel I did better in providing the students (both those in 280 and those in Abstract) with the scaffolding they needed to construct their talks this time around. Next semester I hope to be more intentional still in preparing my students for sharing their ideas with one another orally, both in the formal structured setting of end-of-semester presentations and in relatively informal day-to-day chit-chat sessions (both fora have their merits).

As another day draws to a close, I'll say farewell for now.

Jolly good show...jolly good!

I'm halfway into a Friday filled with student presentations: the 280 folks have offered up three inventive exhibitions, and all did pretty well (especially when one norms out for the nervousness and jitters that accompany what's likely the first semiformal mathematics presentation most of these people have ever given).

The first course on the menu was a deep dish of induction (one within another, like two thirds of a turducken): three of the students worked up a careful verification of the fact that the gamma function generalizes the factorial function, complete with gorgeous LaTeXed slides!

The next offering was a quick course on multinomial coefficients and their usefulness in solving a number of enumeration problems. A few typos aside, their presentation was clear and correct, and interactive! Worksheets are never a bad idea.

Finally, we finished things off with a multimedia derivation of the closed form for the sum of the first N cubes, assuming the formula for the corresponding sums of squares and linear terms. Though there was insufficient time to develop the full proof in class, the method these three folks used was identical to the inductive proof one would use to derive the closed form for the sum of the first N nth powers.

All of today's talks (aside from minor bubbles and burps) were clear, correct, and mostly well-composed. I'm happy. Overall I'm mightily impressed by the maturity of the problems selected by the students in this class, especially since several of the choices made were motivated by inherent interest in one problem or another. (I.e., I didn't have to twist too many arms: a lot of people naturally gravitated towards problems they'd thought up themselves.) The level of preparation has also been exceptionally high, as has the mastery of the subject matter. Though this semester's definitely had its ups and downs, I've got high hopes for a number of the students in this class as they move onward in their math careers. (At least three of them will be doing work with me this summer.)

For now, I must say adieu...I'm off to Abstract II, where I'll be hearing all about lattices, fuzzy groups, and the Sylow Theorems.

It's come to this, has it?

Three days of class remain.

A handful of homework sets, a few exams (most of them take-home), a couple dozen presentations.

There's not much left to do, not much left to say.

We're here now.

It seems like only now some of the lessons I've tried to teach all semester have begun to sink in.

"If you write what you know and what you need, you're halfway...if not further...to a full and valid proof."

I don't just say that to hear it be said. I say it because it's true.

And I think they're finally starting to believe me.

What else? Open my mouth, what other jewels fall out?

"You get out of a class more or less what you put into it."

"The folks who do well on the homework sets are the ones who get started on it right away."

Do I sound a million years old, or what? Who am I to speak?

Was there ever a time when I didn't know how to do a derivative, let alone compute a Galois group? And it's hard to remember that induction isn't a naturally occurring process, that we have to train our brains to mold themselves around its artificial angles and uncomfortable corners.

"...By inductive hypothesis we know that Γ(n+1)=n!..."

By now, nothing could be clearer.

I wonder if those same students who nodded casual assent as we plowed through today's inductive proof would recognize the confusion they themselves faced just two months back when first confronted with that arcane mathematical method. The way's been long and sinuous, and even knowing where it twists and turns it's easy to fall off to its side.

Who knows what takes us to where we are now in our lives?

I used to like to exercise my mind in what I called the "Causality Game": think about a major life event you've undergone, and trace it back, step by step, to its earliest observable genesis: n-1 begat n, n-2 begat n-1, n-3 begat n-2, and so forth and so on and so forth...at a certain point the exercise is absurd, a diagnosis of butterfly burps and other unanalyzable initial conditions.

In the middle of a panel discussion at the WAC/CAC Symposium at UNC Greensboro a week and a half ago I caught myself playing the Causality Game, if only for a minute or so. "What in the world has brought me here?" I asked myself. I was sitting in the center of a room filled mostly with rhetoricians and composition specialists, the people in front of me backed by a graceful arc of windows filled with bright blue sky and overlooking a lush and verdant campusscape. It was a gorgeous day outside.

