Breadcrumbs

I feel like Hansel and Gretel (being both at once would be an apt ontology for this post), following a trail of breadcrumbs as I wind my way through a forest.

In each of the three meetings we've had so far, the students in my MLA course have raised some interesting (and as yet unanswerable) questions. Many of these concern the classic "nature vs. nurture" matter that infects every conversation involving human abilities and achievements. For instance, is it nature or nurture that leads to mathematical (and otherwise) savantism? That is, do "human calculators" owe their skills to some advantageous neural network structure in their brains...or do they develop those skills through hard work and constant application of ordinary neurological machinery?

Opinions were definitely divided on this matter during our last meeting: some accepted Dehaene's explanation that practice makes perfect, and that those who have plenty of time to practice are liable to more closely approach perfection; others weren't convinced. "Maybe none of us are born geniuses," one student said, "but some of us are born with better propensity to achieve genius than others are. Just like most of us will never be professional basketball players, as we lack the physique it would require."

This analogy might remind us that after all the brain is as much a piece of our anatomy as are our arms and legs, and our interpersonal differences in overall anatomy carry over to interpersonal differences in our brains as well. No doubt some of those differences predispose us well to certain kinds of genius inaccessible to others? It's up to each individual to nurture latent talent in the most efficacious way from that point on.

The risk we run in arguing like this is making dangerous generalizations along the following lines: "brain difference X translates into ability difference Y" and so forth. As I quipped mysteriously on Facebook the other day, brain does not equal mind anymore than map equals territory, and if we pretend that we can extrapolate everything we need to know about a person from the structure of their braincase, we anachronize and become phrenologists, feeling for bumps, risking claims about racial or sexual superiority. Remember that it was held for a long, long time that women were less bright than men because their brains were less massive...and that when this point of view was assailed from all sides, its adherents did all they could to rescue it, falling back on more and more convoluted sophistry to save their theory: "well, it's not brain volume per se, but volume of gray matter...or at least the ratio of gray to white matter...well, maybe the degree to which it's all convoluted..." (See Stephen Jay Gould's marvelous The mismeasure of man for a blow-by-blow debunking of such arguments.)

About sexual superiority (and getting back to our trail of breadcrumbs): one of my students turned me onto Cordelia Fine's Delusions of gender: How our minds, society, and neurosexism create difference (New York: W.W. Norton & Company, Inc, 2010), a thorough discussion of modern attempts to pin gender differences in intellect on neurochemistry. Phrenology's not dead, it just looks a lot different than it did 150 years ago: now we don't look for protrusions in the skull, just higher-than-normal levels of fetal testosterone, and we don't come right out and suggest that women aren't as smart as men, we just say they have a greater propensity to empathize than they do to systematize. Fine spends much of her time pointing out flaws in modern phrenologists' methodologies, though not as many she might; I've noted a few flaws she could have mentioned but didn't.

It's a stimulating read, and I plan on sharing a few chapters with the MLA students. It's also leading me to other sources. Some of these are general in scope, like Jan Morris's Conundrum (New York: Harcourt Brace Jovanovich, Inc., 1974), a first-person account of the author's transition from male to female and the worlds she lost and gained in the process. Others are more specific, like Nash and Grossi's analysis ("Picking Barbie's brain: Inherent sex differences in scientific ability?" Journal of Interdisciplinary Feminist Thought 2(1), Article 5) of Simon Baron-Cohen's methodologies...which analysis might doom some of the infant studies which Dehaene cites, as well (O, circularity!).

I also plan on offering up a few excerpts from Richard E. Cytowic's The man who tasted shapes (Cambridge: MIT Press, 2003), an odd novel/memoir/neuropsychology text dealing with the phenomenon of synesthesia. Though not directly related to our course, I think certain passages bear tangentially on our discussions of brain function and will lead to interesting discussions.

More fun to come! Exciting. I hope the students are getting as much out of the class as I am. I may have to offer this course again soon in the Honors Program. Something to think about for next year...

Worth a repost

This morning I received a brief but touching comment on my most recent blog post: "I miss Patrick teaching." I responded to this anonymous post in a manner which I repost here because I think it's worth wider readership:

Please know that I'm organizing this class in a non-traditional manner not because I want to avoid "teaching" (though, believe me, I'm doing as much teaching, in a non-traditional sense, as I would in any other course), but because I truly feel that the Moore method is the best way to approach this material. By asking you all to explain your ideas to one another, it firms up your understanding of those ideas. By asking you to take responsibility for your work, you become the authors (quite literally) of the ideas you're presenting to one another. It's much more learner-centered, and ultimately (I believe, and the literature on pedagogy bears me out) more effective.

Thank you for your kind sentiment! I've not totally disappeared from the scene; as you've noticed, I hope, I'll take my turn "on stage" from time to time.

To elaborate briefly: I know I'm a good lecturer, and I know that I explain things well. But seeing something done and doing it yourself are two different things, and you stand to gain much more from actually solving the problems yourself and explaining your solutions to each other than you do listening to me do it for you. It's a bit more work on your part, to be sure, but the time you spend on that work will be time well spent. Meanwhile, please know I'm still doing a lot of work behind the scenes, arranging problems in a manner I think is effective to help you work your way through the new ideas, including the definitions and theorems I think are most critical to us in our work, and working with you in class as you develop your solutions.

