A tale of two sections

Today it was evident that over the weekend, most folks in one of my sections of Calc III took the time to work through the problem set they'll be presenting on Wednesday. It was equally evident that most folks in the other section didn't.

We'll see how things go on Wednesday. Most of the problems are pretty straightforward, but there are a few that might give pause. If Wednesday goes as I suspect it might, a few folks might learn the hard way that though it always pays (no matter the class) to keep on top of the work, but in a course structured as ours is, it pays double.

Moore is more

Three weeks into the semester, my Moore-method Calc III class has made it through three problem sets (50 problems), treating a substantive review of topics from Calc I and II and five or six sections of the textbook. It's been a few years since I've taught a course in this fashion, so there's been a bit of adjustment as I've gotten back into it.

So far, so good. The students are getting much better at explaining their solutions in front of a large audience (one section has 27 students, and the other 35), and they're becoming more relaxed, visibly. Yesterday's second section was particularly laid back, assiduously focused on finishing their tasks but willing to joke around and have fun in order to set the solvers at ease.

I've been very impressed with students' ability to be wrong in front of each other, and similarly impressed with the audience's willingness to ask questions. They're getting better at asking each other for clarification or elaboration, and not turning to me to ask. I'm letting minor errors slide, perhaps adding a little "does everyone agree?" if the solver's slipped up somewhere. Generally this has been enough to prompt one or two to express disagreement.

How's it helping the students? Hard to say. Several have said they get a lot of the course's design, though one or two have admitted "it's not what I'm used to, and I'm having a hard time adjusting." I've reminded them a couple of times now that in this sort of course they're expected to take on a bit more responsibility than they might in a more traditional course, preparing well and keeping up without my continual exhortation for them to do so.

I'm going to poll them more formally on the course structure at the end of the coming week, after we finish off the fourth set of problems. We'll see where we are.

Meanwhile, if anyone in the class is reading this and would like to comment, please feel free to do so, anonymously if you'd like.

Data mining

I spent a few hours this morning running the numbers on the students currently enrolled in our Honors Program, hoping to get some objective data on students' participation in the program as I move toward my new position in the fall. I made some heartening findings.

Namely, each of the school's three major disciplinary divisions (humanities, natural sciences, and social sciences) is pretty equally well-represented in the Honors Program. School-wide, 9.25% of all declared majors take part in the Honors Program, and the participation rates of the individual divisions range from 7.68% to 9.91%, quite tightly centered on the overall mean. Counting the courses these students take in Honors gives further evidence to this balanced participation: overall, a student in Honors who has declared a major has completed 3.65 Honors courses on average, and the means for the various divisions range from 3.58 courses per student to 3.81 courses per student.

There are certain departments that are particularly well-represented in Honors, including a few that are quite large (and that are therefore somewhat immune to sample-size bias). For instance, five of our seven departments with at least 100 majors can boast that more than 10% of their students take part in Honors, including one department with 122 majors, of whom 17 (13.93%) are enrolled in the Honors Program. At the other extreme, there are four departments, each home to anywhere from 26 to 31 majors, with no students in Honors. (One of these is a relatively new department, one which graduated its first majors just a couple of years ago.)

Interesting.

I don't believe these data to be "actionable" in any way...and besides, the results don't indicate dramatic action. I'll stay the course for now...though it might not be a bad idea to talk with the folks in those four departments to make sure their students are aware of the opportunity...?

We'll see.

Defragging

For the past few weeks I've been hammering away at several different poems, but I've got little more than a couple of callousy handfuls of bent-nail fragments to show for it. Odd images (starry-breasted crows, fog-covered tenement-like brick blocks, concrete tunnels like birth canals), and random thoughts.

Keeping with this fashion, here are a few random thoughts on academics and academia:

1. I wonder at the extent to which we are all isolated in our disciplines, and to which we do most of our work in rooms with four walls and very often no windows.

2. I wonder at the sterility of the Platonist, universalist, formalist conception of mathematics, itself an isolating philosophy, allowing as it does a detachment from the world and from others as we engage in our mathematical work.

3. I wonder at the mechanisms we feel we must make and maintain in order to "deliver" our curricula. The more I learn about the inner workings of the Honors Program, the more I wonder if there are simpler ways to put it all together.

