Constraint and contemplation

I thought I might mention a couple of the activities I asked my students to take part in today.

The first was a simple freewrite, with an even simpler summative exercise at the end. I asked my Oulipo students to freewrite for five minutes on the ways in which they've noticed their writing to be affected by constraint. Once they'd completed this freewrite I asked them each to identify three words which gave some indication of their freewrite's content and tone. They then shared these words on the board:

I joined the students in this exercise. I was intrigued that my three words ended up being "pattern," "unknown," and "death." The first was unsurprising but the last two were unexpected. I reasoned (if one can call analysis through freewriting "reason") roughly as follows: "we seek patterns in everything we see, including heavily constrained literature, because patterns have predictive power and allow us to extrapolate from the known present and past to the unknown future; perhaps the most compelling unknown is death...can the patterns we find in constrained poetry help us understand even this ultimate unknown?"

We'll have a chance on Friday to discuss the words above more thoroughly. So far we've only talked briefly in pairs about our ideas. I'm looking forward to these upcoming conversations.

The other activity I wanted to mention is a contemplative exercise I asked students in both classes to complete. Knowing full well that Wednesday is the longest and most stressful day for many folks in academia (students, faculty, and staff alike), I wanted to do something to help alleviate the stress. Yesterday, while sitting in the first session of my first faculty/staff learning circle since Fall 2011 (it's been too long!), I thought up the following activity, though I'm sure it's not original to me:

1. Take out two scraps of paper.
2. On one of them, write something that's stressing you out or bringing you down.
3. On the other, write something that's bringing you joy or making you smile.
4. Ball the first scrap up into a tiny origami boulder and chuck it into the recycling bin lovingly provided in the middle of the room.
5. Fold the second scrap neatly and tuck it into your pocket, where you're likely to find it once, twice, thrice, throughout the rest of the day, a gentle reminder of something you should be happy about.

It's a simple exercise, but in both classes it elicited a great response. People in both classes gleefully threw their stressors into the recycling bin in a hail of discarded worry. ("This feels really therapeutic," said one of my Oulipo students.) I noticed smiles and nods from many folks as they put their happy thoughts away. The activity took three minutes out of class time, but I believe it went a long way to establishing a sense of community and a more relaxed state of mind in which we would all be a bit more receptive to new ideas. I might have to make this, or something like it, a regular Wednesday occurrence.

I'm new here

Parker J. Palmer's and Arthur Zajonc's The heart of higher education: A call to renewal (transforming the academy through collegial conversations) (the centerpiece of the most recent faculty Learning Circle in which I took part, and about which I've posted somewhat recently) gave me more than its fair share of things to think about. Many of its insights offered theoretical, even spiritual, enlightenment regarding teaching, but other insights were more practical and practicable.

One of the more down-to-earth suggestions offered up by one of the coauthors' colleagues (Patricia Owen-Smith, Professor of Psychology and Women's Studies in the Oxford College of Emory University) was a means of encouraging contemplative practice in the classroom simply by playing music to being each class period. Owen-Smith (pp. 157-161 in the above work) describes how several years ago she began the practice of playing 7-9 minutes of music at the outset of each class. She encouraged her students to "go within, be still, and listen to the self." While she admitted the difficulty of tying this contemplative practice to improvements in students' achievement of cognitive learning goals (as if this would be the purpose of playing the music in the first place!), she reports that

over time, I learned that the music and meditative moments had an impact on many students. Some students began to ask for guidance with their contemplation and reflection....By midsemester several students per class would mention that they looked forward to this nine-minute period of music. Some students began to bring music from their own collections that they found inspirational and important. As we neared the end of the semester, the structure of the class had changed from a group of individuals reluctantly gathered together for study to a community of friends and partners who were creating a space of introspection, quiet, and respect for the process of study and the development of self.

Once or twice a week for the past few weeks I've begun a similar practice in one of my classes. It began with my reading of an excerpt from Rilke's letters to Franz Kappus (a reading which moved one student so much she had to leave the room), and continued with a passage from Dick Leith's history of the English language. At one student's suggestion one morning we watched a scene from Harold and Maude, and the next day I read one of my own recently-written poems ("Ode to Ned Maddrell," which I penned for the last-living speaker of the Manx language). To open off on Monday (at the suggestion of another student) we'll take in one of Gil Scott-Heron's last videos.

This practice has developed naturally, in an easygoing fashion, and perhaps unsurprisingly it's happened in the most natural and easygoing section of any of my courses this semester. From Day One nearly every person in that section of Precalculus has worked well together, helping each other out and asking for help when help is needed. This natural ease with which we've worked together all term has made it hard for me to tell whether the contemplative practice has had a real effect on the esprit de corps of the class...

...but I'm not going to take any chances. The practice has definitely helped me to make connections between the intellectual and the personal, between the scientific and the humanistic. It's helped me to remain focused on what really matters, and it's helped to remind both me and my students that we do well to look for mathematics' usefulness...and to look for its beauty. Much like well-chosen low-stakes writing activities, this practice is well worth the few minutes of class time that it takes.

Beginning right away I'm going to introduce this practice in all of my classes. It can't hurt.

What effect can it have in a class like my current Abstract Algebra class, a room full of stressed-out work-wearied students representing, honestly, probably the greatest range of mathematical ability I've seen in one section since I began teaching at UNCA over six years ago? I admit here and now that I find myself frustrated at how I've managed this course. For some time now I've felt the need to slow its pace to accommodate the most modestly quick learners, but without sacrificing the true nature of the subject, a nature laden with often-abstract proofs. I don't know how much slowing the pace down has helped: several students are still struggling, and as much as I hate to leave them behind (I hate hate hate people who teach to the top ten percent), I simply must move on. Moreover, I sense that the slowness has led to frustration on the part of some of the class's quicker students, and I can feel cliquishness setting in...not a nice way to end the semester.

Maybe it'd be wise at this point to remind everyone that we all have a right to say "I'm new here": we're all free to make mistakes from time to time. After all, no one's lived this life before, and the future hits us all at the same time. But no matter how far you go, you can always turn around.

What say, folks? What are we going to make of our last five weeks or so together?

Where do you want me to be?

Today was, sadly, the last of four meetings of the Learning Circle I took part in this semester, on Palmer and Zajonc's The heart of higher education (about which I've posted recently before)...even sadder was the fact that I'd only been able to attend two of the four meetings, other obligations taking me out of town and off campus. I've found this text eye-opening and enriching, and the conversations surrounding it even more enriching still.

I found myself thinking out loud in the circle today, wondering what our academic lives would look like if we took ourselves out of our offices and started doing more of our work more publicly. What if I started holding office an office hour or two on the quad, or in the student union, or in the glasshouse adjacent to the library? What if my colleagues joined me in this, sharing a table with me as we worked with our students, observing firsthand how we interact with them, witnessing the kind of learning that goes on in one anothers' disciplines?

