Good can come of department meetings

Propositions cut
like glass, bright obsidian
in soft poetic flesh

Welcome!

Yesterday I ran into one of my Precalc students outside of class. I was on my way back to Robinson Hall after teaching my Abstract class over in Karpen; Becky was en route to the library with the rest of her LANG 120 class. There they'd be discussing the use of library resources in conducting research.

We chatted briefly, and I asked her if she was still considering a math major. (She was one of two in that section who, without prompting by me, indicated interest in the major at the outset of the semester.) "I definitely am," she said excitedly, and then a look of worry spread over her face, "do you think I still should?" I was a bit thrown off and only after a few seconds managed to reply with something like "of course!" She went her way and I went mine, but our encounter stuck with me. Why had she asked what she had?

I suspect it may be because she may not feel as confident in her math ability as she did at the start of the semester. Though she's done well on every homework set and on every quiz, like everyone else in the class she's made her share of mistakes and hasn't presented complete understanding of everything we've talked about. Might she believe that only those who can complete Precalculus with flawlessness and perfection are worthy of pursuing a degree in mathematics? (Only later did I think of an apt analogy: as I'm highly unlikely to ever run a four-minute mile, might I just as well give up on one of my favorite hobbies?)

After thinking it over, I found that I could understand Becky's belief, given the traditional structure of mathematics education, home to bell-curve-based grades, punctilious point-based assessment, and lecture-based teaching. There's an air of elitism to the way students are often ranked and ordered, made to fight with one another for a scant few As. The unsaid assumption in classrooms where deep and steep curves guarantee a normal distribution of grades is that only the best need move on, and that the others' services will not be needed. Detailed rubrics with single-percentage-point resolution signal to the student that mastery of fine detail takes priority over authentic understanding. (No wonder students clamber after every point, wondering what it is that separates a score of 8/10 from a score of 9/10!) Fast-paced lectures make sure sure the students who start off slowly get little chance to get ahead; the quickest students (who are often, but not always, the brightest) control the pace in these classes.

All of these factors discourage students who are excited or intrigued about math, but who are put off by the way in which it's often taught. We can't afford to turn these students away. The fact of the matter is we, as a society, need more mathematicians than we can possibly prepare, and we do no good in discouraging anyone who's passionate about the field from pursuing it further. We do well to let as many students through the gate as we can, and to give them all of the support and encouragement they need to develop their skills fully. We do well to eliminate curves and to downplay in-class competition between students. We do well to "coarsen" our grading scales to accommodate "big-picture" thinkers who might miss a detail or two but who grasp complex systems in their entirety. We do well to step away from our classroom's center stage and let students take our place, so that it's not to the top ten percent that we teach, but to the class as a whole.

The bad news is that these practices are not universal; the good news is that they are more popular than ever. They're in use throughout my department and many like it. The youngest math teachers (at every level) are more adept at applying them than their older peers. These teachers are daily developing new tricks and techniques to make these practices more effective, and they're not shy about sharing these tricks and techniques with their colleagues and with their students. The future is bright for math education.

Stick with it, Becky! Welcome to the team. You're in good company. You'll do wonderfully.

Coming soon to your university's bookstore...

Life lurches forward in fits and starts, and all things go in cycles.

My first book (defined as something with an ISBN) was a collaborative effort with several of my fellow graduate students. From here to infinity (A foundation for calculus) (Thompson Learning, 2001) was a slender in-house volume on algebra, trig, and other precalc concepts meant for use in Vanderbilt's Calc I classes to get the students there up to speed on algebra essentials before Calc came along and kicked their butts. We worked together wonderfully well on this project, each writing a chapter, swapping chapters to review and revise during the editing stage, and sewing it together seamlessly before sending it off to the publisher. I wrote the chapter on rules for logs and exponents, a topic I feel very strongly should be taught in a certain way (it all makes total sense if you teach it this way).

I was very happy with that project.

My second book, The Isomorphism Problem in Coxeter groups (Imperial College Press/World Scientific Publishing, 2005) was a one-off, a graduate-level survey of a particular area in combinatorial and geometric group theory. I was approached by the publisher (in, admittedly, a very generic dear-author,-we-invite-you-to-publish-a-volume-in-our-new-lecture-note-series kind of way), and, having nothing better to do with a year of my time, said yes. On and off for almost a year I compiled all of my notes on rigidity problems in Coxeter groups, on automorphism structure in same, on presentation invariants, and on standard combinatorial techniques applicable to Coxeter groups, braid groups, Artin groups, and all other fellow travelers...the result was the text named above. It was something to do. It was a lot of work, and it was reviewed well. In the past six years it's sold maybe 600 or 700 copies, which honestly is more than it deserves to have sold; it's got a very niche audience. I'm proud of the book, but not excited by it; there was little passion in it.

In February 2012, my third book will appear through Jossey-Bass:


This book is a labor of love. It represents true passion, true excitement and ardor. It combines my love of teaching and my love of writing, and the sense of duty I feel toward folks in my chosen discipline. I had a blast writing it, and I'm very proud of the outcome. I hope you'll consider taking a look once it comes out!