4. I wonder at our assessment practices, at every level, from the individual student to the institution as a whole. To what extent are they arbitrary, effective, replicable? To what extent are they doing what we need them to be doing?

5. I wonder at the effects of our educational system, both intended and unintended. How often does a student's passion for perfect grades overpower her passion for learning?

6. I wonder at things as they stand for things, and am reminded of William Carlos William's red wheelbarrow:

so much depends
upon

a red wheel
barrow

glazed with rain
water

beside the white
chickens.

Guinea pigs

I have to admit some trepidation on my part going into the second meeting of my MLA course this past Thursday. Despite my strong record of teaching in mathematics, I have relatively little experience in leading wholly discussion-based courses, and this inexperience coupled with my sense that a few folks in the class are leery of mathematics in the first place made me worry that the conversation we'd have together would fall flat. Though it took a little while for the conversation to get going, my fears soon proved unfounded.

We began in small groups, where I dealt myself into a conversation with the two older gentlemen in the class, both of whom expressed some measure of skepticism about the reading. Uriah seemed appalled by Dehaene's seeming to alternate between making claims of revolutionary understanding of the brain's functioning and retreating to more palatable and defensible observations. Quinn seemed to accept some conclusions but was put off by the relative (to mathematics) lack of "rigor" and less rigid notion of "proof" in psychology literature. I found these reactions heartening, as reasoned skepticism is generally salutary, something to be expected: these two men are among the more mathematically experienced in the class (surpassed only by Bonnie, a former UNCA math major whom I taught in Abstract Algebra back in 2008!), and the insistence on rigor is a more traditionally masculine trait.

When we returned to a full-class conversation, many more ideas came out, primed by my request for each student to identify those aspects of the reading they found most intriguing, most confusing, and most well-received. It would be difficult for me to summarize all of what was said, so I'll focus on a topic we spent much of our time, dealing with the following image:



I crafted this image last spring for my Ethnomathematics course, in order to serve as a Rorschach test of sorts, testing respondents' notion of numerosity. For the longest time I've found it fascinating that as a species we tend to distinguish objects based upon contiguity and connectedness, "topological" aspects, not on color, shape, or other more "geometric" aspects. That is, most people will respond, if asked "How many objects do you see here?" that there are four, for there are four noncontiguous bodies present.

This, however, is the unskeptical answer, unaffected by the sort of "questioning bias" that doomed the Piagetian experiments Dehaene outlines in Chapter 2 of his book. Specifically, when asked to decide which of two rows of small objects is greater in number (the lesser quantity being arranged in a longer row so as to mislead), young children will often respond incorrectly simply because they suspect trickery on the part of the questioner. In our situation, the skeptical respondent, suspecting trickery, might respond (as did Quinn in my MLA class) "one," seeing a single paw, or even "two," differentiating the two objects on the basis of color and not contiguity.

For quite a long time we discussed the evolutionary advantage of enumeration based on contiguity, and various related questions came up: what would have to be true of a species whose members enumerate objects on the basis of other aspects? What could be said of their mathematics? Can a mathematics of "continuous quantity" model our "discrete" mathematics fully and effectively?

If nothing else, everyone in the class wanted to learn the likeliest response to the "how many" question above...if the question could be posed in a less leading fashion. "I've got a sample of over 60 students I can test tomorrow," I said, referring to my Calc III classes. "Why don't I get some more data?" The students loved this idea, and we debated how the Calc III students should be prompted for the purposes of this informal survey. It came down to between "What do you see?" and "Describe what you see." People seemed more satisfied with the second, as it seems to beg for a more elaborate response, increasing the likelihood of a description featuring some kind of numeric content.

Yesterday I began both sections of Calc III with the experiment, providing no context beforehand, so as to minimize bias in the students' responses. (I did inform them of my intent afterward, and gave them all the option of retrieving their responses if they'd prefer that they not be read by others. I hope that no one on our IRB is reading this...) I've not yet looked over the responses, but I'm already looking forward to the analysis we'll do this coming Thursday night. I'll be sure to post some sort of summary results here once we've had a chance to sort it out.

Anyway...I'm counting last Thursday as a success. I think this course is going to run itself. I'm not so worried anymore.

Politics

Who knew there was so much of it in academic administration?