Soon might fall the disciplinary boundaries we're all quick to dismiss but unconsciously eager to maintain. At a university so dedicated as mine purports to be to the liberal arts, we ought to be all about interdisciplinary exchange that blurs distinctions between this field and that. But when it comes down to it, as often as not we retreat inside our hard-shelled departmental silos. "I'm all for interdisciplinarity," we might say, but in practice we add "as long as it happens on someone else's time."

But if we started seeing more of each other on the quad, in the union, in the glasshouse, maybe we'd know more about each other. We'd each know more about other's passions and pursuits, and more about the way the other thinks. We'd be able, at least momentarily, to adopt the other's disciplinary perspectives, and when our students ask us why they're asked to take courses in that field or this, we'd be able to tell them why, honestly, earnestly, and confidently.

If we knew more about each other, perhaps we wouldn't find the sort of territorial entrenchment I'm witnessing right now in various departments' defenses of their current curricula to the Curricular Sustainability Subgroup of CRTF which I'm heading up. Sadly, though perhaps not surprisingly, many departments' chairs are convinced of the inestimable value of their own departments' offerings, and of the waste and profligacy that must go on elsewhere in the curriculum to result in such a burdensome mess. The written responses we've received from most chairs are for the most part meticulously-crafted hagiographies telling tales of the author's department's excellence. Without doubt there is good going on in every corner of campus, but equally doubtlessly I know we're all to blame for the unsustainable burden we've taken onto our own shoulders.

If we were to meet face-to-face more often, there'd be no need for polemic; we could engage in frank but warm discussions. It's hardly a coincidence that the considerable face-to-face work the Curricular Sustainability Subgroup has done this summer and fall has been done so smoothly and so amicably. After all, it's difficult to be defensive (or to go on the offense) when your interlocutor is sitting right in front of you. Face-to-face meetings encourage warmth, empathy, and honesty. The "rocking chair conversations" Parker J. Palmer talks about in Chapter 6 of his book with Arthur Zajonc are the perfect places for real and lasting change.

I'm ready to set out some rocking chairs. I'd like to try this, if just for a few hours each week. Maybe while the weather's nice, the steps of the library might do. When it starts getting colder and darker earlier in the day, I can move to the Pinnacle in the student union. Students, colleagues, let me know: where do you want me to be?

Fishbowl

Today I attended the first of several meetings of the faculty Learning Circle in which I'm taking part this semester, on Parker J. Palmer's and Arthur Zajonc's The heart of higher education: a call to renewal (transforming the academy through collegial conversation), and hot on the heels of this we held the first meeting of the Curricular Sustainability Subgroup of CRTF. The former meeting prepared me well for the latter: big talk and deep thoughts about integrative pedagogy helped inform the reflection my colleagues and I did on the general education programs at several of our peer institutions.

Most such programs differed from ours in one or two fundamental ways: either they comprised significantly fewer required courses (no more than a handful of fundamental classes scattered across the curriculum, generally a few to each major division, and maybe including a writing course or a math requirement) or they placed more stringent demands on the student but offered her a multiplicity of ways of meeting those demands (with copious menus of courses to meet requirements for history, humanities, or "world cultures" requirements).

What could go wrong? Both designs assume students (and faculty!) can pack a great deal of interdisciplinary engagement into a broad array of courses. This assumption seems like a risky one at schools like Wesleyan College (Connecticut), where students must complete only 9 general education courses, 3 from each division, all with different departmental prefixes, and hundreds of courses can count toward the gen ed requirement...wouldn't an optimal interdisciplinary experience require the instructors in all of relevant courses to work in concert to help students realize this experience? Honestly, I'm not sanguine about the ability of an entire faculty to coordinate such a widespread effort.

I wonder, though: if we were to balk at implementing a curriculum like this, would it be out of fear of handing the reins over to our students? Maybe as we contemplate changes to our own ILS program, we should consider that our students may be less risk-averse, more daring, and more willing to try new things than we are: they might not need as much help as we think they do as they muck about in their meaning-making.

In the Learning Circle several of us noted how our students can thrive on the chaos and disorder that ensues when we dare to step away from center stage. Though the exercises we plan for our students might not lead them where we expected them to, they cannot be labeled "failures," so long as they tell us something about ourselves.

I was reminded of a particular meeting of the MATH 280 course I taught a few years back (in Fall 2007, I think?). It was a small section (15 or so students), and full of bold and fearless thinkers. One day toward the end of the semester our conversation meandered from the preplanned course when one student, Quincy, asked a pointed question about modular arithmetic. Rather than finish up the exercises I'd planned for us, we wandered off (all of us, as a class, one student, and then another, and then another, suggesting steps to take) in search of patterns that would help devise solutions to very general equations. The end of class came on quickly (time flies when you're having fun), but not before every one of us in the room (including me) had a far better understanding of modular arithmetic than we'd had half an hour before.

"The Math Department should run a 'fishbowl' course," Quincy said as we walked back to Robinson Hall from our classroom in the basement of Karpen. "At the beginning of the semester, we'd fill a few dozen scraps of paper with brief notes about various topics in mathematics, and at the end of every class someone would pull a scrap at random. Whatever topic was on that scrap would be the basis for the next class discussion, a free-form exploration of that topic."

Though this sort of course would be messy, hard-to-manage, and perhaps occasionally fruitless, to this day I still think it sounds like a hell of a lot of fun.

More to come on the Learning Circle and CRTF...and not so long now before this year's CWPA meets up in the wilds of Wildacres, just yards from the Blue Ridge Parkway. Stay tuned!

Fun Times

I'm smack-dab in the middle of the freest time of year for me, that three- our four-week-long end of May, during which time I'm generally "relaxing" after a successful semester, working like mad to take care of the various reports due from me in the next few weeks, and busily prepping for the REU that starts two weeks from tomorrow.

This year my wife and I were able to get away for our first "real vacation" (defined as "a trip involving neither work nor family visits") in several years, a five-day cruise to the Bahamas. Of course, being who I am I managed to make the most of it by bringing along some highly inappropriate pleasure reading, the thirtieth anniversary edition of Paulo Freire's Pedagogy of the oppressed, the central text for the summer Learning Circle I'm taking part in in June and July. (The only reading material I saw which may have vied with it for the "Most Ironical Reading on a Cruise Ship" award was one woman's copy of Orwell's 1984; I think if she'd have been reading Huxley instead, she might have had me beat.)

I'll likely have more to say in the coming weeks about Freire's philosophy as it applies to math education at the university level, but I wanted to put forward in this post an idea for a new ongoing writing project I hope to implement in at least one of the two classes I'm organizing in the fall (Precalculus and Abstract Algrebra I).

Oddly enough, I got the idea from the cruise company. Every evening around dinner time we were treated to a delightful turn-down service, featuring complimentary mints, expertly folded towel animals, viz.:


and a copy of the Fun Times, the cruise ship's guide to all of organized activities that would be going on on the ship the next day. It was little more than a newsletter, three or four pages in length, just the sort of periodical I think a class (or two) full of students, working together, could crank out at least once a week, if properly prepared to do so.