Beaming...

...with pride! My undergraduate researchers Ned and Ino are making substantial progress on their nutritional analysis project. Fresh out of a meeting with leaders of the Housing Authority of the City of Asheville (HACA), they have a better-than-ever vision of the scope of their project. They'll be working with HACA to develop affordable, nutritious, and pleasant meal plans to distribute to low-income families who rely on the federal Supplemental Nutritional Assistance Program (formerly called the Food Stamp program) for grocery purchases.

This project will be meaningful, rich, and offer technical mathematical challenges. I'm so excited for them, and immensely proud of the work they're doing. They're self-directed, focused, and quick. I have high hopes for this project, about which they're already scheduled to speak at two conferences, including the JMM in Boston in January.

Good stuff! Further bulletins as events warrant.

Where's that voice coming from...?

Today I got to give only my second-ever presentation via Skype, to a small but interested (as far as I could tell) audience at LaGrange College, arrangements made by my colleague Kevin, a faculty member in LaGrange's mathematics department. (Thanks, Kevin!) Kevin and I took part in a Project NExT session at the MAA sectional meeting in Tuscaloosa last March (about which I blogged briefly here), at which time he shared some really neat online networking tools for helping increase real-time student engagement in the classroom. I shared some examples of low-stake writing activities (freewriting, doubting and believing, dialoguing, etc.), indicating their uses in disciplinary classrooms.

Today's telepresentation was an encore performance, to, as near as I can tell, an economist, a writing instructor, a reference librarian, and a couple of mathematicians. (Sounds like the set-up to a lousy joke...) They had great questions, and seemed legitimately engaged.

It's hard to tell from afar. If I learned nothing else from the presentation, I learned how awkward it is working with a group via video feed, dealing with subpar audio and video, and not being immediately present to pick up on subtle nonverbal cues from my interlocutors. In the immortal words of Marvin Gaye and Tammi Terrell, ain't nothing like the real thing, baby.

Also: I've decided to take my show on the road...tomorrow, weather permitting, I'm going to hold a couple of my office hours out on the steps of the Ramsey Library...look for me there from 2:30 to 4:00. If my current students are as excited about this idea as my former students (thanks for reading, Jack!) apparently are, I might get a few visitors. I'd love to see my colleagues out there, too. Let's mingle!

Where do you want me to be?

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

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

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

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

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

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

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

What have we learned?

At some time in my youth I learned that I loved mathematics. Its precision and exactitude suited my budding rational empiricism (though I couldn't have thought to say this at the time). I decided that I'd be a math teacher, because what else could one do with math?

As an undergraduate math major, I learned that I'd most likely teach math at a university, and not at a high school. "Don't waste your talent," my adviser told me. In those words he said it (albeit sotto voce); being a mathematician himself, he likely lacked the social grace to say it less bluntly. I was led by him to believe the dictum that "those who can do; those who can't teach." Though I've learned that it's often the case that our future teachers struggle more mightily with advanced mathematical concepts than do some of their more pure-math-minded colleagues, that struggle is often a fruitful and maturing one, and I no longer keep stock in this saying. Besides, knowing college faculty the way I know college faculty now, and recognizing the importance of a rock-solid middle school and high school education, I'd trade a dozen run-of-the-mill university faculty for one fantastic high school math teacher.

As a grad student, I learned that I loved both teaching and research...in other words, I was born to live a life in the traditional academy. I loved the energy in my classrooms, the excitement of my students. I loved helping them to their personal epiphanies and "aha!" moments that make discovery so fulfilling. (I remember some of my experiences there with remarkable clarity, as in this old post.) As much as this, I loved holing up in my office or in the library and plugging away at the pure mathematical puzzles presented me by Coxeter groups, braid groups, and Artin groups. I loved confronting the unknown, clad only in the armor of socially-constructed axiomatic systems (though I couldn't have thought to call them this at the time), armed only with a few blunt theorems. My doctoral adviser urged me to get a postdoc, to prepare myself for a career at a top-notch research institution, where I'd surely be happiest.

As a postdoctoral scholar, I learned that I didn't want to spend the rest of my life at a research-intensive university. As much as I loved (and still love) research, I could not see myself doing it to the exclusion of almost all else. I had hearty colleagues; they were wise, intelligent, and in their own way fun. But they sacrificed virtually all for the sake of their research careers, publishing so that they might not perish and openly disdaining teaching, for it took time away from their scholarly pursuits. As I grew to be a better and better teacher and came to learn the fulfillment it gave me, and as I came to recognize how much more of an impact an exceptional teacher can have than even the best researcher in pure mathematics, my personal future in academia came more clearly into focus.

As an untenured faculty member, I learned that yes, indeed, teaching is where my passion lies. My research was (and is) fruitful and fun, but had (and has), for me, less meaning than the moments I get to spend working with my students, talking about teaching, and living as fully immersed in an authentic community of learners as I can. I learned that math wasn't all there was, and that my longstanding love of writing could play as big a part in my working life as it did in my personal one. I opened the door to poetry, composition, and rhetoric, and discovered rich new worlds I'm only just beginning to explore. (Much more about that in a forthcoming post on my personal "community of scholars.") Math was moved aside to make room.