I took mental stock.

"I'm 34 years old," I thought, "and I'm a mathematician. I feel strongly about more than math: writing matters, too. I care enough about writing to be here, to be in this room. In fact I'm surrounded by teachers of writing, and they're talking about writing. They're talking about assessing writing. Someone's just asked a question. What did she ask?"

It wasn't a question, it was a comment. It was a divisive comment, meant to be divisive. For an hour or more (before lunch, at which time the combatants no doubt held parley) it was Duke and Wake and Davidson versus all the rest of the room. Old habits die hard.

"What brought me here? What right have I?" I smiled to myself, I almost laughed out loud.

For a minute or so I clumsily fumbled with the knot that tied today to yesterday, trying to make sense of the tangled mess of words and numbers that sat upon my lap, and after a bit I said "ah fuck it" and let it drop.

Oughtn't I be happy enough knowing that I'm happy enough, and leave it at that? Who cares how I got here?

I watched the first episode of Carl Sagan's Cosmos yesterday on Hulu. It's been years since I've seen that, and seeing it again reminded me of just how strong an influence it had on me when I was a child. Sagan was a boyhood hero, and I can safely say that few individuals more strongly than he made me want to be a scientist of some kind. (My dad's about the only one that's got him beat.)

I just looked it up: Carl Sagan's been dead for over 12 years now. I'm glad that I had a chance to see him speak before he passed away.

Just that once.

He was as witty and wise as I knew he'd be.

And now he's gone. And now I'm here. And I've got now what I'd wanted then.

As I'm fond of saying, I'm blessed (there, I said it!) in that I get paid to do something I'm damned good at and that I love doing, so much so that I'd do it even if I didn't get paid. (This is good, because as of today I'm getting paid 0.5% less than last year to do what I do.)

What more could I want?

The wants are never-ending.

I want a roomful of peers and pupils who are in love with learning and who aren't afraid to share that love with each other.

I want unlimited time with which to elaborate every beautiful idea those folks can come up with.

I want unfettered, unfeigned, unrestricted, unlimited inquiry.

I want every who what when where why and how to be asked and answered.

I want to not be so goddamned tired when there's so much more to do, and I want the same for everyone around me.

I want...

...I want...

...I want this semester to be over.

Summer's nearly here, and as my wonderful friend Bedelia hinted on her Facebook page a few days back, next semester's already calling on us to atone for this term's teacherly sins with a drawn-out dunk in the salvific waters of early classroom preparation.

Hallelujah.

One and a half more weeks. It's been a hell of a ride, folks.

Take one last deep breath, we're almost there.

It's come to this, has it? Ever wonder what got you here?

Wonder away.

Newton v. Leibniz, semi-official transcript

What follows is a shadow of the proceedings of the Monday, March 30th trial reenactment, Newton v. Leibniz, performed by my Calculus I class. There is paraphrasis, but I've tried to preserve the most important passages as perfectly as I can. I know that I managed to catch several of the juiciest lines verbatim.

***

9:05 -- 9:07. Leibniz's lead attorney (played by Kent) makes his opening argument. "We plead with you that you look past Newton's fame and popularity, and merely focus on the facts."


9:07 -- 9:08. Newton's lead attorney (played by Silas) makes his opening argument. "We will prove our case through an analysis of our client's paper on infinite series, and we will call John Collins and Henry Oldenburg as witnesses attesting to his character."


9:08 -- 9:32. Leibniz's team mounts a defense. The first witness on behalf of Dr. Leibniz is Ehrenfried Tschirnhaus (played ably by Omar). He testifies to the upstanding character of his friend, and to his mathematical talent.

Tschirnhaus: "I knew him well. We met in 1665, and he visited in 1666. I worked with him further, on mathematical projects. He assisted me in my work on catacaustic curves (involving a geometric form of calculus) and natural philosophy. He was a profoundly good mathematician. He was also an honest man. Though he may have been given the opportunity to plagiarize, he didn't take advantage of this opportunity."