So here's the idea: ask students to put together an ongoing "newsletter" for their class, The Algebra Times, or The Precalc Picayune, if you will. Its content would be flexible, and what went into it from week to week would be left to the discretion of the students (I'd want to have as little to do with its creation as possible). Perhaps, for example, the precalculus newsletter could include

• study tips,
• hints for tricky homework problems,
  
• advertising for study groups,
• applications of course material to areas outside of class,
• games and puzzles,
• "letters to the editor,"
• recommendations for class activities,
• recaps of recent class activities,
  
• personal reflections on math in general,
  
• etc.

Whatever. It'd be up to them. I'd stay out of the day-to-day operations. Maybe I'm being overly sanguine, but I can imagine a handful of particularly eager students taking on editorial and managerial responsibilities (there are a few in every class). To ensure participation by the class as a whole I'd require every student to contribute to the newsletter at least once, twice, thrice, something like that, during the course of the semester. (Jointly-written contributions would count.) It would be difficult to "grade" contributions (I might shy away from this entirely, keeping it a low-stakes exercise), although I'd provide feedback to authors confidentially.

How's this sound? Colleagues: have you tried something like this before? Students: would this be something you'd be all upon?

On your marks, get set...

It's been a while, huh?

As you might suspect, I've had a lot going on, keeping me from posting here for the past, oh...two months? Ouch.

What's going on?

The REU's over, but the work has really just begun: the students this year were fantastic. They were smart, fun, and funny, and they worked tremendously hard, producing a ton of interesting mathematics. (We should get at least five papers out of it in the next couple of months, and maybe more beyond that in the follow-up.) And this year it was "buy eight weeks, get one free": six of the eight students ended up traveling to MathFest in Pittsburgh, PA to present on their summer research.

The REU took up much of my time, but I've had a number of other things going on, as well, among them: (1) working on student learning outcomes for both the Mathematics Department and the Writing Intensive program, (2) putting together the ILS Oversight Committee's report to the Academic Policy Committee (still not submitted, but finally almost finished), (3) planning my contribution to the short course I was helping run at MathFest, (4) working with my College of Charleston peeps on our paper on the rhetoric of mathematical writing, (5) looking ahead to my round table discussion of program assessment for this year's CWPA (coming up in a few weeks), (6) laying the groundwork for the book I'm now slated to write for Jossey-Bass (w00t), and (7) trying to get ready for my Fall classes, which start on Monday.

(1) Student learning outcomes: although I see the purpose, ultimately, of programmatic assessment, too highly institutionalized assessment tends to become bureaucratic and corporate. I don't want to say more about my role in this university-wide process this past summer other than that it was frustrating at times, had its minor joys, and though it was ultimately fulfilling in that I feel like some good will come of it, I worry about the intentions some people in administration may have for codifying the university's student learning outcomes as rigidly as is being done. 'Nuff said.

(2) The ILSOC report to the APC: this is more of the same. I see the purpose, and the writing of this document is a worthwhile activity, but one which runs the risk of being overly parliamentary and pro se. 'Nuff said.

(3) MathFest short course on Sage (an open-source computer algebra system): I actually wrote to my colleague Oscar (one of the co-organizers of the short course) in mid-July, a few weeks before the course was to take place, expressing my insecurities about putting together an hour or two of meaningful material.

"Nonsense," was the substance of his reply; "you'll do fine. Write about what you know." So I threw together several worksheets full of Sage code purporting to implement a number of higher-level graph theoretical algorithms and called it good. The worksheets actually went over really well in the short course and filled a rather comfy niche in the program. I felt good about what I'd done.

The moral of the story: just do it.

(MathFest, by the way, was a blast. Pittsburgh's a nice city with lots of pretty runs and nice architecture, the REU students were fun to hang out with, as always, and the MathFest program was replete with fun stuff to see and do. A good time had by all.)

(4) Math rhetoric: as I mentioned in my last post, oh so long ago, Bella and Damian came up from the coast to spend a couple of days with me, hammering out a plan to finish the paper we'd begun (with Nicola, who couldn't make it) on the rhetoric of the writing my REU students had produced in 2008 and 2009. They had a lovely time while here, visiting with and interviewing the REU students from this past year, working on our draft at the time, and just hanging out. I'm happy to say that as of last night we have a draft that's nearly ready to submit to the comp/rhet journal Across the Disciplines. It's the first of what we hope will be...well...at least more than one article on the rhetoric of mathematics and its learning, and the role writing plays in inducting students into the scholarly mathematical community.

(5) CWPA: I'm already looking forward to the Wildacres retreat! It's going to a be a full house this year, several dozen of us crowding into the nooks and niches of the Wildacres North Lodge to join in round table discussions of assessment with various forms and functions. I plan on presenting a brief "natural history" of the assessment that's gone on in our WI program during the past three years, exposing to public view the layers of rock comprising the "pilot" assessment Lulabelle headed up in AY 2007-8, the refinement she and I performed in the following year, the plans for assessment she and I laid out in the lead-up to the university's reaffirmation of accreditation (notice I haven't mentioned SACS yet in this post...), and the modifications those plans underwent as I limned the WI program's assessment of student learning outcomes this past summer.

It's sure to be riveting.

Actually, I'm more looking forward to hanging out with Damian and Bella, and with them planning the next phase of our study of the REU students' writing.

(6) Book: Oh yeah...I'm now under contract to write a text (working title: More Than Numbers) on the role of writing (specifically, writing to learn and writing in the disciplines) in mathematics and the mathematical sciences. Jossey-Bass received well the proposal I pitched to them several months back (in March, I think?), and after some fine-tuning of the project, I was brought on board. I'm excited, but at the same time terrified. I'm confident it'll all work out well, though. I've already done a ton of enlightening reading (if you get a chance, check out Scott L. Montgomery's The scientific voice...it's a fascinating read that, along with several other sources, has helped me put together a theory of the schizophrenic nature of undergraduate mathematical writing), and I have some great ideas. Furthermore, my consulting editor for the project is someone a few of whose books I've read and whom I respect very much; I'm sure she'll be very helpful to me.

I will say this, however: if between now and next March I start randomly babbling to you about reader response theory or the role of writing in the mathematical finance classroom, just smile and nod your head.

(7) Fall 2010: yup, we get underway in two days. Yikes. I'm actually really excited: I've got the first week mapped out in all three of my courses, and I've plotted out what I think are clear and navigable problem-centered paths for both of the lower-level courses (Calc I and Linear Algebra I) to follow.

Calc I is going to begin with a brief review, and then the students will work toward developing the skills needed to solve an optimization problem from economics. They'll need to "invent" the method of secants and tangents, the definition of a limit, and the definition of the derivative; and they'll need to learn how to compute some simple derivatives in order to solve this problem. Once all of that groundwork is laid, we'll spend some time working with the textbook and picking up some useful formulas for limits and derivatives, but everything we do is going to be application-driven and learner-centered. I hope to not have to lecture for more than 10 minutes at a time for the whole damned semester.