As a tenured faculty member, I'm learning more and more each day that I can do what I want to with my career, and that I can follow my ever-changing passions. I'm learning that my most meaningful collaborations are most rarely found among people in my own discipline. I'm learning to adopt and adapt ever more reflective and integrative practices into my classes. Some of these practices are decentering and unsettling, both for the students I work with and for me...but ultimately such practices are the most rewarding. I'm learning that I stand to learn the most from those who are "supposed" to learn from me, and that only at a liberal arts institution like UNC Asheville would I have the chance that I have to engage in that learning.

I've learned enough to know that I don't know much at all, but that that's okay, because no one really does, and I know as much as anyone I'm likely to run into. We all know different things, though, and when we meet at the crossroads and shelter at the inn there for the night, we'll have wonderful stories to share with one another before we move on to make our way in the world once more. Sharing travelers' tales: that's how we learn best.

Here's a tale. This semester I'm working on an outside project with two of my favorite students, Ino and Ned, with whom I've shared several courses in the past. They've signed on as "undergraduate research" students because initially we expected that the project would involve a good deal of digging into linear programming and other linear algebraic methods of optimization, requiring us to perform original research at some point. (It's an offshoot of a miniproject Ino and Ned worked on with a couple of other students in my linear algebra course last fall, in which they performed a feasibility study of several different diets from both a nutritional and a budgetary standpoint. Wonderful stuff!)

The deeper down the rabbit hole we go, the more it becomes evident that we're unlikely to break any theoretical ground, but the more we realize the transformative potential of the work we're doing in the larger sphere. The work these bright kids (and they are among the brightest our school is privileged to serve) are doing will positively impact our region by helping community outreach organizations assist low-income families in planning healthy and affordable meals. Our off-campus partners are excited about the budding collaboration my students are forging, and I predict great things coming of our engagement with one another.

You might still call this "undergraduate research," writ large...more accurately still, you might call it "service learning." Whatever it's called, it's learning of some sort. Moreover, it's authentic, it's transformational, and it's real. It represents the sort of learning I hope to do more of as I move forward from here. It's the sort I hope to help my students and colleagues do more of as well.

Let's wait until morning and then move on, and make up more travelers' tales to tell at the next crossroads we meet.

Be afraid, but don't be afraid of your fear

Today I had a very contemplative morning. I had a good run, and it couldn't have been a nicer day for it: the air has an early-autumn crispness, and the trees a golden-green that connotes an ageless seasonal change.

I gave thought this morning to an opportunity I've recently been granted, one of which I'll soon take advantage, and which I hope will bear fruit. I can't say much of it publicly yet, but I will say that I became fully determined to make a move this morning when I realized two things:

1. I'm a little afraid of taking this chance, and

2. it's that little bit of fear that's convinced me the chance is worth taking.

I know that if things work out the way I hope they will I'll be presented with entirely new challenges I've not yet faced in my career to this point. I'll be doing a lot of learning on the fly and a lot of playing it by ear. I'll be bearing a great deal of responsibility, but also relying to a greater extent than ever before on others to help me carry out the tasks I'll be responsible for. I'll be delegating, relegating, moving and shaking, and working my tail off.

It's a bit frightening. I've almost balked once or twice because I know that though I'm qualified to take this on, and though I'm as ready as I'll ever be (and as ready as anyone could be expected to be), it'll still be a rough road. I'll definitely be outside my comfort zone. It was only just this morning that I admitted to myself that I've been a little afraid of moving down this path much further.

But you know what? That's a good thing. If we don't put ourselves in that "zone of proximal development," as Vygotsky put it, we don't put ourselves in a position to do much learning or growth. If I only ever do things that I know that I can do, that I'm utterly unafraid of doing, I'm not going to get much out of them.

These twin realizations: the presence of fear and the healthfulness, the appropriateness, of that fear, have moved me forward. I feel stronger and more whole.

There's a parallel in one of my courses right now. Yesterday I spent roughly 10 hours grading (9:00 a.m. to 7:00 p.m., almost nonstop), about 7 hours on Precalculus alone. Their most recent homework sets (especially Homework 6) were challenging ones, involving complicated problems which had to be broken down into simpler subproblems. The students had mixed success in seeking solutions to these problems. Some patiently broken them down and crafted careful solutions; others were less successful, impatiently attempting to swallow the problems whole.

The problems are meant to push the students forward, to move them from a place where they feel comfortable to one where they feel challenged, and maybe just a little scared. I'm confident that the students can do what's asked of them, though, and that they have the skills needed to solve the problems I give to them if they take their time and work carefully. I'm confident that if they take time to contemplate the problems piece by piece, they'll grow in confidence and competence.

I sent the students an email just now, including a model solution to the toughest of the homework problems. Here's some of the text from that email; I hope it helps them place our work together in a healthy context:

...I also recognize that the problems I'm asking you to complete are not easy ones. Each of those on HW 6 likely took you 45 minutes apiece (maybe more) if you did them clearly, capably, carefully, and well, as many of you did. I was impressed with the neatness and precision of some of your answers!