This I promise you: my explanations are still here for you if you need them, and I will be delighted to help you work your way through any problem you might struggle with. All I'm asking is that you give it all you've got to come up with solutions on your own first. Believe me, you'll get much more out of it that way.

So, let's stay the course, y'all. I'm enjoying class so far, and so far you're doing a marvelous job. Keep it up!

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.

Close to home

Why is it that, of all of the Occupy activities that have gone on for the past two months, the pepper-spraying of peaceful UC Davis students strikes me so much more strongly than any other?

The extremity of the incident is hard to ignore: riot-geared police officers, decked in body armor and military weaponry, approach and attack unarmed and peaceful demonstrators seated in a position of pure passivity. Whether or not you agree with the reasons for the protest (I do, but I'm certain some of my readers do not), the level of force used against the sitting students is clearly out of proportion by several orders of magnitude.

The diffidence of the university administrators (in particular, UC Davis Chancellor Linda Katehi) is as striking. Afterward she is coolly dismissive of the protestors, adopting a detached and thoroughly establishmentarian "well, that's what happens when you don't do as you're told" stance in the wake of the attack.

The immediacy (and ubiquity) of the images plays a part, too: they're arresting and unavoidable...and every video of the attack clearly shows a dozen or more others actively filming the incident, a self-replicating postmodernist pastiche of views, every one showcasing unmitigated brutality.

All of these aspects of the attack factor into its effect on me, but I think what's most striking to me is how similar the students attacked are to my own. How easily those Davis kids could be the ones who come to my classes, who hang out with me in the Math Lab, who joke with me in the hall and who ride with me to conferences! How easily my kids could have been the ones writhing in agony as they gasp for breath and when they finally find it spend several hours coughing up blood! They're interchangeable. I find myself putting my students' faces over those of the kids in the video clips, and I shudder.

I wonder: how differently would my own institution have responded to actions like those of the peaceful students at UC Davis?

History lesson

Some might say this is not a "teaching" post.

They would be wrong.

I was chagrined several weeks ago when I learned how ahistorically many of my students live their lives: one of my brightest students showed her historical ignorance by not knowing, within ten years or so, when the second world war was fought.

So it is that I worry that my students might not know what's going on around them now.

I join the chorus of academic voices expressing not disappointment, not chagrin, not tut-tutting and head-shaking sadness, but rather disgust and horror at the events taking place recently at campuses in the University of California system.

All that I might say has been said more fully and more eloquently than I can here (here, for instance, and here), so I'll say little more then to say that Chancellors Linda Katehi and Robert Birgeneau must resign...as should any other academic "leader" who condones or supports the actions (or inactions) they've taken.

I cannot say more that's not been said already. I want only for my students not to miss this moment.

To them I say: this is what history looks like.

State of mind

A few months ago Zima, one of my grad school colleagues who now teaches at Kennesaw State University in Kennesaw, Georgia asked me to present at an undergraduate research conference for which she'd just received MAA funding. (Said conference is this coming weekend; it's the one Ino and Ned are presenting their findings at.) I'll be giving a run-of-the-mill plenary talk on some of the graph theory I did with a couple of REU students this past summer, and I'll be presenting in a workshop on inquiry-based learning (IBL) at the outset of the conference.

I offered Zima a title that's so generic I really could talk about anything: "Guided discovery in the mathematics classroom." I feel confined by this generality. Indeed, when I actually sat down a week or so ago to try to figure out what in the hell I needed to say about IBL, PBL (problem-based learning), Moore method, etc., I had a hard time coming up with much to say other than expressing my feeling that all too often these techniques are too "formalized." That is, I get the sense sometimes that the people who apply these techniques look on them as an all-or-nothing process: "if it ain't straight-up Moore method, it ain't anything at all" or "I use guided discovery every single day to address every one of my students' learning outcomes." So I put together a half-hour laundry list of things to say along these lines: be open to using guided discovery in moderation; it's not the be-all-end-all any more than any other pedagogical paradigm may be.

Then just now, while lying in bed unable to sleep (though admittedly probably needing to), I realized that I can say more, for I realized of a sudden why I've had such a hard time trying to come up with something practical (and original...I suspect that the folks I'll be addressing in this workshop are going to make up a choir to whom I won't really need to preach) to say about guided discovery: in my mind, guided discovery is not so much a pedagogical process as it is a state of mind.

I find more and more that in teaching it's not so much what I do with my students as how I do it that matters most. Put another, perhaps more practical way, effective teaching comprises a gestalt-like complex of actions and not a single action individually. Guided discovery is what might be called an emergent operation which cannot be broken down into its constituent parts without losing much of its energy and effectiveness. So it is that I don't necessarily engage my students in singular activities, each of which forces students to lead themselves to original, new-to-them, conclusions, so much as I try to treat them in every way, in everything I do, as co-learners, co-discoverers, seekers of authentic knowledge.