Newton's attorneys have nothing to ask in cross-examination.

The second witness called to the stand is a noted historical and mathematical expert (played by Olaf), who testifies as to the differences between the two scholars' mathematics works.

Leibniz's counsel: "So Newton and Leibniz used different techniques?"

Witness: "Yes."

Leibniz's counsel: "Is it likely that their methods were developed separately?"

Witness: "Yes."

Newton's counsel [in cross-examination]: "What, specificallyt, leads you to believe that the methods are different?"

Witness: "Though Newton's work [on calculus] was performed in the years 1665 and 1666, he didn't publish his work until after [Robert] Hooke died. Leibniz had no exposure to Newton in that time."

Newton's counsel [after a good deal of conversation with ihs client and colleague]: "If Leibniz had seen a manuscript of Newton's, is it reasonable that he would have stolen Newton's work?"

Witness: "Yes, reasonable. He did see Newton's work, though it didn't relate to calculus."

Newton's counsel: "Did Leibniz see those papers? Did he use them?"

Witness: "I don't know for certain."

The third witness called in Leibniz's defense is Leibniz him(her)self (played impeccably by Francine).

Leibniz's counsel [holding Exhibit A, a diagram of Leibniz's derivation of infinitesimals]: "Is this involved in your derivation of calculus?"

Leibniz: "Yes."

Leibniz's counsel: "Can you explain it, please?"

Leibniz: "This illustrates my formulation of calculus through the use of infinitesimals."

Leibniz's counsel: "When did you perform this work?"

Leibniz: "The work was completed by 1675."

Leibniz's counsel: "Is it true that you were in London around that time?"

Leibniz: "I was indeed visiting John Collins in 1676. At that point he showed me a copy of Newton's De Analysi, but I didn't take any notes on it."

Leibniz's counsel: "Why were you holding off on publishing your own work?"

Leibniz: "The Holy Roman Empire was a tricky place to publish at that time. Catholicism was on the outs, and I didn't want to do anything to draw attention to myself."

Newton's counsel now begins their cross.

Newton's counsel: "The word 'infinitesimal' appeared in Newton's work in 1665, did it not? Are we to believe that you got no information from Newton's manuscript?"

Leibniz: "I can't prove that I hadn't stolen from Newton, but I ask that you take me on my word that I did not do so. Infinitesimals are integral [no pun intended] to calculus, so the word should appear. On the other hand, I came up with the word 'calculus.' Moreover, many of our terms are different, like 'fluids' and 'fluxions.' After all, I'm not going to call it a duck if it's an infinitesimal triangle."

Newton's counsel: "Would there have been time for you to have found a new route to calculus, given the time delay between your reading my client's work and your publication of your own? Or maybe you changed the date on some of your manuscripts?"

Leibniz: "Why would I do that? Absolutely not."

Newton's counsel: "Could you have, though?"

Leibniz: "I don't date all of my notes, so I can't be sure of when they were created."

Leibniz's attorneys call their final witness, Sir Isaac Newton himself (played by Knut, who shows outstanding mastery of Newton's work on calculus).

Leibniz's counsel [holding a diagram illustrating Newton's derivation of the Fundamental Theorem]: "Do you recognize this?"

Newton: "That looks like my rate of change method. I began my work in 1665, and published it in 1704. In 1687 my first publication appeared."

Leibniz's counsel: "Why did you wait so long to publish?"

Newton: "I was afraid of criticism, as usual, and I wanted to touch it up."

Leibniz's counsel: "Did my client come to the Royal Society in 1711 to plead his innocence?"

Newton: "Yes."

Leibniz's counsel: "You don't think that your position as President made you a little biased?"

Newton: "..."

Newton's attorneys begin the cross-examination of their own client.

Newton's counsel: "You were the President of the Royal Society. Don't you think that's quite an honor?"

Newton: "That is the case."

Newton's counsel: "When did you say you began your work?"

Newton: "In 1665."