Linear's going to be the same: Day One features a variation on the "Markov Dance" that started the semester off the last time I taught this course, in the semi-mythical beginnings of this blog. The limiting behavior of this system is going to be the motivating problem for everything we do in the first half of the course, as we work our way through everything we'll need to do to get to the fun and useful stuff (eigenvalues, diagonalization, and Markov processes): solving linear systems, inversion of matrices, solvability criteria, and finally eigenvalues. I'm going to downplay theory and upplay (what a cool word!) functionality. I hope to get to the fun stuff by the sixth or seventh week, at which time we can spend almost all of our time on applications. It's going to be fun.

I should mention that I've modified my grading schemata somewhat, but not substantially, from last semester: students in both of the above courses will still have an opportunity to perform potentially unlimited revision on most things they hand in, but I'm tempering the grading scale so that it's no longer possible to earn full points back with each revision. I think the system I've set up is a generous and reasonable one, though, and I believe it'll still serve to motivate students to learn rather than to grub for a good grade.

Meanwhile, the Senior Seminar's speaker schedule is chocked full, with four visitors (all versed in speaking to undergraduate audiences) coming to share their knowledge and spend a little time with the students. It's going to be a good semester in that course.

Okay, that's all for now...I'm going to try to post more regularly during the coming semester, although I can't promise it'll be more than anxiety-tinged stream-of-consciousness rambling about this or that idea for the book.

Fare thee well!

Weather's great, wish you were here.

I haven't had much time to write, but I thought I'd send a postcard from this edge.

We've just finished three weeks of the 2010 REU, and things are chugging along smoothly.

Incredibly smoothly.

Wow.

The students' first presentations (yesterday) were masterful, already conference-worthy: their talk mechanics were spot-on, they displayed superb mastery of the concepts and computations they presented, and their slide designs were eminently professional. In every way their presentations were superior than many of those I've seen given at major conferences by seasoned mathematicians.

They're ready for the "minisymposium" we're playing host to next week, to which we've invited the faculty and students from two other area REUs.

I hope that I can take some of the credit for the students' success to this point: I've been yet more intentional in my design of the program's activities than I've ever been in the past, providing even more and more timely instruction on writing, presenting, use of scholarly sources, reading of math papers, etc. than in past iterations of the program.

And I've had help in this, too, that I've not had before: Bella and Damian, two of my Charleston-based partners in composition-theoretic crime, came up the mountain a little over a week ago to talk to the kids about writing and math. The visit was twofold...perhaps even threefold: (1) to give the students the rhetorician's point of view on mathematical writing, (2) to interview (though this term was never used) about their past experiences with academic writing in general and mathematical academic writing in particular, and (3) to get our shit together and make some progress on the paper we began back in February when I joined them for a working weekend down in South Carolina. (To this last point, briefly: I was tasked with coauthoring sections on the students' mastery of visual rhetoric, specifically with regard to the visual effects on the reader of white space and mathematical formulas; and on the students' use of sources. My analysis of the latter was very enlightening to me, and helped me quantify the fact that this year's students are well ahead of their predecessors when it comes to making effective use of the literature they find in the course of their investigations.)

The two-hour-long "interview" with the students was wonderful. In the discussions that arose the students revealed rich and robust histories with writing, including a good deal of relevant and highly intentional instruction I might not have expected. These students are very well-prepared (which helps to explain the success they've had so far in this summer's program!). The interview also gave us a great deal of data we can use as a foundation for our next article, in which we hope (already!) to trace the development of students as communicators of mathematics through the course of a summer research program one of whose foci is on writing in the discipline. Bella's already kicking around the idea of coming back toward the end of the program to interview the students again and see if anything's moved.

Mathematically, the students are fantastic. They're very self-directed and intrinsically motivated. They're displaying superlative creativity and originality of thought, to a greater extent than did many students in previous years of the program. They're making terrific use of the sources they find, plying them effectively to serve multiple purposes: to find other sources, to support their own propositions and theorems, and to situate their work in the context of what's come before. And every one of them has a clear sense of purpose in his or her project: no one of them is "stuck," as I've seen happen in the past.

More important, perhaps: they're getting along exceedingly well with each other. From Day One they've bonded remarkably well. They do almost everything together, not because they feel like they have to or because they're afraid to do thing alone (both of these have been moving forces in previous iterations of the program), but because they want to. They clearly like each other. A lot. It's good to see.

It comes across in simple things, too: yesterday, during the students' first round of talks, this year's group proved themselves to be the most attentive and responsive audience members we've yet had in the program. They listened with clear and unfeigned interest, they gave encouraging nonverbal feedback (nods and smiles) at appropriate moments, and they asked a number of good questions of their peers at the end of every talk. Midway through the talks Tip (my sole faculty partner in running the program this year) and I marveled at the fluency of the students' presentation skills and remarked at how though the students had all chosen to work on individual projects (something we initially bemoaned) it was evident that to a greater extent than in previous years the students are collaborating with one another behind the scenes, sharing ideas, talking about LaTeX and Beamer, and helping each other through rough spots.

Ladies and gentlemen, we have established an effective learning community.

What else is up?

I've only been to one meeting of the Learning Circle (having missed last week's to work the run-off election) I chose to take part in this summer, but I have a good feeling about the direction it seems to be taking.

I've received good feedback on the proposal I sent to Jossey-Bass for a text on writing in mathematics.

I've finally submitted a reasonable draft of an assessment plan for the university's Writing Intensive student learning outcomes to the powers that be. I doubt that's the last I'll hear of that, though.

It's been a busy summer, but a good one. I'm excited to see what the coming week will bring.

For now, I'm going to kick back and watch Ghana get the better of the U.S. soccer team.

Day Two is in the can

We've wrapped up two days of the 2010 REU, and things are running smoothly so far.

The students are gelling tremendously quickly into a friendly and supportive learning community. As early as the welcome barbecue a few nights back it was obvious these folks were going to get along really well, and the last couple of days have borne that observation out. "I was a little worried before coming here," one of the students told me at the barbecue, "but everyone's so nice that I know I'm not going to have trouble working with them. It's going to be easy to ask them for help if I need it."

Although I can't yet claim to know them all incredibly well as individuals, each is beginning to make known her or his particular strengths and weaknesses. Three of them are particularly eager to demonstrate their skills and hesitate not at all in traipsing to the board to show off their solutions. A couple of the others are more retiring, willing to whisper the answers (almost always correct) from where they sit but less enthusiastic about taking to the board. The remaining three are quieter still and it often takes a little prodding to get them to share their ideas. Notably, to an extent not seen since the first year of the program, one of the students is clearly emerging as the "social director." She's the most outgoing of the group, though a couple of her peers aren't far behind when it comes to eagerness and enthusiasm.

I should note that the students needed little convincing to complete the first day's freewriting exercise, which was designed to tease out of the students their personal goals for the overall program. I've posted the resulting list of student goals here; happily, most of the learning goals coincide very well with my own.

The students are making quick progress through the long list of graph theoretical definitions I bludgeoned them with yesterday afternoon, and we're already over halfway through the graph theory notes I hope to complete before calling it quits. We've also hit on several topics from dynamical systems and geometry, and tomorrow brings more graph theory, dynamics, LaTeX, and Mathematica. It's going to be a full day, but I think we'll get through it.