These are not easy problems; they are challenging and probing. It's for the best: I believe that challenging problems are those most worth doing. They push us to our limits and force us to confront fully our understanding of the ideas we come up with together. I'm just relearning now (relearning from you, as much as from any other source) that those things that are most worth doing are those things that are difficult to do, that challenge us, and that, perhaps, even scare us a little.

My reflective morning's brought me other thoughts as well, about which I'll be posting throughout the week. Several stem from my ongoing reading of Parker J. Palmer's and Arthur Zajonc's The heart of higher education: A call to renewal (transforming the academy through collegial conversations) (San Francisco: Jossey-Bass, 2010), the centerpiece of the Learning Circle I've been taking part in (when possible) this semester. The book has great richness, and has led me to reflect deeply; as I wrote to myself at one point "there's poetry on every page!"

In the next few posts I'll talk about what I've learned from an ongoing project about which here I've yet said little, about resistance to curricular change on the part of even the most well-intentioned (and change-oriented) faculty, and about my own elusive "community of scholars" Palmer and Zajonc extol on page 128 of the book I mentioned above. About all of these I've thought today.

As I said, I had a very contemplative morning.

Just flew in from Little Switzerland, and...

This year's CWPA was, as always, fulfilling, friendly, and enlightening in the extreme.

The focus was on grant writing and all it entails, from planning projects, to finding funding sources, and on to starting and polishing proposals. Several new ideas for medium- to long-range plans popped up, not the least of which involves finding financial support for the rhetorical analysis Bella and Damian are still performing on REU students' research writing.

Beyond the professional fulfillment the conference gave me, I benefited from the trip personally, as well. Once again the warmth and welcome of my colleagues in composition and rhetoric reinforced my belief that I've made the right decision in moving my career in this direction.

Meanwhile, back at the ranch, I'm really enjoying both courses I'm teaching this semester. As is to be expected of a course in which everyone in the room knows well ten or more of the others, my Abstract class is often rowdy, but engaged. The same can be said of the second section of Precalc. I made the mistake of calling them "squirrelly" the other day, in response to their giggliness, and this only made them giggle more. It's a good group, though, and they work hard. Their primary strength as a class, as I told them today, is their willingness to ask questions: they're completely unafraid to "look dumb" in front of one another, and this fact often makes them quick and effective learners. In contrast, my early morning section is quiet and focused, different but equally bright.

I'm going to take a few minutes at the end of class tomorrow to do a one-third-of-the-way class survey, to get the goods on the students' point of view. We'll see how much my take on our classes matches up with theirs.

Also: too early to tell what'll happen for sure, but there are some big opportunities drifting my way. More to come as time permits...

Math...who needs it?

As a low-stakes exercise at the end of my first section of Precalc this morning I asked students to write down the most math-related activity in which they took part over the weekend. I got several quotidian responses ("I attempted to complete HW #5," "Chemistry HW," and the like) but I got some more exotic ones, too:

"I explained what a TI-83 was to a 6 yr old"

"The most math related thing I did this weekend was weigh the amount of almonds I wanted at Earthfare and calculate the price in my head."

"Calculated my availability for a client given all my meetings and classes that I have this week."

"I worked at the hospital all weekend, and at my job, I utilized math to calculate the number of calories and carbohydrates patients had been consuming."

"Averaging pace/mph/minutes per mile while running 1/2-marathon."

"I got kinda bored, so I figured out every time of the day where the angle of the clock hands were exactly the same as those of 4:00"

Fun times! We'll see what my second section can come up with.

On another topic, this afternoon I'm off to my fourth Carolinas Writing Program Administrators conference at the Wildacres Retreat Center just off the Blue Ridge Parkway. This year's theme is seeking external funding, and participants will spend a bit of time hammering out grant proposals as they bounce ideas off of one another. (Our current Writing Center director and I are planning some sort of regional writing-themed conference that would bring in not only university students and faculty but also K-12 educators and their students, and members of the community at large.) Of course, I'm sure I'll find time to work on my ongoing research with the College of Charleston crew (we're presenting this work at the Four Cs in March), and to cut loose with my comp-rhet buddies from all over the Carolinas.

Further bulletins as events warrant!

UPDATE!

Because I know you've all been on tenterhooks since my last post, here are some of the most interesting weekend uses of math from my second section:

"I built a table and had to measure where to cut. I also wired a dimmer switch which required me to use some math."

"The most mathematical thing that I did this weekend was to figure out how many CDs my band sold based upon how much money we had in our 'money jar' at $15 an album. Also in figuring how much we owe our bass player when paying her 22% revenue earned per gig."

"I was playing a power chord on a mandolin and my friend asked me, how do you do that? I told him it was the same as a power chord on a guitar but reflected over the origin."

"I was actually bitching last thursday about how I would probably never use the research from our homework ever, but this weekend I got into a discussion about Dow Jones & was proven wrong. I laughed and will no longer bitch. :)"

Progress! :)

Out of the wilderness

This morning I finished reviewing and responding to my MATH 461 class's homework set, the one featuring several very open-ended problems related to Fibonacci sequences, Euclid's Algorithm, and greatest common divisors. Their work was fantastic, and the sense I got from most students' solutions was that they'd achieved genuine understanding, something I honestly don't see present in most responses to the cut-and-dried prove-this-theorem sorts of questions one might ask in an upper-level mathematics course.

