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!

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!

Author! Author!

Every person is the author of her own adventures.

This is a point I try to make to all of the students in all of my courses, in which I downplay my own authority and up-play the students'. "I've got no more claim to the truth than you do. The only difference between you and me is that I've been doing it for a few more years."

It's a point I've tried to make to my colleagues, most recently this afternoon at yet another CRTF meeting. This one was a meeting of the "Big Picture" Subgroup, at which the leaders of the other subgroups (including yours truly) were asked to make presentations on our ongoing work. I had a bit to say about our review of department responses to our "information request," and about our intended review of various ILS components.

I hope that our review will be guided by a handful of basic principles:

1. Our curriculum will function most efficiently and effectively when ILS learning outcomes and departmental learning outcomes (and the means of achieving those outcomes) are brought into fullest alignment.

2. Our curriculum will be most sustainable when the resource demands it places on faculty, staff, and students are minimized.

3. Our curriculum will offer the most rich and most meaningful learning opportunities to our students when they are allowed to plan and pursue their own courses of study, navigating course requirements that are rigorous but flexible.

This last principle places a high value on non-prescriptive curricula, featuring both general education programs and degree programs with relatively few specific requirements...programs that ask the students to play an active role in putting their own academic houses in order. I don't feel that our current curriculum features such programs.

The other day, in a hall conversation with a colleague, I referred to our role in the current system as "helicopter professors": our requirements are structured in such a way that our students' academic careers are micromanaged stringently. Students are tended to carefully, led from year to year in flocks, protected and prepared (for graduate study or real-world employment), but rarely challenged to set out on their own. Based on analysis of student behavior over the past several years, the Research and Evaluation Subgroup of CRTF discovered that only 18.5% of the courses our students take count as "free electives," taken for no purpose beyond academic exploration (such courses satisfy neither major nor ILS requirements). All other courses, all but little more than a semester, go toward putting a check in some bureaucrat's box.

"We need to make sure that our students who want to do graduate work are at least well enough prepared to get into a decent masters program," one of my department colleagues insisted at tonight's meeting. I agree, wholeheartedly. But I disagree with the means he suggests we must use to get them there. Many of our peer institutions offer much more flexible programs, with far fewer explicit course requirements, and still manage to send higher percentages of their graduates into prestigious programs. (The fact that this friend of mine is shortsightedly using graduate school enrollment as the be-all-end-all measure of an academic program's success is a topic for another post...)

More important, students completing more self-directed courses of study gain authority over their own actions. They grow in competence and confidence as they're asked to take on more responsibility for their own lives. They mature more quickly. They learn how to ask and answer important questions concerning their coursework and their careers. Forced to connect the dots for themselves, they become more authentic experts in their own disciplines.

This isn't to say we shouldn't offer our students some kind of guidance: nothing can supplant informed academic advising. Good advising can take the place of stringent requirements. If a student should wish to pursue graduate study, she should be encouraged to take courses that will most well prepare her for that study. If she fails to follow up on the advice her professors give her, she might be sunk...but she might not. She may succeed in her ambitions, but even if she doesn't...so what? Even if she doesn't end up where she'd originally set out to be, she's had a chance to plot her own path in the meantime, learning from whatever mistakes she's made on the way. Life is what it is, and each of us is who each of us is.

In my second section of Precalc (and again in my Abstract Algebra class), I read an excerpt from Rainer Maria Rilke's 6th letter to the poet Franz Kappus (Letters To A Young Poet, translated by Joan M. Burnham, Novato, CA: New World Library, 1992, pp. 53-55):

You should not be without a greeting from me at Christmastime, when in the midst of festivities your feeling of aloneness is apt to weigh more heavily upon you. Whenever you notice that it looms large, then be glad about it. For what would aloneness be, you ask yourself, if it did not possess greatness? There exists only one aloneness, and it is great, and it is not easy to bear. To nearly everyone come those hours that we would gladly exchange for any cheap or even the most banal camaraderie, for even the slightest inclination to choose the second-best or the most unworthy thing. But perhaps it is exactly in those hours when aloneness can flourish. Its growth is painful as the growing up of a young boy and sad as the emergence of springtime....Think, dear friend, reflect on the world that you carry within yourself. And name this thinking what you wish. It might be recollections of your childhood or yearning for your own future. Just be sure that you observe carefully what wells up within you and place that above everything that you notice around you. Your innermost happening is worth all your love. You must somehow work on that.