Practically, this realization makes it possible to grow opportunities for genuine discovery in the most infertile academic soil, for every simple textbook problem becomes, if viewed from the right angle (like anamorphic art) a chance for authentic "research-like" engagement. Guided discovery is an "angle" from which these problems may be viewed.

I'll try to say a bit about this on Friday when I'm leading my portion of the IBL workshop. Are these views original? Meh...perhaps not. But they're more original, and, more important, more meaningful, than whatever else I will have to say. We'll see how they're received.

More big picture stuff

In regards to another recent post, it looks like we're about to beat up on the student body again. I say once more: what in the hell are we doing?

I don't usually get political here (because my pedagogy, writ small, doesn't generally intersect directly with politics outside of my own institution), but I can't help it now: if you support any leader opposed to fair taxation and rational government spending and stimulus, you support the eventual decay of our educational system. Don't re-elect these fools who are destroying our future.

Impending sense of something

Well, all my classes are prepped for a couple of days' absence from the scene. My Abstract students are hard at work on their latest problem set, and the Precalckers are plugging away at their second take-home exam.

All is calm.

Ever get that feeling that something big is about to break?

I've got that feeling right now. I can't put my finger on it. Maybe it's just that the next week and a half is frightfully busy: something's bound to happen.

We'll be ready, my friends. We'll be ready.

Big, big picture

For several years now, I've been a "big picture" kind of guy: I paint in broad strokes. I see forests and not trees. I'm more satisfied with hand-waving proofs than most of my colleagues would be. I'm more concerned that I and my students get the overall idea, the intuition, the gut-level understanding, than that we get every last detail right.

I'm not sure when I took this viewpoint on, but I think it's a relatively recent phenomenon. After all, I'm not sure you can make it through a graduate program in mathematics without punctilious attention paid to the merest minutiae of theorems and their proofs. It's come on more recently, as I've grown as a teacher and scholar, and as I've come to grips with bigger and bigger problems. It's gotten "worse" lately. I mentioned only half-jokingly to the Associate Provost for Academic Administration, with whom I've been doing a lot of work this semester for CRTF, that I'm starting to think like an administrator. For good or for ill I no longer consider curricular issues as they'll affect a single instructor or even a single department, but rather as they'll affect the school as a whole.

Perhaps the widest view one can take I took for a moment today, and it scared me: if you look at our nation's higher educational system as a whole, there are terrifying trends.

One incident: this morning I got an email from a student who's having to take a break from it all to maintain sanity. Literally. This person is going off the map for a little while and heading home to be with friends and family and to deal with mental health issues.

A second: yesterday I spent a tearful half hour with another student who's clearly dealing with stressors beyond finishing up an essay for Humanities or prepping for this week's coming Abstract Algebra exam. This is a very good student, one I know well, one whose performance this semester has declined noticeably from past semesters. There's much more going on behind the scenes.

A third: last week one student spoke of a narrow escape from dropping out and returning to military service just to make ends meet. Fortunately the financial aid office and the counseling center were able to help make an end-run around a vindictive ex-spouse whose uncooperativeness was preventing disbursement of much-needed funds.

These aren't just isolated events. Dozens of my students, in addition to managing a full load of difficult classes (often in excess of 18 hours dominated by high-level mathematics coursework) also work 20, 30, even 40 hours a week, and deal with manifold family issues (I can think of several students whose entire families depend on them for nearly everything), and still manage to keep it together...or not, as the case may be. The cracks are showing: I've had more people cry in my office than in any other semester I can remember. It ain't getting better.

These students shouldn't have to deal with all they're dealing with today. They should be free to be students, to be free from having to support themselves with back-breaking (or at least time-sucking) labor.

It hasn't always been like this. I speak from experience.

I had it far easier than these folks, barely more than a decade ago. I had student loans and a scholarship, and my family'd done well in socking some away to help me go to school, so the only work I needed to find to get me through my college years was as a work-study assistant in the Math and C&S Department's computer lab. Big. Deal. This was a sinecure. I "worked" 10, maybe 20, hours a week, passing out boot-up disks (yes, we still used those back then), trouble-shooting simple software problems (usually involving nothing harder than MS Excel), and shooting the shit with my geeky friends. Most of the time I got paid to do my homework. The rest of my time (that not devoted to hanging out, eating pizza, running, and annoying the crap out of my dorm mates by playing Pink Floyd's "One of These Days" really frickin' loudly on my monster tower speakers) was spent on schoolwork. I managed to make straight As (aside from a couple of stray minuses) and learn a lot. I was a slacker my first year of college, because I could afford to be: I quickly learned that I didn't need to put too much effort in in order to do well, so I took it easy. My second year I started taking courses that offered legitimate challenge, so I buckled down. That's also when I first realized that I could teach myself as much as anyone else could teach me, so I started studying on my own, everything from additional math and physics and computer science to philosophy, literature, languages, and history. On my free time. (Holy crap...free time! What's that, again?)

I could reminisce forever, but let me get to my point: I'd love to launch into a cliché and crotchety "kids these days" diatribe, but I can't. I had it easier than these kids. Far easier. I didn't have to work my ass off just to stay warm and well-fed, let alone well-educated, and I had time enough (and more) to do well in and learn from all of my classes.