Seriously, though, I have a hunch I'm going to have to practice a good deal of diplomacy in the coming years as I inch closer to administration.

Week Two's been a blast, with Moore-method Calc III moving right along (no major hitches so far!), the second meeting of my MLA course in about two hours (one student withdrew after Week One, and I'm curious to see what the others think of Dehaene), and my first crack at putting together the Honors Program's course schedule underway. That last one's a balancing act, but I can't imagine it's nearly as hard as programming a large department's schedule. Let's just say I appreciate the work my chair's done on that task for the past several years.

One of my lovely Charleston colleagues just gave me a tip on the following book on delivery, apropos of our conversation about the role LaTeX plays in mathematical writing: Rhetorical delivery as technological discourse: A cross-historical study, by Ben McCorkle (Southern Illinois University Press, 2012). I ordered a copy. Just call me Mr. Spontaneity.

Okay, off to prepare for a meeting to discuss funding for this year's Conference on Constrained Poetry!

JaMMin' in 2012

Aside from a brief post on my presentation on writing research, I've not yet had a chance to say anything about this year's Joint Mathematics Meetings (JMM), from which I returned a little over a week ago. It was a fruitful affair, marking my first ever JMM where I spent more time in meetings than I did at talks. (Avoiding administration: ur doin it wrong.)

My first full day at JMM began with a two-hour meeting of the MAA's Committee on Undergraduate Programs in Mathematics (CUPM). I was recently appointed to this body, a group charged (as you might expect from its name) with making recommendations regarding the form and content of undergraduate mathematics programs across the country. Of course, this is a very loosely-defined directive, and mission-creep inevitably sneaks in. We spent a good deal of time talking not only about the undergraduate programs themselves but also their interface with K-12 education.

What struck me most about this meeting was my sense by its end that even the most well-informed of college mathematics educators are at a loss when it comes to solving some of the biggest problems facing math education today. Why is this? It's not like the problems are new ones: for decades we've dealt with student recruitment and retention, students' transition to higher mathematics, and imperfect transfer of skills from AP coursework to college coursework. It's not that we don't have proven pedagogies and time-tested methods of math education at our disposal...and it's not that we have a shortage of talented teachers to put those pedagogies and methods into practice. Maybe it's simply that the student body we're dealing with is diverse enough to foil any attempt at applying one-size-fits-all panaceas: more than ever before we serve a population whose members differ from one another ethnically, economically, socially, spiritually, intellectually, and in every other way we can think of...to extremes heretofore unimaginable.

The next meeting I had to make was a one-on-one with one of my colleagues in the AMS. After receiving a note I'd sent a few months ago to the Project NExT list regarding my forthcoming book, Flora had expressed interest in meeting with me at the Joint Meetings to talk about it in a bit more detail. It seems that earlier in her career she'd gone down a path much like the one I've followed recently, leading WAC and WID efforts at her home campus (DePauw University) before heading over to the AMS full-time. Flora and I shared an hour or so together talking about the importance of writing (and other modes of communication) in the teaching and learning of mathematics, and before long she invited me to take part in a morning meeting of the AMS's counterpart to the CUPM a couple of days later. Though she couldn't guarantee me the floor, she mentioned that one of the members of the committee had brought up writing as a potential topic for further elaboration by the AMS's Committee on Education (CoE). Topics selected for such elaboration become the focus of discussions and workshops at the CoE's fall meetings.

So I made it to that Friday morning meeting (still bleary-eyed after a night of revelry with several of my Vanderbilt friends). Less focused than the CUPM meeting had been, this one consisted of a loosely-knit (and often heated) conversation on several topics related to undergraduate math education. We spent about twenty minutes each topic: potential certification (by the MAA, AMS, or both jointly) of undergraduate mathematics programs, facilitating students' success in calculus courses, and the necessity for upper-level "elective" coursework like point-set topology.

The first of these was the most controversial issue, on which there was much disagreement. For my part, I brought up a concern that's faced the members of the Curriculum Review Task Force this past year: those major programs which face accreditation by a professional body are among the most rigid and time-consuming, placing heavy demands on both students and faculty. In this way they are unsustainable and resource-intensive. One of the other folks present at the AMS meeting countered that the "accredited" majors at her school are the ones that receive the most attention and resources from administration, and that this alone is reason enough to pursue the adoption some sort of certification procedure.