Newton's counsel: "What did this work concern?"

Newton: "It was on fluxions and fluids. And rates of change, with infinitesimals."

Newton's counsel: "And this work was done in 1665?"

Newton: "Yes."

Newton's counsel: "Did you do anything else prior to 1687 that Leibniz might have seen?"

Newton: "No, but I wrote letters to Leibniz that contained veiled references he could have pieces together."

Newton's counsel: "You claimed to be able to calculate the tangent to any curve?"

Newton: "Yes. I referred to that in my letters."

Newton's counsel: "Is it reasonable to assume that Leibniz saw your letters and plagiarized your work?"

Newton: "Definitely."

Newton's counsel: "Thank you, Sir Isaac Newton."

Leibniz's attorneys ask a single question in redirect:

Leibniz's attorney: "The Presidency of the Royal Society is a position of power, correct?"

Newton: "Yes. I had excessive power in that regard."


9:32 -- 9:38. The court recesses for a brief break. After the break Newton's side mounts an offensive.


9:38 -- 10:02. Newton's attorneys present their case. The first witness called to the stand is John Collins (played astutely by Mary Ellen).

Newton's counsel: "You and Sir Isaac Newton are colleagues, correct?"

Collins: "Yes, since 1670."

Newton's counsel: "Can you describe Sir Isaac?"

Collins: "He's peculiar, but brilliant. He has a sensitive soul. He's very averse to criticism and often withdraws into depression."

Newton's counsel: "Would you call him vindictive?"

Collins: "No."

Newton's counsel: "Is he honest?"

Collins: "Yes."

Newton's counsel: "Perhaps the most honest person you've known?"

Collins: "Not more honest than my mother, but Newton's definitely up there."

Leibniz's attorneys begin cross-examination.

Leibniz's counsel: "You were personally involved in Newton's promotion to the Presidency of the Royal Society, correct?"

Collins: "Yes. I voted for him."

Leibniz's counsel: "And you were partly responsible for introducing Leibniz to Newton's work?"

Collins: "Yes, I got them together. I had no idea that Leibniz would plagiarize Newton, though."

Leibniz's counsel: "You're positive that Leibniz plagiarized?"

Collins: "Yes. We have the letters proving it."

Leibniz's counsel: "Have you seen Leibniz's techniques?"

Collins: "I've seen his methods, and some of them look different, but that doesn't take away from the fact that he saw Newton's work."

Leibniz's counsel: "Are you aware of the publication dates that show my client's work appeared before Newton's?"

Collins: "It was publicly distributed work, even if it wasn't formally published."

Leibniz's counsel: "How well do you know Leibniz? How would you describe him?"

Collins: "He's brilliant."

Leibniz's counsel: "Brilliant enough to come up with calculus on his own?"

Collins: "Yes."

Leibniz's counsel: "If he could develop calculus on his own, why steal it from Newton?"

Collins: "That's a good question!"

The next witness to take the stand on behalf of Newton is Henry Oldenburg (played by Kevin).

Newton's counsel: "What is your relationship with Newton?"

Oldenburg: "I'm the Secretary of the Royal Society. I've been in frequent correspondence with him, and have had many personal interactions with him."

Newton's counsel: "What is Sir Isaac's character?"

Oldenburg: "He's easily discouraged by criticism from other people. He's never satisfied with his method, and he always tweaks his experiments over and over to make sure he's got it right."

Newton's counsel: "I read about that. When he was working on On Optiks, he deformed his own eye to learn how it would effect his vision." [At this point, for the only time in the trial, Silas broke character: "No kidding. I read that. It was insane."]

Oldenburg: "I pushed Newton to publish. I talked with him often as a friend and not as a scientist. He was kind of withdrawn. At one point he dropped out of correspondence for 19 months. But I personally saw his work develop in his letters."

Newton's counsel: "When was this?"

Oldenburg: "In the early 1670s, I think."

Newton's counsel: "Was the manuscript available by 1675?"

Oldenburg: "Yes."

Newton's counsel: "Would it have been available to Leibniz?"