Further bulletins as events warrant. For now, bed calls. My bedtime reading? The basis for the Learning Circle in which I'm taking part this summer, Helping college students find purpose: the campus guide to meaning-making, by Robert J. Nash and Michele C. Murray (2010, San Francisco: Jossey-Bass). We'll see what comes of it.

Back to the basics

I'm feeling a bit less stressed-out than I was this afternoon when I put together that last post. A good run always helps me out.

While handing back exams in both my morning and afternoon Calc I sections today I brought up the idea of using portfolios as a means of assessing student learning in mathematics courses. This idea was couched cozily inside of a conversation about the shock of receiving a "bad" grade on an exam (as some of the students no doubt experienced today). "I hate having to grade y'all," I told them. "I'm more and more opposed to grading in general, and to the simplistic distillation that goes into assigning a single letter grade to such a Gestalt as the sum-total of a student's learning activities throughout an entire semester."

There were many nods of agreement when I described how I'd like to be able to supply them with all of the same feedback I give them already...without the numerical rankings, the stigma-making marks that say "she's more highly-ranked than he is."

"I'm not going to do it this semester, since it wouldn't be fair to any of us, you all or me, to change the system midway through. But I'm seriously thinking about it for future semesters."

More nods of agreement. I'm convinced that students are not against this.

But if I were to move to portfolios, the first question would be, What goes into those portfolios? Clearly students would be asked to submit materials of various sorts that purport to demonstrate mastery of course learning goals. Ultimately, then, the question becomes twofold: What are the learning goals of the course? and What course activities (projects, exams, written assignments, homework assignments, etc.) would be sufficiently rich to demonstrate clear mastery of the learning goals selected?

As I reminded my 280 students today (quite forcefully, I hope), when you've got no idea what to do, you go back to the basics. In 280 in particular and in mathematics in general that usually means you'll want to take a long, hard look at the definitions. In course design, it means you'll want to take a long, hard look at the reason you want the students taking your class in the first place.

My current learning goals for Calc I (as stated in this semester's syllabus) are as follows:

1. Be able to explain to a peer the concepts of limit, continuity, and derivative.

2. Demonstrate how basic problems in physics, engineering, chemistry, and other fields can be couched in math terms using mathematical models.

3. Be able to follow confidently the course of a simple proof.

4. Be able to perform and properly interpret derivatives.

5. Demonstrate (through informed question-asking) a healthy skepticism regarding mathematical and scientific arguments.

6. Demonstrate how to approach a (not necessarily mathematical) problem effectively by breaking it down into smaller problems, arguing by analogy, and applying other basic problem-solving techniques.

I think it's clear that mastery of some of these would be very difficult to assess using "traditional" assessment instruments. While (1) and (4) could be got at with a well-designed traditional test, assessing (2), (3) and (6) would require a more robust (and likely highly nonstandard) project of some sort, and (5) would require something extremely atypical...maybe a dialogue of some sort, or some other "creative analytic practice" (to use Laurel Richardson's term).

Of course, the above learning goals are merely my own...I'd love to see what students could come up with for learning goals of their own. Maybe I should ask them? Yes, I think I shall.

Clearly there's a lot of thinking left to do, on many persons' parts.

For now, I'm off to eat dinner. I hope that if you read this, you'll reflect on it for a moment or more and offer me a few thoughts of your own in the comments section.

Will the (Learning) Circle be unbroken

I'm in Clemson for a couple of days, very much enjoying the 24th Annual Miniconference on Combinatorics and Algorithms. I'll be speaking tomorrow about a topic I've thoroughly enjoyed working on for several months now, with the help of one of this past year's REU students. Despite the research-heavy atmosphere I'm breathing down here in Tigerville, I've been reflecting deeply on teaching philosophy and practice in the last several hours.

You see, I've decided that I'd like to offer to lead a Learning Circle for Spring 2010, and I'd like to base it one of two books, both by the same author. I'm trying to decide between Alfie Kohn's No contest: the case against competition (1986) or the same author's collection of 1990s essays, What to look for in a classroom. The former I read a couple of years back when it came up in the reading I was doing to supplement the reading a Learning Circle I was taking part in (I don't remember which Circle it was now); the latter I'm reading now. (I took in about 50 pages as I whiled away a couple of hours this morning by the giant fountain at the center of Clemson's lovely campus.)

Both books are compelling and offer frank examinations of questionable classroom practices that leave our students in the lurch. I think both would make for wonderful multilogues between faculty and staff.

Which to choose?

While working my way through one of Kohn's more challenging essays in What to look for, I began to envision a way in which Calc I could be taught authentically and in a purely problem-based and student-centered manner. It might go something like this:

Day One. "Holy shit! Here's an economics problem we've got to solve! XYZ Megacorp, LLC needs to find all of the maxima on this profit curve!" [Shows students real data, with very messy numbers...so messy that guesstimating by eyeballing a graph ain't gonna cut it.] "What's all that mean?" "Well, let me brief you on XYZ's portfolio..."

Day Two. "So, how are we going to find these maxima, y'all? Any suggestions?" "I dunno...those are the places where the graph goes up and then comes down..." "Yeah, if we could find the places where the slope goes up and then down..." "Well, how do you find slope?" "Uhhh..."

Day Three. "Okay, so we need to find slopes, right?" "Uh...yeah." "How?" "Uh..." [A timid voice, as yet unheard, comes from the back.] "It's rise over run...right?" "Cool...can you compute those?" "I guess. Let's try to get some formulas..."

Day Four. "So now we've started computing rises over runs. But so far they're giving us rises over runs on intervals, not at specific points. How's that going to help us find the exact point where the slopes go from increasing to decreasing?" "Uh..." "We've got some formulas we can work with..." [Another timid voice, this one from a different student.] "What if we take two points that are really close together?" "All right, what does that do to the formulas?" "Is that the right way to go?" "I don't know. What do you think?" "Uhhhhhh..."

Day Five. "Okay, so you've got these formulas...I give you two points...two points that are really close together, and you can give me the slope. Let's try it out for a bunch of different functions. Which ones would you like to try?" "How about just y = x?" "That's too easy, man...let's do y = x²." "And square root of x!" "Dude." [There is much scribbling and crushing of paper.]

And so forth.

This whole enterprise would be messy, sporadic, anxiety-inducing, and would require the patience of a boatload of Jobs to do it effectively. Moreover, I can't see it working with a class of more than about 12 students.

I wanna try it!

More later, to be sure. And I'll update you on the Learning Circle situation once I've figured out which Kohn I'd like to roll with.

"You can call me 'Patrick...'

But if you're uncomfortable with this, 'Dr. Bahls' is fine."

These are the words with which I begin most classes on the first day of any given semester. They're uttered shortly after I rock on in wearing vividly patterned shorts, Chaco sandals, and a plain uncollared T-shirt.

To my more dressed-up and buttoned-down friends in management or accountancy departments, or to my colleagues from cognate math departments in more conservative universities, I look a caricature of liberal academic hippiedom, a stereotype of the left-leaning professor. My dress is casual, my manner with my students more casual still. I'm unconcerned with formal titles, encouraging my students to think of me more as an equal than as an expert. I'm laid-back, easy-going, and down-to-earth.