Let us reflect, my friends. What is it you find within yourself? How can you make your life your own?

Graph Theory: Day 2

As the snow storm descends on the Asheville area, I'll take a moment to briefly chronicle this afternoon's mathematical goings-on.

I felt a bit out-of-step in my first section of Calc II today. I never really got into my stride, somehow, and I felt awkward. The awkwardness carried over into the second section, with whom I felt more at ease, but still stretched thin. I'm looking forward to Friday in both of those sections, I'll be leaving much of the work up to them. Then Monday will bring the first of several food-based exercises, always favorites with the students.

These two classes were more than made up for by Graph Theory.

Right away the atmosphere was a positive one: before class, as people were still trickling into the classroom in dribs and drabs, everyone was chatty, jovial, open. The students joked, compared solutions. Everyone seemed relaxed, ready. I put some colored chalk on the front table and went to the side board, where I wrote "Correctness / Completeness / Clarity / Composition," urging the students to intone these words as a mantra as they prepared their presentations.

Then we began.

Things went well from the start: when called, each student took to the board to the sound of applause from her or his colleagues. Everyone was quiet and respectful during presentations, and each success was met by another round of applause and cheers.

The first few presentations went smoothly; it was Problem 4 that caused a bit of hullabaloo.

"Problem 4. Draw as many fundamentally different graphs as you can, each having order 4 and size 3, also writing each as a triple."

Its the fourth and fifth words here that brought down the house: there was (understandably! I'd somewhat hoped that this problem would provoke a discussion) a great deal of disagreement regarding what was meant by "fundamentally different"; it'll be another week, at least, before we define graph isomorphism. (Brigitte actually said a few words about "bijections" that were very close to the mark, but her quiet voice didn't carry so well amidst the hubbub.) The chimerical nature of this phrase, coupled with the immense number of graphs having the properties desired, led to uproar. Poor Joachim, attempting to answer the problem as fully as he could, was interrupted by a chorus of overly helpful classmates: everyone wanted a piece of the problem, and the next ten minutes were spent in taking unruly turns at trying to pin down the meaning of those elusive words, "fundamentally different."

Ultimately it became clear that we all had more or less the same idea as to what those words meant.

The discussion was lively, even heated, but ever respectful and supportive: no one attacked anyone else, corrections were friendly ones, and even when there was disagreement, the disagreement was civilly made.

The next three problems were relatively humdrum; Problem 8 caused a bit more furor, though without the controversy attending Problem 4. Quincy was called on the complete Problem 8 (asking for an enumeration of the maximal number of edges in an order-n graph without multiple edges), and he offered a nearly-complete proof of his (correct) formula, the sum 1 + 2 + 3 + ... + n.

"Did anyone have a different proof?" I asked. Sylvester offered that he did, and he went to the board to provide an inductive proof of his (equally valid) formula, C_n,2 + C_n,1. Throughout both presentations, everyone was quiet, attentive. Sylvester's proof brought us to the end of the period, midway through the first problem sheet.

Afterward Quincy characterized the mood of the class as "fun, but serious." "We all mean business, we're taking it very seriously," he said. "But we're having a good time with it." He had a blast, as did his friend Norbert, and as did Nadia, who spent some time after class trying vainly to convince Olivia to join our class.

I am positively delighted with the way class came off today: the students took control. They constructed their own mathematical meaning while engaging in lively, sincere debate about deep mathematical issues. If we can replicate today's success over and over again for the next several dozen class periods, I'm going to end this semester as the happiest man on Earth (not that I don't already hold claim to that title).

I'm already looking forward to Friday.

I'm also looking forward to tomorrow: barring too-hellish weather, I'll be trudging into campus to fulfill a number of bureaucratic commitments, and to meet with Sieglinde and Trixie, my budding freshperson graph theory research team. Trixie's progress on the problems I pitched her over break has been nothing short of astounding: I met with her yesterday and she showed me the pictorial essence of the results she's come up with, and they look solid. Sieglinde's indicated progress too, and I can't wait to see what she's got in store. They're both sharp are tacks and a kick to work with.