Many of my students have no such luxury. They have to work, and often work hard, just to afford the modest cost of the education my public school provides to them. (It may be worth noting that I went to a pricey private school for college.) If they don't work (and often even if they do) they have to go into debt up to their eyeballs to pay the bills, for they cannot always rely on family to help them out (many of these students are first-generation college students and come from families who can ill-afford to give financial aid). Moreover, they face dim job prospects on graduation, with the economy lagging the way it is. There are no sure signs of long-term improvement. Thus many of my students are going to saddled with crushing student-loan debt and little opportunity to find the work they'll need to get to help pay it back.

Do you think a student with 18 hours a week of coursework and a 40-hour-a-week job who's drawing several hundred dollars per term in student loans and trying her best to hold her family together really has time to learn (or care) what a derivative is? Do you think such a student is going to be completing her homework in full, passing her exams, and finishing her oral presentation on subgroups of groups of symmetry? Even if she's meeting these superficial measures of academic achievement, do you really think she's going to be getting much meaningful out of it?

It's become cliché to put someone in "The 99%," but my students, almost every last one of them, belong there. Bless 'em all, they belong there.

And my students amaze me. Even with 18 hours of coursework and 40 hours on midnight shifts at motel desks and family foibles and student loans and squabbles with landlords over week-late rent checks, they do still learn what a derivative is. And they do still care. They don't just finish their homework, they polish it to perfection. They don't just pass their exams, they ace them. They do incredible work, and they do it without complaint. Somehow, despite having to do everything else we demand that they do, they're learning, and they're helping each other to learn. They're there for each other, even when the system's left them behind.

But this can't go on forever. A college education can't be what it used to be if we don't give these people a bit more breathing room. What in the hell are we doing to them?

The system needs to change. Otherwise, in ten or twenty years we'll look back and notice that though we've stocked our universities with the brightest scholars and the best teachers, and though we've built the shiniest classroom buildings and equipped them with the sleekest high-tech gadgetry, we've miseducated an entire generation simply because we asked too much of them while they were trying to do what we want them to be doing before they do anything else. We just wanted them to learn.

Well...I'll keep doing all that I can, and I'm sure I can count on my students and colleagues (some of the most admirable people on Earth) to do the same. It's a mountain we've got to move, and the only way we're going to move it is if we all get behind it together.

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?

Research update

Ned and Ino are chuggin' along. Their research project, the culmination of which will be a pamphlet detailing the means of making a week's worth of healthy, affordable meals is flying forward. We just overcame a major obstacle today: we had to figure out how, once we'd found the ideally affordable combination of foods needed to obtain the right amount of each nutrient in our analysis individually, to combine this information to give us the most affordable combination of foods to give us the right amount of all nutrients at once. We made it over that hump, and I truly think it'll be the last big one before our projects reaches its end.

So proud!

Partners in crime

I know a number of people (many former and current students, a good number of colleagues, and assorted folks I've never met) read this blog, but not many often comment publicly. Quite often, though, I get comments about it on Facebook or in my in-box.

This past week I got a note from a fellow who's teaching precalculus at an inner-city high school in Boston, using inquiry-based learning. He wrote me asking about the methods I'm using in my own precalc classes right now and shared some of his own (he's asked me not to post his notes, as they're very much works in progress). I'm very impressed! His notes are clever and engaging, offering students a scaffolding students can use to climb from the barest basics up to properties of advanced functions, logs and exponents, and trigonometry. You can read about his exploits here.

By comparison, my methods that are considerably less purely inquiry-based...he's doing pretty much straight-up Moore method with high-school students! Inventive and impactful.

Dead Zone

Today I felt like every one of my classes was a bit stuck in the mud. (Oddly enough, my morning Precalc class, often my quietest, was the most lively today.) It was all we could do to keep making forward progress in a couple of the classes, and I felt like I was beating a dead horse more than once.

What to do at such times?

The classical response, at least from instructors in content-driven disciplines: "well, we've got so much to cover that we've got to keep going..."

...but this is wrongheaded. If you press on without pause, it's not like students are going to get any more engaged, and it's not like they're going to get much out of whatever you do together, anyway. In pressing on you'll be doing so only for the sake of pressing on, and any progress you make will be illusory.

This is exactly how it felt in my second section of Precalc, and in Abstract Algebra, in both of which classes we worked with (what I thought were) some pretty neat mathematical ideas: in Precalc we solved a nontrivial optimization problem involving rational functions, and in Abstract Algebra we looked at the subgroup lattices of a couple of groups and examined the asymptotic behavior of Euler's function φ. I don't feel that either class picked up on the subtle beauty in a way they would have had they been in a more receptive mood.

Don't get me wrong: I'm not blaming my classes. Both of these classes are full of wonderful students who are generally unafraid of working together to make a healthy and supportive learning environment. (I've bragged on my Precalc peeps enough this semester for you not to know how much I care about them.) Rather, I think we've just hit that point in the semester, about 50-60% of the way through, where everybody's just DEAD.