Moreover, the students made remarkable progress in proving some nontrivial mathematical results they themselves got to formulate. Several presented solid proofs of one direction of the equivalence I mentioned in my last post, and though no one successfully proved the converse (I was only able to do it myself last night, after several false starts), a few students made earnest attempts at so doing and a few finished just shy of the mark.

Furthermore, two of the students noticed that, though I'd not intended it, the fourth problem on the homework set had close ties to the previous three (all of which were similar). This last problem asked them to characterize the numbers which caused "worst-case" performance in Euclid's Algorithm when divided into 99. A bit of thought (after examining a mess of data) will convince you that the worst case is achieved when the numbers you select give the most "Fibonacci-like" sequence of quotients when divided into 99, numbers for which most of the "q" values stemming from Euclid's Algorithm are 1, so that the corresponding remainders remain as large as possible. Thus, these two students pointed out, Euclid's Algorithm should perform most poorly when you use to divide one Fibonacci term into its successor. One student even presented several pages of numerical evidence for this worst-case behavior, building off of the generalized Fibonacci sequences we'd just worked with above. It was splendid.

Finally, one student made an observation regarding the frequencies with which each "run time" occurred when Euclid's Algorithm is applied, noting that when plotted, these frequencies traced out a very normal-looking curve. "What might happen with other values for b, besides 99?" he asked. No doubt there's some nontrivial number theory lurking just below the surface: primality plays a role, for sure, and I'm sure Euler's φ comes into play.

All in all, I get the feeling that the students got far more out of this set of exercises than most (any?) I've ever assigned in my career. I'm going to see that all of my homework sets for the rest of the semester are in a similar vein.

Into the woods

It wasn't until I wrote yesterday's blog post that I realized the extent to which I'm pushing inquiry-based learning in both courses I'm teaching this term. In both Precalc and Abstract Algebra I, the majority of the homework problems students are being asked to complete are what can legitimately be called research problems, and I'm posing them as such, guiding the students through an initial "data collection" stage, leading them then to a "conjecturing" stage, and from here to a point where they should be ready to offer at least a partial proof. The questions I'm asking are very open-ended, and in a few cases already this semester I'm not even sure I know the answer.

Example: I've got the Abstract students making conjectures about the relative primeness of consecutive terms in generalized Fibonacci sequences: for what natural-number pairs (α,β) is it the case that any two consecutive terms in the sequence defined by s₀ = s₁ = 1, s_n = αs_n-1 + βs_n-2 are relatively prime? I admitted up front that I don't know the answer to this (though I have some guess as to what might be true), but I asked students to try out several cases, formulate a conjecture based on the data they gather, and try to prove their hypothesis.

What fun! I'm having fun, anyway. And what a way to learn! I have no doubt that the students are apt to become more talented mathematicians (and more generally, problem-solvers) when asked questions like this than when asked to complete cut-and-dried textbook proofs for which the answer is already told to them.

How NOT to graph a shifted function; or, The nature of mathematical research

It happens to us all at some point: we're confronted with a problem to which we simply don't know the answer. It's a problem never posed to us before, one we've never seen...it may not even look much like anything in our prior experience.

It's happened to me. Many times.

For example, for the past year or so I (and, off and on, several undergraduate researchers and a couple of colleagues) have been struggling to find answer to a seemingly simple question: where on Earth does the mode of the independence polynomial of an arbitrary 2-regular caterpillar lie?

Okay, so maybe it's not that simple of a question...but it's one that's resisted analysis of every kind we've attempted for well over a year.

However, undaunted, we have tried several different means of cracking this muthah. We've tried geometric methods, combinatorial methods, algebraic methods...even analytical methods. We've tried it all, to no ava...well, to some avail: we've learned a lot about the structure of the objects we're studying, and though we don't yet know what method will work to solve the problem, we can tell you several methods that won't work. Hey, we've tried.

And that's what matters: the fact that we've tried. In the end, it's okay to not know what the answer to a particular question is. After all, none of us are born with inherent knowledge of algebra and calculus and combinatorics: we're going to be asked questions the answers to which we simply don't know.

Put another way, ignorance is inevitable; what matters most is how we confront that ignorance. Inaction gets us nowhere. Action of any organized kind is preferable, and more preferable still is action of a sort our experience suggests will give us a means of responding to the problem we're posed. This kind of confrontation with ignorance is called learning...or even research.

Yes, research: it doesn't cheapen that lofty term at all to use it to refer to the simple actions we undertake when we, for example, try to graph a simple function we're unfamiliar with.

Allow me to demonstrate.

When confronted with a truly unfamiliar function, here's what not to do:


You can't be expected to be familiar with every function ever invented...there are too many of them! But don't just sit there! Don't let ignorance get you down! If you're not sure of what to do, maybe try something that's worked in the past, like...