As you might suspect, I disagree. From the point of view of a math department member, this move might make sense: why not try to carve out a bigger chunk of the pie by forcing your school to support your attainment of accreditation benchmarks? But from the point of an administrator, the move appears more questionable: the pie's only so big, and with the economy the way it is, it's likely to get any bigger. If every department's trying to carve out bigger and bigger pieces for themselves, there's not going to be much to go around. We'll starve each other out if we don't cooperate more meaningfully at higher levels than the department.

To be continued, I'm sure.

The conference wasn't just one meeting after another. I had a lovely time reconnecting with several past REU students (including Wilhelmina, from way back in 2007!), grad school friends, Project NExT buddies, and a bajillion other people I'd not seen in a long time. I spread the word everywhere I could about my book, shamelessly leaving flyers on tables all over the convention center. I made it to a dozen or so talks on graph theory and group theory...and to several posters and presentations by my students, past and present. Most outstanding was Ino's and Ned's talk on their ongoing research into nutrition, given in the MAA's session on the mathematics of sustainability. They nailed it. Several folks had great questions afterward, and they received at least three invitations for collaboration and further presentation. We'll be following up on those shortly. Well done!

Much more work to do! But it's great fun. I'm looking forward to seeing what the coming weeks and months bring. 2012's gonna be a good one.

Lesson #1

One week down, a whole bunch to go. My MLA course has met just once, as have both sections of MATH 480 which I'm team-teaching with my colleague Timon this term. Calc III's had three chances to get together, and I'm very happy with how those meetings have gone.

The students have shown no hesitation whatsoever in getting together in groups and hammering out solutions to the problems posed to them, and they've shown similar eagerness in getting up to the board to strut their stuff.

What's impressed me most is the quickness with which they seem to have learned the most important lesson one learns in a Moore-method course: it's perfectly okay to be wrong.

"You know what happens when you make a mistake at the board?" I ask. "Does the sky open up, bolts of lightning raining down from above, smiting you where you stand?" Despite the inevitable one or two students who deadpan sardonic yeses, they get the point. Not only is it okay to be wrong, it's necessary, even salutary: often only in being wrong can you eventually be right, as trial and error often lead to full understanding. The process by means of which we proceed from error and ignorance to understanding is called learning.

I've got profound admiration for the several students in both Calc III sections who made mistakes at the board the past few days, every one of whom recovered almost instantly, retaining dignity and respect. My thanks go to all of my students for a wonderful first week, and especially to those who showed the others that being in error is just not that big of a deal.

Back to school

UNCA serves a large number of nontraditional-age students who are returning to school after taking time off for other things, and many of these folks are older than I am. Therefore I'm used to not being the oldest one in the room when I'm teaching a class; I've only been the eldest in maybe five or six of the 50-60 class sections I've led at this school.

But I'm not used to being one of the youngest in the room.

My MLA (Masters of Liberal Arts) course, Number sense: The philosophy and psychology of mathematics, met last night for the first time. I've got eight students in the class, four or five of whom are my senior in age and in life experience. It's a great bunch, and I can tell I'm going to learn more from them than they're going to learn from me.

We started off with some freewriting, through which I asked the students to probe into their own mathematical pasts. I hoped to find out what it is that makes these folks tick mathematically and to determine what they perceive to be the most basic and fundamental of mathematical operations. If we can get at the these operations, we'll be in a position to start our study of mathematical cognition where our brains begin, with approximations of enumeration.

I'm delighted to report that there's considerable diversity in the class when it comes to mathematical background. I found it interesting that the two gentlemen in the class reported more facility and familiarity with mathematics than their feminine counterparts (with one exception). Both of them described delight at working with statistics and geometry, and obsession with game-lake mathematical puzzles. The one woman with more mathematical experience is a former UNCA math major with whom I had the pleasure to work when I last taught Abstract Algebra I (in Fall 2008). This is her first semester of study in the MLA program.

The other five folks have considerably less mathematical background, but will provide perspectives from other points of view, reporting interest and expertise in psychology, history, and philosophy. I'm excited to learn from Samantha, who is taking time off from her work as a teacher for special-needs children. She mentioned how frustrated she is with mathematics education, and hopes that our class will give her the skills to help improve the way students are taught math at a young age. More power to her! Her high expectations for the course will definitely keep me on my toes.