It's the Dead Zone. Wearied by exams and essays and due dates and deadlines, overburdened by homework and quizzes and lit reviews and response papers, we're tired and jaded and not having much fun. (Believe me, kiddoes, this "we" often includes me. I apologize if I'm sometimes snappy around this time of the term; it's hard to be irrepressibly chipped every hour of every day!)

On reflection, I've thought of a two-fold healthier response to these Dead Zone doldrums:

1. Play. Put down the pen or pencil, put the paper away. Let's just think of something fun we can do with whatever it is we're working at this very moment. Optimization problems? Let's come up with some kind of crazy variation on the theme, whether we have any idea how to solve it or not. Let's set it up and see if we can work it out, like Ariadne weaving a web through the labyrinth. Are we sick to death of subgroup theorems? Let's break it all down with an explicit or example, or two, or three...let's dissect the dihedral groups until there's nothing left but individual elements...let's take it apart!

2. Reflect. The next step comes as no surprise to those who know me well: once we're done playing, let's take a minute or two to write to ourselves, if only to reflect on what we've been able to discover. Did we reach an end? How? Did we run aground? Why'd we lose our way? What's our play got to do with other problems we might encounter in mathematics and beyond? Write about it, write about how we feel about it. Hell, write about how we feel in general: why are we so dead today?

Too often affective learning goals get lost when we focus too heavily on cognitive learning goals, and that goes double for content-laden quantitative sciences. Let's try not to lose sight of our humanity, and the fragility that comes with it. Let's take care of each other as we come together to learn.

By and large all of my classes this semester are doing a marvelous job at this, and I admire them for it. I never cease to be amazed at the quality of students with whom I get to interact and learn. You're terrific people, all of you!

A somewhat schizophrenic conversation

[Note: this post includes a homework assignment for my readers, toward the very end. If you're a teacher, student, or alumna/alumnus, please take a moment to respond when you're done. Thank you!]

A few years back we graduated one of the brightest students I've yet to work with at UNC Asheville. Sedgwick was a soft-spoken and deep-thinking environmental studies major with whom I had only one chance to work, in a Calc II course he enrolled in just before he graduated. He and I shared some pleasant conversations during his studenthood here, but we've shared many more (often from afar) since his moving off to broader pastures.

He wrote a few days back indicating that he'd had some thoughts (which he'd written down) about my CRTF-related posts, and wondered if I had any interest in reading them. Knowing his perspicacity, I knew they'd be well worth the read, so I told him to send them along, by all means, asking if I might repost them here, as I've done in the past. He's granted permission.

Sedgwick's comments concern the ILS Topical Clusters in particular, which are considered by many (myself included) to be the weak point of ILS as a whole. Here's what Sedgwick has to say:

I will preface these comments by saying that I was one of the last students to graduate under the old General Education requirements, so I have no first-hand experience with ILS, despite being a recent alum.

I did take a look, however, at the clusters on offer. Currently, a cluster appears to be nothing more than an arrangement of existing courses that fit some nebulous theme. This situation seems to be the functional equivalent of forcing all students to declare a 'mini-minor,' albeit less useful because the promise of interdisciplinary depth seems hardly fulfilled, given how little time one can commit to a cluster relative to other requirements.

The main part that confuses me is the fact that there is a theme at all; it seems like a needless restriction. The structure of a student's education comes from their major, which offers the technical, career-focused classes they need. Asking the liberal arts portion of the curriculum to follow a cluster's pre-determined path is like asking a journey for directions: clusters rely on the false premise that students can (or should) connect the dots in the present.

To illustrate that last point with an example (that will no doubt become cliche): the reason that the original Macintosh debuted with multiple fonts and typefaces was because Steve Jobs took a calligraphy class at Reed College years earlier, a course that interested him but had no real-world usefulness to him at the time. My concern is that by requiring students to adhere to a theme in the 'liberal arts' part of their studies, they could be missing out on experiences that may be of use one day, in a manner that's impossible to conceive of while still in college. The way Steve put it: "you can't connect the dots looking forward; you can only connect them looking backwards." UNCA's advantage, as our state's liberal arts institution, should be in providing the broadest array of dots for students to connect in the future, as they need them. In this light, restricting the ILS experience to a small subset of available courses does not make much sense.

If all ILS wants to do (or can do at the moment) is force exposure to other departments, then get rid of clusters and just say that students should take X number of courses outside their major. However, I think UNCA's goal is to emphasize the 'integrative' part of ILS. Clusters were an important first step, but I believe the 'integration' was too high-level to have the intended effect. Ultimately, integration needs to permeate the coursework itself, which why I would suggest 'cross-up' courses instead of clusters.

A cross-up course would be a deliberate collaboration of at least two departments. What would these cross-up courses look like? It's hard to say: I trust faculty to have a better eye for how their chosen discipline can interact with another. I know the synthesis of mathematics and writing is an important part of your teaching, so that seems to be a natural fit. My own background and interests can easily see collaborations between Computer Science and Environmental Studies. I think the possibilities are only limited by the interests of faculty and their willingness to work together.