And I can get away with this because...

...is it because I'm a straight white male?

I talked about this with new friends I met during a visit to Clemson University yesterday (shout-out, Nanette!): I've known for a long time now that a number of my equally-if-not-more-well-qualified female colleagues have a hard time dropping the word "doctor" from the phrase by which they ask their students to address them: the word is an extra layer of armor plate that protects them from charges of academic inadequacy. ("Of course she's qualified to teach this course, she's a doctor!") I've only recently begun to think about the privileges bestowed upon me by other markers of my sociological makeup.

In the "plus" column: male, straight, white, non-disabled, holder of an advanced (terminal) degree.

In the "minus" column: atheist, untenured, left-handed.

If it were a matter of toting up points to determine my net level of privilege and power, I'd probably come out ahead: my pluses are more numerous than my minuses, and generally exert more force. I'll never have to worry about going through my career being defined as a "white male mathematician," while some of my finest colleagues have had to put up with appellations like "one of the best female research mathematicians in the game." (Allan G. Johnson, in Privilege, power, and difference, p. 33: "People are tagged with other labels that point to the lowest-status group they belong to, as in "woman doctor" or "black writer," but never "white lawyer" or "male senator.")

What assumptions, fair or unfair, intended or unintended, privileging or oppressing, am I taking with me into the classroom?

For instance: having a septuagenarian student in one section of my calculus classes reminds me that I can't make generational assumptions, and even the laid-back level of discourse with which I often interact with students of more typical college age in an attempt to convey tricky technical ideas in a down-to-earth fashion might confer unearned advantages to them to the disadvantage of someone less familiar with today's slang and pop cultural references. This same highly fluid "code-switching" that I do in my classroom, which puts many (most?) students at ease, may also put up unintended obstacles for students whose first language is not English.

What about race? At a school like UNC Asheville (which is woefully undiverse from a racial point of view), it's easy to fall into normative assumptions about the ways in which people of various racial, national, and ethnic backgrounds perceive and receive new ideas and new information, even ideas and information as abstract and "objective" as those found in mathematics. Am I unwittingly privileging the vast majority of my calculus students, who are white, to the disadvantage of the meager few (no more than five or six out of sixty-five this semester) who are of color? Obviously I'd like to think that I'm not, but privileging assumptions are often subtle. What would such assumptions look like in the mathematics classroom?

More to come, certainly.

On maturity

I'll likely be putting up a rather lengthy post later today concerning thoughts I had during the first meeting of my latest faculty Learning Circle (on Allan G. Johnson's Privilege, power, and difference, 2nd ed., 2005), which took place yesterday.

For the time being, I just wanted to share a simple observation I made last night as I worked my way through this week's Calc I homework: almost unfailingly, seniors in any major (no matter how closely related to math, no matter how great their raw math talent) will do well (B or better) in a Calc I class. Nowhere is this more true than on the homework, on which seniors typically outscore first-year students by a substantial margin.

Why is this?

I believe it's because to do well in a relatively low-level course like Calc I, one need do little more than pay attention in class, take good notes, and complete homework cleanly and correctly and punctually. These things done, quizzes and exams will generally take care of themselves. Seniors excel at these things because they're mature, they're well organized, they're adept at managing their time. (In fact, I would say that the number one difference between first-year students and seniors is their ability to budget their time effectively.)

I've got two seniors in my Calc I classes right now (one an Economics major, the other a Health and Wellness major), and despite their relative inexperience in calculus, they're both riding roughshod over most of the classes' first-years. Why? Their homework is gorgeous; it's clear that they both take the time to do it right, and I know at least one of them does it in multiple drafts. In doing the homework well, they're solidifying their understanding of every last concept we discuss in class, and thus they're able to do pretty well on the quizzes, too.

They're no exception: in classes past I've noticed that most seniors exhibit similar success.

A tip for the young 'uns: start the homework right away. Seriously. I don't say that over and over and over for the fun of it. Take time to do it neatly, legibly, correctly. The homework really is where you do much of your learning, as you puzzle through problems and sort the ideas out from one another. (This is especially the case in classes less like our Calc I classes, wherein you wouldn't see as much hands-on activity as we do.)

For now, farewell. Next up: "The privilege to call myself Patrick."

Update on my conversation with Sedgwick

After reading my open-letter response to his e-mail regarding Don Tapscott and the ILS Program at UNC Asheville, Sedgwick wrote me again with a link to an on-line version of the Tapscott interview to which he'd referred: http://aolradio.podcast.aol.com/twit/TWiT0197H.mp3.

Enjoy!

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.

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!

Jump

Tomorrow has me jetting off to Austin (from what I'm to understand just about the only livable place in Texas, and just about the only place in that state I've never been) to take part in the Ninth Biennial International Writing Across the Curriculum Conference.

That's a mouthful.

It's also mildly frightening, since it'll be the first time I've ever presented at a decidedly non-math conference before. Frightening but exhilarating at once. The feeling I've got going into this shindig is not unlike the one I had when I made my first research presentation at a non-group theory conference a little more than two years ago now. (As a group theorist by formal training, I felt a bit of a dilettante in dabbling with graph theory, so I felt it a pivotal moment when I started going to/presenting at graph theory conferences.) It's a jump.

It's fitting that I've started reading Mihaly Csikszentmihalyi's Flow: the psychology of optimal experience (Harper: New York, 1991), the centerpiece for the summer Learning Circle in which I'm participating.

A relevant passage, read just last night (p. 61):

What people enjoy is not the sense of being in control, but the sense of exercising control in difficult situations. It is not possible to experience a feeling of control unless one is willing to give up the safety of protective routines. Only when a doubtful outcome is at stake, and one is able to influence that outcome, can a person really know whether she is in control.

This conference should be fun.

Never too many cooks

Howdy, folks!

Yesterday's installment of Super Saturday set a "Math Discoveries" record for most student volunteers, with six stalwart students showing up, breaking the mark of five, met earlier this semester and originally set in Fall 2006 when five students came out to help organize and oversee a series of mathematical games. Many thanks go to nearly-omnipresent Beatrice and Belladonna, the irrepressible Tallulah and her roommate Betty Sue, who isn't even in my class but thought it might be fun to come along, Sieglinde, admirably representing my morning Calc I class, and first-time shower Trixie, whose hand-painted polygons were a hit with everyone (I likened them to stained glass, and one of the little kids thought they bore a favorable resemblance to wood chips). Thanks also go to the eleven Calcsters who, though unable to attend, each contributed a hundred or more poster board polygons to the effort, they were very much appreciated.

As with any successful Super Saturday, I'm not sure who had more fun, the college kids or the young 'uns. Both big and little fingers had a hard time at first in managing the assembly of cubes, dodecahedra, and icosahedra. Once we got the hang of it, though, it was smooth sailing, and it was hard to stop. Several of the little kids left for home with their own polyhedra and handfuls of unwed polygons, some just took a few to use as templates to trace their own. Tallulah and Betty Sue vowed to spend some time this weekend making more polyhedra with which to adorn their room (maybe they'll start a craze?). Several of us old-timers hung out well after class was over, finishing off particularly large polyhedra and chatting about next semester's schedules.