Ah...now we've got some traction...a few more values...


...and we're starting to see results...now let's plot some points...


Huzzah!


Ignorance dispelled! Or at least held at bay for a bit. Congratulations: you've now learned something new. Put another way, you've completed a miniature research project. Seriously, you've just done research, applying known methods to approach an unknown problem. That's how you do it.

At the risk of sounding repetitive, let me exhort you once more: please, don't just sit there. It won't simply "come to you" if you're not doing anything at all, but it might if you try something out.

P.S.: photo credits go to my former student, and awesome stats major, Karl. Thanks, Karl!

Cause 'n' effect

I've already heard from several former students who are now in K-12 education that the scenario depicted by my friend Kyle is more than ever a reality. I've always known the high-stakes testing, NCLB, and all that nonsense were driving our public schools off of a cliff, but I don't think I was ever aware of just how bad it had become...and Buncombe County's schools are by no means among the nation's "worst."

What in the hell are we doing? I see two immediate and profoundly negative effects to this shortsightedness.

First, by drilling our students on "basic skills" (i.e., reading and math) to the exclusion of all else, we're ensuring that our students will grow to loathe those subjects more than they already do. Many but the most math-motivated students come out of high school hating math and reading little, and being unable to do more than solve pointless decontextualized machine-gradable math problems and extract topic sentences from unchallenging tendentious passages of "literature." This isn't an inevitable outcome: my experience in working with my department's Super Saturday "Math Discoveries" program has taught me that there are many bright 8- and 9-year-olds of all backgrounds who positively love math, and that if that love is nurtured they'll stay focused on it for a long time. Moreover, my experience in teaching Calc I, Calc II, and even Precalculus has taught me that it's not impossible to reawaken dormant affinities for math if it's presented in a way that makes it interesting, challenging, relevant, and fun. But drills and standardized tests are neither interesting nor challenging nor relevant, and they're certainly not fun.

Second, by neglecting less readily-assessable (or at least less readily-assessable-through-machine-graded-standardized-tests) subjects like social science, Earth science, literature, music, etc., we are ensuring that our students will have virtually no ability to contextualize whatever minimal understanding of reading and mathematics they're able to eke out of their loathsome experience with these latter subjects. Their knowledge of math will be disembodied, inapplicable to any other science. "Word problems" will always be "word problems," impenetrable blocks of quasi-mathematical lingo, and not opportunities to apply mathematical understanding in a meaningful setting. Worse yet, reading (and writing) will always be a soulless enterprise, undertaken in order to find a topic sentence, identify three (precisely three) salient pieces of evidence in support of that sentence, and pen a five-paragraph response containing both an introduction and a conclusion (god forbid you forget either).

I'm well aware that there are many more negative effects than these, but I've not got enough time to list them all here...nor am I the most well-trained to make this list. (I'm sure many of my recent grads who now teach in the public schools could do a better job than I could, and I hope they find their way to this post and chime in in the comments section.) I'll close by quoting a colleague/Facebook friend, who said on my link to yesterday's blog post "why is it getting so hot?...why am I in this hand basket"?

Rilly?

If you didn't already know it, it might tickle you to find out that I bowl on a pretty laid-back bowling league on Monday evenings, called the S.I.N. (Service Industry Night) League. The folks I bowl with are almost to a one friendly, caring, compassionate, and often very intelligent individuals. I never cease to be amazed by the warmth and wisdom these people show to and share with others. It's fair to say that S.I.N. League helps me maintain my sanity.

One of my teammates, Kyle, recently took a job working in one of Buncombe County's public schools, as a teachers' aide. He's loving his job, and it's giving him great insight on our nation's educational system. I love that his insight is untempered by educational lingo and unfiltered by official administrative fiats. From him, you get the dirt.

Kyle keeps a blog himself, a collection of biographical blurbs tangentially related to the webcomic he and his wife (also a teammate) write about the exploits of our bowling team (I'm Parker!). While checking the comic (updated every Monday!) this afternoon, I found the following interesting comment on Kyle's blog:

One somewhat shocking thing I learned is that our teachers have been told not to teach science and social studies because they don’t have enough time, and are primarily tested on reading and math. I had noticed that every class I went to was either doing reading or math, but I thought it was simply because I was there early in the day and maybe they did other things in the afternoon, but no! There is so much pressure from the state and county governments to preform better in math and reading that they simply skip over other subjects.

Rilly? I'd be interested in learning more about this phenomenon...I mean, I know that a good deal of emphasis is being placed on certain subjects because of high-stakes testing and other accountability measures, but this seems extreme. Are teachers (at least in the 2nd/3rd grade level where Kyle helps out) being discouraged from teaching science and social studies at all, or are they simply being told to put emphasis on reading and math? And to what extent is this discouragement/recommendation official? Is it an unwritten rule that's clearly understood, or is there a paper trail somewhere?

I'm a bit shocked by this.

Thoughts?

Fishbowl

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

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

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

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

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

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

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

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

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

[Redacted]

I wrote a brief post in which I offered up more explicit complaints than ever before about the pedagogical practices in one of my school's departments before thinking better of it.