There could be a simple rule that each department must form a cross-up with at least X number of departments. With a pool of cross-up courses available, just have students take X number of them to fulfill the ILS requirement. That's it. With the Intensives requirements still in place, the curriculum would not suffer in rigor. Like custom clusters, cross-ups could also be student-initiated with proper coordination. What better way to give UNCA students an edge in cross-disciplinary work than by taking classes that are actually cross-disciplinary by design? I cannot help but imagine that this type of setup would also confer a degree of market separation from peer institutions.

In short, exposing students to several unique instances of cross-disciplinary work seems to be a more pragmatic use of the limited time students have to devote to ILS electives. Cross-ups can also align faculty more towards collaboration than rivalry, encouraging departments to think about what cross-disciplinary experiences will work for students once they leave the academy and face an unforgiving job market. Of course, given the fiscal situation up there, asking faculty across the campus to design and teach a dozen or so new courses is likely a non-starter. But doesn't it sound exciting, something that really fits with the purpose of UNCA?

My open-letter response (which is almost identical to the response I had to Sedgwick's last letter to me, linked to above): I agree...ideally. I actually think the cross-up courses are a great idea, and if implemented would lead to a much more flexible, manageable, and student-authored learning experience that would replace (and substantially improve upon) the current system of topical clusters. The primary problems I see (as does Sedgwick himself) are logistical.

Namely, cost, in person-power and faculty time, if nothing else, is a prohibitive factor. Given our current budgetary climate (if I had a dollar for every time I've typed that word in the past few months...), we quite literally can't afford to ask all faculty to take time out of their schedules to design new interlinked courses. Moreover, we lack the administrative power to begin giving faculty appropriate credit for leading the many team-taught courses the cross-up system would entail...

...But wait a minute...Even as I was typing those last two sentences I began thinking to myself..."what?!?" As Sedgwick pointed out in the post I linked to above, universities are, though many who staff them would be loathe to admit it, among the most conservative of institutions around today, and change comes very slowly to them...I often think that we often make up excuses (too expensive, too time-consuming, administratively infeasible, etc.) for doing things we, institutionally, simply don't want to do.

To the first point: how does maintenance of the admittedly flawed and unpopular ILS Topical Cluster system demand any less faculty time and resources than would implementation of a new program that would likely require considerably less oversight and administrative overhead? On reflection, the faculty claim "I just don't have time to sit down and design this course" is wholly ridiculous...faculty are designing new courses all the time. Who among us isn't thrilled and filled with pride when first given the chance (in, maybe, our second or third year on the faculty) to teach a special-topics course related to our research? And how many of us, especially those of us in our first, second, and third years of teaching, find ourselves teaching one or two new preps every year? Though these courses are often not new, they're new-to-us, and take a fair amount of time to tweak and tone as we make them our own.

The last paragraph points out an obvious "in": our newest faculty are likely to be the most willing and able to implement a new curricular component like cross-up courses. Not only do they expect to have one or two new preps any year anyway, they're also less entrenched in their disciplinary positions and are more likely to be open to cross-disciplinary fertilization. I may just have to talk to a few of my younger colleagues about these ideas...

To the second point above: the argument is often made that we don't team-teach much here because it's simply too difficult to give faculty the appropriate "credit" for teaching such courses. The system as it exists, supposedly, allows us only to give credit for teaching half of a course for such courses, and in order to meet various benchmarks for faculty activity (the infamous Delaware study among them) faculty teaching such courses would have to teach far more than an acceptable load to appear on paper as though they're being productive. I can't buy this argument; if I did, I'd be as shortsighted as the folks I've been ranting about in my recent CRTF posts.

I can't buy it because I'm just not sure I've shopped around enough yet: might it be that the problem is one of "vision"? Several of my colleagues on the Curricular Sustainability subgroup have remarked that resistance to change may be predicated on a lack of understanding of other ways we could do things than the way we're already doing them. That is, maybe we're encountering so much insistence on doing things the way we've been doing them because folks just don't know how else these things can be done. Our response on CRTF has been to try to come up with models. Just this past week I asked the folks on my subgroup to identify institutions offering "model" majors and degree programs in their respective disciplines, suspecting that these programs will likely prove more sustainable (e.g., more flexible and less prescriptive) than their cognates on our campus.

Maybe what we need is more models. This brings me to the homework I mentioned at the outset of this post:

1. For those faculty reading this post, it would delight me to no end if you could comment on this post with a paragraph or two (or more, if you wish) about the nature of team-taught interdisciplinary courses at your school. How are they organized? How are they overseen and assessed? How does the administration grant faculty credit for teaching in these courses? How are they received by the students? Are they required, recommended, or simply part of the body of electives students might opt to take? All of this information would give me ammunition I could use to make the case for these courses.

2. For those students reading this post, it would give me similar delight if you could comment on this post with a paragraph or two about how you would receive such courses. Would you be interested in taking them? Among what disciplines would you like to see more collaboration? Would you find it helpful if such courses were required...and would you take them even if they were not? For UNC Asheville students in particular: would you prefer this kind of system to the current system of ILS Topical Clusters? Why?

Obviously I can't require you to respond, but even just a few words would be of such tremendous help to me that I really hope you'll consider writing back.