A moment of crowing: I've won two of these six over as Math majors, and I've still got hopes for a third!

Another member of my morning section approached me last weekend regarding a Math major, and I was heartened to hear while speaking with him after class on Friday that he was convinced in part because of this blog. Orville, you have no idea how happy that makes me! Welcome aboard! Any questions you've got about the major, please ask. I like to think that my approachability and others' is part of what makes our department and our major such a popular one, and such a strong choice. I really do believe that ours is not only one of the best programs on the campus, but also the friendliest.

Going backward in time...on Friday afternoon I spent an hour with a couple of my colleagues in talking over Chapters 3 and 4 of Bob Moses's Radical equations. Both remain somewhat cynical, one regarding the entire venture, another regarding the relevance of the Civil Rights Movement in all of this math talk. For instance, one doubted the strength of the "ball bouncing" analogy invoked by Moses: if you want to get their (the kids') attention, says Moses, go to the corner and start bouncing a ball. At first they might not take notice, but gradually they'll come, and they'll ask questions to find out what it's all about, and before long you'll have a game going. "I can't really see myself bouncing the ball," my colleague admitted.

"I don't think Moses's point is that every person reading this book has to be a ball-bouncer," my other colleague pointed out. "Everyone has a part to play in this, and many people will be acting behind the scenes in some organizational capacity, you don't have to stand on the streetcorner with a ball."

"I don't think Moses anticipates that every person reading this book is going to rise up and become a part of the movement," I added. "If for every ten people who read the book only one hops aboard, then that's fine."

Something is better than nothing, someone better than than no one.

I feel that our discussion was a good one, and it's helping me to understand the weaknesses of Moses's approach, as well as its strengths.

This past week I approached the director of the Teaching Fellows program at UNCA, asking her if she thinks she'd be able to interweave a reading of Moses with her program, much as she did with Jonathan Kozol's The shame of the nation last year. I've yet to have a real-time conversation with her on the matter, but from our one e-mail exchange I think she might be up to the collaboration. I'd love to get some of my students on-board with the reading circles. How 'bout it, readers, are you up for it? (Don't try to hide: too many of you have outted yourselves as regular readers for me to think you're not out there!)

I've had some more thoughts about what a more learner-centered Calc I class might look like...having been reminded this past semester just how loathsome "word problems" are to math-minded freshpeople, perhaps it would be best if they spend a semester never seeing a problem that isn't a "word problem." That is, from Day One through Day Sixty (or however many days there are), every example considered would be embedded in the context of some application, no matter how simple or straightforward. No computation would be without at least some interpretation requiring a modicum of extractive analysis. Sure, it would be a bear at first, but the students would grow stronger and stronger as the semester wore on, and by the time they'd get to topics in which "word problems" are traditionally replete (related rates?), it would be nothing new to them, and they'd breeze on through.

Something to think about.

The difficulty, clearly, would come in fitting computations classically done for their own sakes out with realistic applications. I ain't despairing. It can be done, it'll just be difficult.

By the way, for those keeping score at home: I don't think I'm arguing for a return to a "reform" curriculum for calculus. I'm not, for instance, suggesting that the formal definition of a derivative be put off for weeks beyond its natural point of introduction. I'm suggesting that a more "traditional" curriculum be retooled to accommodate meaningful and motivating examples.

Before I close this post, I should mention that tomorrow we try Newton v. Leibniz. I'm happy in that I think both of the primary parties in the trial, in both classes, have done a good job in preparing. The worst-case scenario would have involved Newton and his team damning the torpedoes and cruising ahead with full sail while Leibniz et al. drifted around in front of them on a chunk of creosote-soaked flotsam.

We'll see how it turns out. For the time being, I've got some Mathematica code to write to help me out with my research, and I promised a few Calc I kiddies that I'd try to slap together a practice version of the third mid-semester exam, to be handed out this coming Thursday.

Super Saturday IV

I'm not sure why I got up at 6:30 this morning...at least I got a head start on the grading. I'm halfway through the Calc HW at this point, but 280'll call this afternoon.

At this moment I'm sitting in Zageir 227, 45 minutes remaining before our fourth Super Saturday session begins. I've got nothing to do for a bit, so I thought I'd take a moment to check in here.

How'd the week wind up?

In 280 we talked about injectivity and surjectivity, concepts that are substantially more puzzling in the abstract than they are in the familiar context of real-valued functions. We had a bear of a time working through a proof of the equivalence of the two most frequent characterizations of injectivity. There was something about the subtlety of the contradiction that was tripping up even the best and the brightest in the class. The confusion likely had to do with the convoluted structure of the statement we were trying to prove. Roughly speaking, the statement looks like (A ⇒ B) ⇔ (C ⇒ D), for various statements A, B, C, and D, so at one point we're dealing with a hypothesis that looks like (A ⇒ B) ∧ C. Say huh?

In Calc I we rapped about the way a function's graph is determined by its derivatives. On Monday I'm going to have them do a few exercises as a class to drive home the meaning of properties like concavity and inflection, and how they relate to increasing and decreasing.

Oh, by the way: they made substantial recovery on their exam revisions, raising the class average to roughly 77%, up from the woeful 68% last week. Several students offered touching apologies or narratives on their submitted revisions. Magdalena, for instance, mentioned how she saw herself in the inner monologue I included in the post following last week's grading frenzy. "Oh my god, he's inside my head!" she said. She vows never to become so reliant on the solutions manual again.

They're good kids.

Newton v. Leibniz takes place a week from Monday; this coming Monday I'll get the rough drafts of their "arguments," many of which I will share with opposing teams so that each legal team may adequately prepare a defense having had access to the other's evidence and data. So far most of the teams seem to be functioning pretty well together, with a couple of exceptions. There's been a little tension stemming merely from personality differences present in one or two of the groups. In a perfect world, perhaps...

...you know, when I was a student, I didn't so much like working in groups. I didn't have to do it all that often, but when I did, I got it done with as soon as possible and moved on. I can't say now how I must have come across to my teammates. Bossy, maybe? Headstrong? I hope not. I know that I'm the sort of person who wants to get it done "right," even if that means that I have to take the entire project onto my own shoulders and do every last bit of the work myself. Better that than that the project turn out shoddily because one of my teammates had the audacity to not be as smart as I was.

I suppose I might have been bossy, without meaning to be.

And I wasn't always my sweet current self, either. Though sarcasm is my native language, over the years I've learned to tone it down with others whom I don't know so well. Sarcasm, spite, vitriol. My mother loves telling the story of how I frustratedly called one of my kindergarten colleagues "stupid" because she couldn't grasp the nuances of the game I was trying to play with her. I was insufferable as a young 'un, but I've gotten more mellow with age.

Why can't we all just get along?