Oldenburg: "Well, he was elected a member of the Royal Society in 1670."

Newton's counsel: "Was he a big name at that time?"

Oldenburg: "Yes."

Newton's counsel: "As big, intelligent, important as he was, does that mean he wouldn't plagiarize?"

Oldenburg: "It's possible."

Leibniz's attorneys begin their cross:

Leibniz's counsel: "Newton showed Leibniz documents pertaining to calculus. Why did you encourage this?"

Oldenburg: "I wanted to help further knowledge in mathematics. It was unfinished work, but Leibniz requested to see it.

Leibniz [from his/her seat at the defense team's table]: "I did not request that!"

Me [as presiding judge]: "Order."

Newton's counsel [on redirect]: "The issue, Mister Oldenburg, is not that Newton shared his work, but rather than Leibniz plagiarized it, correct?"

Oldenburg: "Correct."

Newton's team calls their final witness, Isaac Barrow (played enthusiastically by Bernice).

Newton's attorney: "How did you meet Sir Isaac Newton?"

Barrow: "It was 1667, in an optics lecture I was giving. He was a geometry student. It was in the early 1670s when I saw his work on calculus. He took a lot of scorn and criticism. He was concerned about his work, since it was not concrete like geometry was."

Newton's attorney: "As a professional thinker, if you had ever published a paper in which you'd made use of another's work, would you acknowledge the other in your work?"

Barrow: "Yes. To not do so would be plagiarism."

Newton's attorney: "Knowing Newton as you do, would Newton have accused Leibniz of plagiarism if he'd not been guilty of it, if he'd merely collaborated with Newton's full knowledge of it?"

Barrow: "No, and he would have been happy with Leibniz if he'd given credit where credit was due."

Newton's attorney: "Is it possible that Leibniz used Newton's work as a stepping stone to complete his own?"

Barrow: "Yes."

Newton's attorney: "And Newton would have been all right with this had Leibniz given him credit?"

Barrow: "Yes, there'd be no argument."

The defense had no questions for Barrow.


10:02 -- 10:04. Leibniz's team offers their closing argument: "Both of these men invented calculus, and invented it by different means. As a remark, note that the two men never met face-to-face, never collaborated, and neither likely knew what the other looked like. I would also like to point out that Newton used fluxions and fluids and not infinitesimals, and we have never accused Newton of stealing the ideas of another."


10:04 -- 10:06. Newton's team offers their closing statement: "Sir Isaac Newton, President of the Royal Society. Undeniably both of these men have great minds, but the point of this trial was to establish whether ot not Gottfried Leibniz stole Newton's work. Newton began his work prior to Leibniz's beginning his own work, and this work was available to Leibniz, so it could have been done. Leibniz saw Newton's manuscripts, and there's no way he could have ignored what he'd seen."

***

The things I like best about this re-enactment:

1. The students' robust preparation: every single one actively involved in the court proceedings showed they'd practiced their roles carefully and had mastered the material they'd be asked to discuss.

2. The students' arguments: while the Leibniz team did all they could to drive a wedge between the two scholars' methods (no doubt hoping the jury would be convinced they were different enough to have been, undeniably, developed independently), Newton's attorneys focused on the character of their client, arguing (more subtly at times than at others) that he was a trustworthy man, honest and upstanding, surely incapable of making false accusations of plagiarism.

3. The students' performances: with only one or two (understandable!) exceptions, none of the students broke character, and every one offered sterling deliveries.

Coming up next: in their own words, students' reactions to the activity, and a little commentary of my own.

Field extensions photo

Here's a shot of the Venn diagram we made for yesterday's exercise (my apologies for the crappy quality of my cell phone's camera):



As we decided in class, the sets of pure, radical, finite/algebraic, and arbitrary extensions make up a rough map of the inner solar system, while the normal field extensions act as a short-period comet orbiting the rational sun on an elliptical orbit pointing to the upper left corner of the picture. (The field of complex numbers, clearly visible in the upper left, marks the comet's aphelion.)