I realize now that this conversation, once between Sedgwick and me and then just between two mes, has turned out slightly schizophrenic. It's really helped me to get these thoughts out of my skull, though: I'd not before now seen the untenability of the "we don't have time to..." argument. I needed to write it to see it. (It's writing-to-learn, y'all!)

My thanks for those who've read this far! I'll soon be posting on a conversation with another reader, a high school teacher in Boston who's making use of IBL methods in his precalculus course. Stay tuned!

Reflect

After having started off two different meetings of my second section of Precalc with readings from reflective writing of some kind (see this post and this one), I invited the students to join me in bringing contemplative readings into class. Several have indicated that they enjoy this start to the class, and I look forward to seeing what the others will bring in.

"Teaching"

I noticed the other night, just before leaving campus, that in Abstract Algebra I'm about two weeks "behind" (about three handouts, each roughly equivalent to two days of class) the place where I was at this time in Fall 2008, the last time I taught the course.

My immediate reaction, which, fortunately, dissipated almost immediately: "Holy crap...how can I catch up?!?"

I looked over the handouts separating now from then. They were filled, for the most part, with technical lemmas and other minutiae about groups, facts like "the condition that g^-1 be a two-sided inverse is redundant" and "to check that H is a subgroup it need only be shown that gh^-1 for all g and h in H." I thought about my students, and their aims and ambitions, and I realized almost immediately that they could do without "covering" these lemmas. At best, they'd memorize them for several days, work a contrived homework problem or two meant to test them on the results, and then forget them.

Meh. We'll skip those handouts. Instead, we'll move on to the good stuff: subgroups, homomorphisms, more meaningful structural results that are powerful, intriguing, and beautiful.

Meanwhile, the homework is clearly kicking my students' butts. Even the more experienced students are struggling with it. It was only after thinking about it for a bit that I realized why this is: instead of asking them to pantomime the proof of some result that differs only slightly from something we've talked about in class or put the polish on a theorem I read out loud to them, I'm having them build from scratch most of the canonical examples; I'm having them introduce and analyze the most important definitions.

It would take me fifteen minutes to "teach" them all there is to know about the subgroups of the integers, but in doing so I'd guarantee that nine out of ten of them would smile (or scowl) and nod their heads, take careful notes, commit what I'd said to memory, and not understand a lick of it. I'd rather they take a few hours outside of class to do it for themselves, struggling with every step, pounding their heads on the Math Lab's countertops in frustration, cursing me under their breath, gaining intuition all the way. The best human computers of all time, from Napier through Gauss to Ramanujan, built their skills by living with numbers, loving them, spending their days with them cheek to jowl. It's hard work, but you'll gain so much for it, mathematically speaking. You'll gain intuition, and, ultimately, understanding.

I spent an hour or two with several of these students in the Math Lab this afternoon, and the progress they made was incredible. I'm so impressed by their intelligence and determination.

Keep it up, folks! I'm proud of you all.

Grades, schmades

Tonight I talked for several hours with one of my best friends on Earth, and one of my most reflective fellow teachers of mathematics, Griselda. (Tired of this and that, and longing to move closer to friends and family in the Northeast, Griselda recently left her job teaching at a public liberal arts college in a nearby state to teach at a boarding school in Pennsylvania.) As usual, our conversation was broad and far-reaching, and dealt with issues in every corner of education.

My most meaningful self-realization of the evening: the only reason I still give grades in my courses is because I have to; the school requires them from me at the end of the term. And the only reason I give grades throughout the semester is to provide substantive justification for those end-of-term grades.

Don't get me wrong: I love responding to my students' work. My responses make up my half of a conversation with the students about the discoveries they're making. It's an ongoing dialogue, and often an exciting one. They say something to me, I say something back, and after one or two iterations we might make some sense of what it is the other is trying to convey. Eventually we'll come to a consensus regarding the meaning of whatever matter we're faced with. I love these conversations. They're where learning takes place.

But grading? Uh uh. I'm over it. I'm tired of this competitive academic economy based on artificial and extrinsic rewards.

How can I break the cycle? Maybe if I rebel and refuse to assign grades...?

Something to think about...

...Griselda always leaves me something to think about.

Thank you, my friend.

Running in reverse

Recently I've had a chance to feed my undying love of linguistics. I've been reading up on the history of English and its antecedents (like Angl0-Saxon) and victims in the clash of tongues that's taken place on the British Isles since the early common era (like Cornish and Manx). The text I'm currently reading is Dick Leith's A social history of English (London: Routledge & Kegan Paul, 1983), an interesting book offering a glimpse of English's development as a social, as well as a purely linguistic, phenomenon.

More interesting than Leith's treatment of English per se are some of the observations he makes about the codification of language, and the role of "authority" in the preservation and propagation of language across time and space. A central thesis of his book is that all too often we forget that language is very much dynamic: it is ever in flux, constantly changing...and that in the end that change is not driven by grammarians or the intellectual or economic elite so much as it is by the ways in which every member of society chooses to use the language.

These are points that even the most perspicacious language-lovers among us tend to overlook. The reminders Leith offers have made me think of new (to me, at least) and "subversive" paradigms for poetry (a post on that soon, perhaps)...but they've also recalled for me the central role every member of a learning community plays in that community's advancement of knowledge, while issuing a reminder as to just how dangerous it can be to trust blindly in the authority of a textbook.