After we probed our mathematical pasts for a bit, I presented a few exercises and experiments I hoped would whet their appetites for the material we're about to study (from Dehaene's Number sense). With no promise that this link will be evergreen, you can find Mathematica files for these exercises on the course website under the entry for January 12th.

In the first exercise I challenged students to hold in their memory progressively longer randomly generated strings of digits, demonstrating the means by which we tend to use our linguistic faculties to store such strings in short-term memory. (This use underlies linguistic differences in the ability to memorize and compute with numbers: the brevity of number words in many Asian languages allows native speakers of those tongues to outperform speakers of other languages in basic memorization and numerical manipulation.) The second exercise demonstrates how the arrangement of objects affects our sense of their number: more densely packed objects tend to appear more numerate than those that are sparsely spaced, and more orderly-arranged objects appear more numerate than those that are randomly placed. For the third activity, I asked the students to report any sort of synesthesia they've ever experienced: do they sense that numbers have specific appearance, texture, color, smell, relative spatial position, or gender? I think the students found it odd when I reported I've always had a very well-defined sense of the gender of every digit.

We wrapped up with a short discussion of formal expectations for the course, a topic I'm trying to de-emphasize as much as possible. I'm looking forward to next week's discussion. I'm curious to see if the students will find Dahaene's book as intriguing as I have.

Swimming!

Two class-days into the semester, and things are going swimmingly. I'm putting the course together with a modified Moore method, cycling (roughly) through the following steps:

1. handing out problem sets,
2. giving the students time in and outside of class to work through solutions in groups,
3. asking the students to present their solutions in class,
4. asking the students to write up solutions to selected problems as homework, and
5. quizzes the students on completed problem sets.

As of now I've met with the first section of the course twice and the second section once, but every meeting has been lively and energetic, and I've needed absolutely no coaxing to get students to work in groups and to come to the board. The atmosphere in the course has been supportive and friendly. I'm delighted that this morning's class gave us an opportunity to note that, as I'd promised, no lightning bolts came down from the sky when a student made a mistake at the board.

It's been a long time since I taught course using anything close to the Moore method (my special topics course in graph theory, run in Spring 2008), and I notice that I've grown considerably as a teacher since then. I'm more confident, and that confidence has enabled me to feel less awkward taking a peripheral role. In particular, I find that I'm much more able to sit in silence than I was in the past. Silence in a crowded classroom is disconcerting, and it's all one can do to keep from saying something after ten or twelve seconds of quiet have elapsed. I've grown accustomed to such silences, though, just as I've grown accustomed to (or, more accurately, enamored of) the thrumming of three dozen voices trading tricks as the students work in groups in class.

I'm confident. It's going to be a good semester.

Day One...and WHAT a Day One!

It's been a while since I last wrote a very substantial post. I had to look back at the blog just now to see what was going on, and what state my mind was in, when I've checked recently.

Most of my last few posts have been abstruse, political, and maybe even metaphysical, dealing with everything from the affective learning goals met through inquiry-based learning to the horrors of administration-sanctioned crackdown on students' rights to peaceably assemble on their own college campus. I fear this post might continue this trend toward the abstract.

Roughly three months ago I posted about an opportunity I'd been asked to explore which, quite honestly, scares me more than a little. I feared this opportunity because I knew if I offered to take it on and if my offer were successful it would present me with an entirely new set of challenges and that I'd slip and stumble and make a mess of it every now and then as I got into the flow. But hey, no one of us is born knowing how to swim. We've all got to pick it up somehow.

Well, it's time to swim.

It's official: in Fall 2012 I will become the University of North Carolina, Asheville's new Honors Program Director.

During the next several months I'll be learning all that I can about this new responsibility. I'll be tailing the current director like a second shadow. I'll be surveying the program's current students and surveying the faculty. I'll be making new plans and making new friends. I'll be reflecting on my own ideas and I'll be reaching out for others' ideas I'd never dream up myself.