More later...I'll let you know how my first session of the departmental Moses Learning Circle went yesterday afternoon (short version: I'm not so sure my colleagues are sold on the claim that mathematical literacy is a civil rights issue, or the claim that improved numeracy is a societal panacea), but for now I've got to get ready for the SS kiddies who'll be here in a few minutes (Tallulah and Deidre are already here, ready to lend a hand).

Avanti!

Bile and bluster

I've now had nearly 72 hours to come down from my last post.

In retrospect, perhaps I was a bit harsh.

I mean, I'm pretty sure that my students aren't out to depress me with their eight-o'clock apathy and their poor performance on the exam. ("Did we really make you cry?" asked one of the students in the second section. I think the answer I gave was sufficiently vague so as to leave the matter unresolved.)

At the end of the day, they're good people, they had a rough week last week (didn't we all?), and enough of them have expressed enough remorse over their collective blanking, biffing, and blooping on the exam to make me regret my vituperative post in the wee hours of Saturday night.

Meanwhile, my 280 class has been doing wonderful things I fear I've not yet lauded enough. I'm very happy, for instance, with their Professional Proof Analysis papers, particularly insightful comments from which I've selected and compiled to share with the class as a whole. A sample:

"I think [the structure of the Hass and Stewart proofs] reflects a contemporary view that introductory Calculus is more about using Calculus than deeply building an understanding of why it works....[T]o the exploring student, it suggests the text is offering statements like this is true and here is why, instead of by using these ideas we arrive at this helpful result to introduce the material."

And:

"Along with the actual analysis of the proofs, I believe it is worthy to acknowledge that the Weir and Thomas proof was composed by three authors, as opposed to the one author each of the other had. This is important because it allows for immediate revision of each author’s ideas to produce a tighter, more inclusive proof (I now see why work in groups during class)."

We've been working on order relations for the past couple of classes, and a couple of the students seem to have a particular knack for this stuff. Timofei has even expressed interest in signing up for a reading course on lattice theory with me next semester. I think that would be wicked awesome. I pulled Davey and Priestley (Introduction to lattices and order) off the shelf to browse through it with him after class yesterday, it would make an excellent text for a reading course, and he could get it cheap cheap cheap. I don't think it'll take much convincing to make him take the plunge. I've got a soft spot in my heart for order theory, I don't think I'll ever leave it alone for long.

So what's going on in the pedagogical scene, besides the day-in-day-out of my classes?

Yesterday we had our first post-season meeting of the self-authorship Learning Circle. I and three others assembled to share our "homework": each of us was to produce a short narrative describing what self-authorship meant to us (perhaps from the point of view of a practitioner within our respective discipline), illuminating it for the benefit of one of our peers who may never have heard of it.

My narrative on self-authorship in mathematics was informed heavily by my ongoing experience in this semester's 280 course. My description came off sounding a bit clinical in comparison with Thibault's downright hortatory "manifesto" (his word, not mine) on self-authorship in theater. Echoing Goethe, he indicated the fundamental need for a student of theater to be true to one's self. Meanwhile Nola's narrative struck me as a bit more catholic and all-encompassing, with an emphasis on the mutuality of the relationships arising in Baxter Magolda's Learning Partnerships Model. I liked aspects of all three narratives. In discussing them, I found particularly insightful Nola's observation (as I described to her my experience of watching my older students interact with my younger ones) that Vygotsky's "zone of proximal development" is arrived at in the Math Lab.

I see the same dynamic in the conversations between the students taking part in our Math Problems Group (which convened a couple of hours ago this evening): the difference in ability between the weakest regular attendees and the strongest is noticeable but not overwhelming, and all of them possess the background needed to interact proficiently (if not fluently) with one another, yet a few have perceptibly stronger skills than their peers and often serve as coaches for the others.

Tonight I served up a particularly nasty Putnam problem from last year's exam (one that only a handful of the top solvers in the country scored well on), and I let 'em at it. It took 75 minutes to arrive at the rudiments of an argument, though by well before then they'd convinced themselves that they had the right answer. It was fascinating watching them at work, watching them scratch away on their paper, listening to them communicate with one another and share their ideas. Three of the attendees independently arrived at the same conjectural solution, allowing for minor variations on a theme. While Nadia continued to work on her own, Nikolas met up with Beulah and Simon and formed a consensus on a particular formula for the number we sought. Soon Nadia gave up on her computations and joined the others, and they spent another fifteen minutes clarifying their thoughts, and then another fifteen minutes spinning their wheels before we decided to table that problem and start another easier one, a simple problem on graceful graph labellings. As I suspected, this last problem only took Nikolas about five minutes to solve, and we ended the night on a high note.

Now?

I.

Am.

Tired.

Well...

...we'll see how the calc kiddoes are doing tomorrow. I've talked with a few of them about test corrections, including a couple more who agreed that it wasn't that hard an exam, they just had the mother of all brain farts last Thursday.

I'm not sure what it was about last week (something in the air?), but I don't think there was a single person worldwide who was in top form.

Except maybe Matt Ryan.

But that's a different story.

Anyway, before this becomes a football blog, I'd better call it a night.

I'll try to get another post in tomorrow to talk about seven gajillion and one ideas I got from Quimby in Fayetteville. I'd also like to talk about my interview tomorrow with one of the new Writing Center student consultants, the status of the Robert Moses Learning Circle I've put together for our department, the NSF grant I'm nearly done writing, and my final decision to run my upcoming graph theory course using the Moore method.

Stay tuned!

Random thoughts

No time to get much coherent down, but before it slips my mind...

1. Faculty Learning Circles are where ideas come to life. Before today's meeting I spent a good deal of time thinking about what self-authorship would look like in mathematics students: how could it be assessed? In what manner would a self-authored math student behave? Are there warning signs? Once one knows what to look for, how can one go about designing the appropriate activities to facilitate and promote self-authorship? I raised these questions with the small group that convened this afternoon, and it was decided that it might not be a bad idea to start thinking about a conference on Self Authorship Within and Across Disciplines. (No good job goes unpunished!)

2. A few minor homework committee woes creep in: after it came to my attention that a few people had felt steamrolled by forceful personalities, I felt it necessary to send an e-mail to the 280 folks reminding them that there is almost always more than one correct proof to any given proposition. When serving on a committee, this must be kept in mind so that one doesn't turn a blind eye to alternative correct proofs one isn't expecting; when receiving feedback from a committee, this must be kept in mind so that one doesn't feel obligated to thoughtlessly undertake a committee's suggestions: if you're pretty sure your proof is right, perhaps the committee misread your argument, or misunderstood your intentions. Stand by your proof, and take it up with one of the folks on the committee. They're human, too, and every one of us is capable of error. (God knows I've demonstrated that over and over and over and over and over and over and over and over and over...)

3. ...I could have sworn I had a 3. Never mind.

Everything else is groovy. The Calc I folks are off and running with their team projects on specific heat, those are taking shape before my eyes (love those Mathematica graphs, huh?). In Foundations it's sets, sets, sets, and we're getting ready for Round Three of the newly-rechristened WNC (to includ Western Carolina University and Warren Wilson College as well) Mathematics Problems Group, tomorrow evening at 5:30. Pizza 'n' Putnam, what better combination?