The following passage from Leith (p. 68) struck me (cf. the comments some of my precalculus students made on their last exam):

Unfortunately, many people tend to treat dictionaries with reverence: rather than being seen as a record of usage, the are often regarded as the arbiter of it, a source of enlightenment for the ignorant non-specialist. In fact, the traditional arrangement of words in dictionaries gives people a strange idea about language. The alphabetic arrangement disassociates a word from the company it keeps, presenting it as a unit isolated from context and words of similar meaning. More important, many dictionaries give the impression that words have only one meaning, to be found on the right-hand side of the page. Even the fullest dictionary, the Oxford English Dictionary (OED), which shows the whole range of meanings by citing examples of a word in use at different periods in its history, puts the meanings first, then lists the examples, thereby obscuring the process involved in deriving the meanings; for we learn the meanings of new words most efficiently by hearing them in a wide range of contexts....It is not surprising, therefore, that people often misunderstand them.

How often too we ask our math students to use their textbooks in the same way they'd use a dictionary, placing theorems and proofs before (or, more often than not, simply in lieu of) the intuition and arguments that led to those theorems and proofs in the first place? How do our textbooks obscure the many long hours of exploration and discovery that went into the derivation of the theorems that pepper the textbooks' pages? Without access to the discoverer's process of discovery, the reader is apt to feel as though a given fact or formula arises ex nihilo, and that they, the uninitiated, are not privy to its inner workings.

Food for thought. By me, it's better to let the students stumble around a bit, piecing things together for themselves as they author their own textbooks. That's just what I'll be doing when we talk about general rational functions in Precalc tomorrow...strap yourselves in!

More words of wisdom from my precalc students

Yesterday afternoon I worked for an hour with the student consultants at our university's Writing Center, helping them to understand what mathematical writing might look like, preparing them to work with students in the quantitative sciences who might come in with writing assignments from their quant courses. I was happy with the conversations we had together, and delighted by the students' energy and enthusiasm.

I shared with the consultants one precalc student's response to the midterm question in which I asked the students to describe the most meaningful learning outcome they've achieved so far this semester (see here for one response, not the one I shared at the writing center), and promised that I'd share a few more.

Let me throw a few more out there, all with a common theme. Several students, including the three quoted below, indicated experiencing the realization that mastering math is more than just memorizing formulas, and that if you stop trying to memorize every last formula but rather try to break each down and understand its inner workings, you gain immeasurably through your effort. That understanding is strengthened if math is put in a contextual matrix, placed alongside other disciplines so that its relevance becomes more apparent.

Katarina had this to say about her personal revelations:

This course is unlike any other math class I've taken. We actually discuss math in English at a level that I believe everyone in the class can understand....Instead of memorizing formulas, we play them out on the board, multiple ways, so that it is almost illogical not to understand their function and there is no need to actually memorize them....We write our answers in paragraph form, truly explaining the reason for doing them and our end result....It is so unusual to me to have math and other subjects (such as writing) overlap. My initial response was to avoid it but now I'm beginning to embrace the idea. After all, isn't UNCA's liberal arts program all about integrating many subjects in order to have a broader education?

Thomasina (who took a year off from high school before coming back to college) had this to say, upping the ante by acknowledging not just understanding, but enjoyment, and even aesthetic appreciation:

When I graduated high school, I decided that school was pointless....Read, memorize, regurgitate. That is all I was ever taught. But to understand? To break something down to its very core and build it back up, seeing every piece as they’re placed together to form a whole again. It's actually beautiful. I never got what you meant when you’d say math is beautiful, but now I get it. To have the ability to look at something complex and make it simple and tangible - it's art. And it's not just with math. It applies to everything: decisions, work, other people. Everything is a complex formula waiting to be taken apart and understood, and then put back together in a way that makes sense and feels right.

Kurt reports an experience similar to that of Katarina and Thomasina:

I've found more interest and enjoyment in something that I previously found tedious and boring and have found that I actually can relate math (including calculus) to the real world and my everyday life in ways I'd not before considered or imagined....I see this insight as far more valuable to me as a person than any one mathematical concept on its own. This is the kind of insight that changes people's lives, gives the potentially brilliant scientist a view into the potential locked up inside, or even just changes a fundamental attitude or a paradigmatic shift in thinking altogether....It's a great feeling to realize suddenly 'hey, I GET this!' and even better to find 'hey, I actually LIKE this!' I truly wish there were more classes like this one which, if not persuading one to major and work in the field, to at least open one’s eyes to the possibilities.

If a precalc course can move students to wax this rhapsodically about their learning, how much can students get from still deeper and more meaningful courses? It's up to those of us who teach to make our courses as relevant as we can.

Moo...?

As I mentioned in a post not long ago, I recently asked my Precalc students to draw comics in which the characters explain how to multiply two complex numbers. I'll showcase a few here and there in the next few days. Here's an almost sickeningly cute one by Thomasina and Urban:


Also soon to come: more excerpts from the students' midterm essays on their most meaningful learning experiences so far this course.