Suffice to say, I was livid this morning on hearing from some of my students about the way one of their courses is being "taught."

Livid, and impotent: what can I do? (Bitch about it on my blog, apparently...)

Another CRTF-related post, in which I begin to sound more and more like an administrator

One of the issues we on the Curriculum Review Task Force are facing as we go about our business is the rather ill-defined identity of UNC Asheville as a university. We're finding it difficult to put a pin on our strengths, weaknesses, and other salient characteristics. What is it that best defines us, our commitment to undergraduate research? Our emphasis on service learning? Our small class sizes and opportunities for one-on-one teacher/student interaction? These all play a role, but it's becoming more and more evident that certain unintentional, often demographic, institutional characteristics shape our school more than these do.

Foremost among them is the predominance of transfer students at UNCA. Roughly half of those students graduated from UNC Asheville in any given year spent only half of their undergraduate careers at this school. Many transferred in after spending time at one or more regional community colleges, while others have bounced around from school to school, sometimes visiting as many as six or seven schools (in extreme cases) before coming here. These students often have more "real-world" experience than their peers who matriculated at UNCA as first-year students, and in any case they almost invariably contribute rich multifaceted perspectives to every class they attend. (In my experience many UNCA students with a community college background are "nontraditional" students, returning to school after several years in the workforce and several years of growing up. These students are mature, organized, diligent, and earnest about their learning. Even when they're not as "book smart" as some of the kids fresh out of high school, these other strengths more than make up for that shortcoming and make them fantastic students.)

Moreover, these students' presence has a profound impact on the structure of our curriculum. Consider, for instance, students' completion of the ILS requirements. A number of those students coming in from community colleges may arrive having completed the 44-hour core, an articulation agreement that suffices to knock out most ILS requirements; many other such transfer students, as well as those who've bounced around from school to school before landing at UNCA, have not completed the core and may arrive with little more than a long list of elective courses, numerous ILS core requirements outstanding. These students may therefore end up having to take three or four years at UNCA to complete their degree requirements, despite having spent two or three years elsewhere, and once done they may have taken as many 160, 170, or more credit hours, well more than the 140 beyond which tuition costs half again as much. (Every hour past 140 hours costs 150% the base tuition rate.)

Thus these students take a disproportionately long time to graduate, pay more for the privilege of enrollment, and often fail to see the point of the school's liberal arts mission in the first place. To this last point: many of these students come to UNCA because it's relatively cheap, it's convenient (Asheville's mild climate, high quality of life, and welcoming atmosphere make it a destination for many), and they believe it will provide them with the skills needed to fulfill the functions of whatever career they aspire to. Few transfer students come here specifically because of the interdisciplinary learning experience UNCA purports to offer through its liberal arts curriculum. More often, such students are annoyed by the burdensome requirements that curriculum imposes upon them. "I just want to take my major courses and get out of here."

Surely these students can't help but feel like suckers, stuck playing a game in which every hand is stacked against them. Yet, as I mentioned above, these students make up, and, indeed, for a long time have made up, a large percentage of our undergraduate population. (In each of the AYs 1995-6, 1998-9, 2004-5, and 2005-6, they accounted for a larger percentage of the graduating classes than "native" students did.) This isn't likely to change, given our status as a relatively low-cost provider of a fairly high-quality education.

How, then, can we restructure our curriculum to avoid treating these students like second-class citizens? For one thing, we should ensure that whatever core requirements we impose, they can be efficiently and effectively completed by all students, without sacrificing their goal of providing an authentic interdisciplinary learning experience. We should also take care to structure our major and degree programs flexibly, permitting students to fulfill these programs' requirements in a timely fashion and not get mired down by labyrinthine class sequences in which key courses are offered no more than once every year or every other year.

These are, in fact, among the principles which have been guiding our work on CRTF this summer. Time will tell if we succeed.

Students, what's your take? We've solicited woefully little input from students as we've begun our review, and I'm curious to know what you all see as the strengths and shortcomings of UNCA's curriculum. Feel free to comment.

Nudge

I've mentioned a few times (see here) the ongoing work of UNC Asheville's relatively new Curriculum Review Task Force, on which I'm serving as "point person" for the Curricular Sustainability Subgroup. We've spent the summer reviewing the structure of our own curriculum, breaking it down into little bits and comparing those bits with the corresponding bits of other schools' curricula.

The first component we examined were the majors and degree programs, writ large. After finding data that suggest these programs are considerably more prescriptive (in terms of the specificity of courses required) and burdensome (in terms of the number of courses required), we compared several of the most prescriptive and burdensome programs with cognate programs at several peer institutions, including both a few other COPLAC schools and a few institutions which serve as "role models" in one way or another (Bates College and Davidson College, among others). Our initial findings were confirmed: with a profusion of highly-specialized concentrations whose completion requires, on average, a relatively hefty amount of coursework, our programs are byzantine beasts, looking more like miniature graduate programs in some regards than the sort of programs you'd expect to find at a liberal arts institution. Notably, these findings suggest that those who place the blame for high times-to-graduation on the ILS program (of whom there are many at UNCA, especially among the old guard) are at least in part misguided; the majors are just as much to blame as the general education program.