And I'll be looking for support, from the hundreds of wonderful colleagues, students, family, and friends who have helped me to get to this point in my career...and who will continue to help me as my career takes this new turn. I thank every last one of you for all that you've done for me until now. I thank my colleagues and students, for your suggestions and support and for helping me to craft the communities that have helped us all learn together. I thank my friends and family, for the time you've given me, and for the shoulders you've let me cry on and lean on. I wouldn't be where I am today if it weren't for you, all of you.

With peace and love, I thank you.

Now, let's get to work. It's dawn again, with a brand new sun shining. There's much to be done!...including getting to my first class, which begins in 12 minutes...

Blowing up the interwebs

I'm not sure this won't cause a cataclysmic loop of self-reference, but here's a link to a very nice little Chronicle write-up on this past Wednesday's JMM talk on our rhetorical analysis of the REU students' writing.

Nice.

I hope the folks that make it here from that site will take the time to poke around a little bit and read a bit more about my work with math and writing. Please think about picking up a copy of my forthcoming book, Student writing in the quantitative disciplines: A guide for college faculty, available in March from Jossey-Bass.

I hope once I'm settled back in at home tonight or tomorrow to write a run-down on my whole JMM experience (minus much of the after-hours goings-on), but for now I've got a few more talks to make.

On deck

It's a new year, and we're only a few days away from a new semester (beginning Monday, January 9th). There are big, big things in the works: my book comes out in a couple of months, the work of the Curriculum Review Task Force should be coming to a head this term (with concrete recommendations to the Faculty Senate due by April), and...well...other news items about which I'll be able to say more in a few weeks' time.

What's in store, teaching-wise? I've got three preps this term, one for two (large) sections of a course I've not taught in almost six years (Calculus III), one for a single (small) section of a course I've never taught (a Masters of Liberal Arts [MLA] course on the cognitive psychology behind mathematics), and one for my section (of two) of our senior seminar.

In the seven years I've been at UNCA we've not taught two concurrent sections of that last class, so I'm not sure exactly how we're going to manage it. I've yet to talk to my colleague Timon, who'll be teaching the other section. I imagine we might hold many activities together, splitting when it comes time for the students to present. There's simply no way we'll get through 26 student presentations in the five or six weeks we can offer them to speak. We'd have to have an unprecedented four talks per period to make it work, and that's simply unworkable. Thus splitting into separate sections for that part of the course, though not ideal, is about the best we'll be able to do.

Meanwhile, I've got plans for the other courses. Calc III, which I've not taught since Summer 2006 (the last summer I wasn't running the REU), I'll be running with a modified Moore method: one day each week will be devoted to discussion of new definitions and discoveries, a second day to small group work on the current problem set, and the third to problem presentations. Both sections of this class are big enough (roughly 30 students apiece) that I'll probably ask students to "present" simultaneously (two to four at a time) whenever feasible. This'll ensure that we make it through problem sets in a somewhat timely fashion, and that each individual student gets more opportunities to present. I've already worked with about half of the students in both sections, which familiarity will help me ease into the new term.

I can't say the same for my MLA course. It's been nearly a decade since I taught a graduate-level course (a special topics course on Coxeter groups and related groups which I led at UIUC back in Spring 2004), and this course differs dramatically from that one. We'll be exploring the learning and cognition of mathematics, and I plan to inject a good deal of philosophy and sociology into the mix as well, drawing on a number of sources to paint a picture of mathematics most people never see. We'll begin with Stanislas Dehaene's marvelous book Number sense: How the mind creates mathematics, about which I've blogged a bit before (see the "Dehaene" tag at the right), surveying the psychology of mathematical discovery, before moving onto Imre Lakatos's Proof and refutation, a philosophical treatise designed to lay bare the workings of what might be called the "mathematical method."

I'm not sure what to expect from this course. I suspect there'll be a week or two of me feeling out the students (currently there are seven students enrolled) to see where their interests and aptitudes lie. Likely none of them are straight-up mathematicians; I'll be curious to learn what they're hoping to get from the class, and I'm certain they'll help me give it more direction.

Ah, well...one week to go. Before then I'm off to Boston for this year's JMM, at which several UNCA students (and a few past REU students) are presenting. I'm particularly excited to see how far Ned's and Ino's work on nutritional data has come since their presentation at Kennesaw State in November. (They're presenting in a special session on mathematics and sustainability.)

Further bulletins as events warrant, likely soon.