We've just begun to examine the ILS program, drawing on data produced over the last several years (from NSSE, from perceptions surveys conducted on our faculty, etc.) in order to understand what parts of the program are working and what parts need work. Although there's more data-digging to be done, my hunch is that we're merely going to end up corroborating all of the findings made in the past few years: the intensive programs are doing quite well, and though the Humanities program, the first-year and transfer seminars, and a few other less-controversial components of ILS may need a little tweaking, it's the ILS clusters that'll need to be overhauled.

Several weeks ago my colleague Tyrone, a member of my Curricular Sustainability Subgroup, noted in a meeting that we may do well to consider our curricular issues from the point of view of "choice architecture," the art of arranging options for people in ways that help them make decisions more easily and to greater benefit. To learn more, Tyrone recommended the book Nudge: Improving decisions about health, wealth, and happiness by Richard J. Thaler and Cass R. Sunstein (New Haven, CT: Yale University Press, 2008), which I picked up at our library a couple of days back. I'm about halfway through, and so far it's proving an interesting and enlightening read. It's full of the sort of observation that makes one think "well, duh, that's obvious" about something one likely wouldn't have thought of oneself. Example: how dumb is it to arrange the range elements in a square on a stovetop, while the knobs that control them are almost invariably organized in a row (pp. 83-4)? A better choice architecture would arrange to place both the elements and their respective knobs in similar geometric configurations.

To the matter of ILS clusters, how might close attention to choice architecture help us craft a more easily-navigable system of courses which still ensures that students enjoy a meaningfully integrative interdisciplinary learning experience? Right out of the book, we might keep in mind the following points:

We should expect students to err in their choices, and design our system so that those errors won't prove catastrophic. Example: the cluster program should be flexible enough for students to plug in a single missing course at the last minute if they've let it slip their minds until then.

We should give students feedback on the choices they've made in completing their course requirements. Example: our advising software DegPar goes a long way in this regard, letting students know what requirements remain to be completed at any given time. If we move toward a system in which students design their own interdisciplinary clusters (as we've discussed doing several times now), perhaps we can require students to craft reflective essays after each prospective cluster course is completed, ensuring that they understand the ways in which the courses interrelate.

We should help students produce "mappings" which help students translate the complex benefits of their choices into more easily understood terms. Example: in a sense, we do this already by presenting several prepackaged cluster options to our students; "complete any one of these clusters," we tell them, "and you're apt to receive a meaningful integrative learning experience." For instance, the title "The Science and Politics of Human Health and Illness" makes it clear that the contents of the cluster so named will give students a multifaceted perspective on issues related to health and wellness: though the courses may be complex and challenging, the cluster's name is an easy index that guides interested students to complete that particular cluster without having first to complete every course the cluster contains.

More to come on this matter. I do recommend Nudge; it's an easy read, but a useful one.

Back

We're back. It's been a while.

While the REU seemed to eat up less of my time this summer, revisions on the book (to appear early in 2012 under the title Student writing in the quantitative disciplines: a guide for college faculty) and work on the Curriculum Review Task Force seemed to take up every last bit of whatever was left.

We're now four days into the Fall 2012 semester, and I already feel as though I've found a groove in Precalculus. I've not taught this course for three years, and I must say that I've been looking forward to teaching it again. I've thought a bit about how I would approach the course, I've come up with some new activities (like this one), and I'm coming at it with renewed energy. So far the class has been great. (It doesn't hurt that the department's choice of text is not catastrophically awful, like the text we'd adopted the last time I taught that course.)

We spent today motivating relations and functions, and I ended class with a low-stakes writing exercise (who, me?) asking the students to work in small groups to come up with several examples of relations or functions which have real-world relevance, expressed as "pairings" between sets of numbers. They came up with some fantastic ones, some of which could the basis for interesting statistical surveys. A sampling (all verbatim):

• The decrease of the temperature paired with the increase of the elevation
• The number of texts you send paired with the time spent on your phone
• Pair the childhood obesity with each child's level of poverty
• The profit of the lemonade stand paired with the amount of sugar used
• The speed limit of an area paired with the number of car crashes in the area
• The amount of wildlife disturbances compared to the average of the new developments
• Pair the number of baseball ticket sales with the baseball team's winning record
• Pair the profit made by jacket companies based on temperature

I'm stoked. Both sections of this class seem to be gelling already, and they're full of outgoing individuals. Already obvious "leaders" are emerging from among the ranks of those students who are unafraid to ask questions, suggest answers, and put solutions on the board before the whole class.

Abstract Algebra has yet to get into the same groove, but as yet we've only met twice, and yesterday's class meeting was dedicated to an intentionally chaotic consideration of a boatload of multiplication tables I'd asked them to construct. In asking them to analyze and explain the patterns these tables exhibit, I'm leading them to begin thinking about what salient features the most "well-structured" algebraic objects (sets equipped with a binary operation) might possess. We'll make that more explicit tomorrow when we define monoids, groups, and semigroups.

More to come soon, I promise! On CRTF (oy), on the QEP (oy oy), on many more things...