On that note, it is wearily but happily that I bid you a good night, I'm off to do some relaxing reading before calling it a day. Adieu!

Write or wrong

Hey, hey, hey! It's a brand new year, folks!

So far this year I've done little related to my teaching, I've spent most of my time reading (gasp!) for pleasure. I've capped off seven books in the last two weeks...golly, it's nice to have free time. This morning, for instance, I finished Wangari Maathai's Unbowed: a memoir, a recounting of her life in Kenya and the founding of the Green Belt Movement, the organization primarily responsible for her receipt of the 2004 Nobel Peace Prize. Before that it was a collection of short stories by Guy de Maupassant, Georges Bernanos's The diary of a country priest, John Griffin's classic Black like me, and a pair of books by Jonathan Kozol and Kurt Vonnegut, reviewer elsewhere in this blog. It's been great to be free to read again, something I'm sadly unable to do much of during the school year.

I've also been mulling over what I'd like to accomplish through my teaching during the coming year.

Last year could be characterized by consciousness: I believe that more than anything else I learned to become fully conscious of my pedagogical efforts, and cognizant of the effect my deliberate actions would have on my students. I made conscious efforts to structure my assignments developmentally, to engage students in meaningful, conscious discovery. I believe that my conscious focus paid off, I feel as though my 280 class benefitted enormously, for instance, and the effort that went into Newton v. Leibniz was repaid tenfold by the students' growth through the project. (By the way, I heard back from Prof. Bornstein, she was delighted to hear from me, and wrote me a wonderful letter on her own ideas on teaching. She looks forward, as do I, to continued correspondence. I need to write back to her...)

So what is it that will characterize my teaching in the coming year?

Discovery, perhaps? That will certainly be a central theme of my upcoming graph theory course, in which I'll be challenging students to rebuild the discipline from scratch.

Or maybe authority? Might I focus my energy on encouraging my students to take the reins in their own studies, to ask the questions that need to be asked, to take responsibility for their own futures?

The line between these broad territories is an unclear one. I look forward to seeing how my classes take shape in the coming weeks.

At present I'm ready for Day One (now a week and a half away, on Monday, January 14th), freshly printed syllabi, worksheets, problem sets, and project outlines covering my desk. All I've got to do now is get some Skittles for the candy machine, in order to be ready for the third installment of Calc II's Confectionary Conundrum, an exercise whose execution I've now got down to an artform.

Good news came yesterday in the form of an e-mail from Texas: both my individual presentation and the panel presentation I'm putting together with a couple of my UNCA colleagues (one from the Writing Center and a second from Sociology) were accepted by the organizers for May's 9th International Writing Across the Curriculum Conference at UT-Austin! This is exciting. It'll be the first time I'll have had a chance to speak at a non-math-related conference, about a subject that's quickly becoming my second specialty (writing in the mathematics curriculum). In my individual presentation I'll be talking about the use of the "homework committees" and other structured peer-review exercises to encourage student self- and peer-assessment and self-authorship. Our panel will discuss the ways in which discipline-specific writing is taught, nurtured, and evaluated in the liberal arts setting. My portion of that program will invite non-mathematicians into the world of mathematical writing, indicating the similarities between math writing and writing in other disciplines. By highlighting the grammatical structures, syntactical rules, stylistic conventions, and assessment criteria that characterize mathematical writing and by comparing these aspects with corresponding aspects of writing elsewhere, I hope to dispel the notion that math writing must be an alien enterprise to non-mathematicians.

This is going to be an exciting conference.

Meanwhile I'm only a couple of days away from departing for San Diego, site of my fifth Joint Mathematical Meetings. A lot going on there (judging an UG poster session, presenting in the expander graphs and Ramanujan graphs special session, glad-handing every mother-lovin' person I can find to drum up support for my REU), but I'm already looking beyond it to May, bringing not only the Writing conference but also my next graph theory conference, to which I hope to drag a few students. (One of my freshpeople is making great progress on graceful labelings over this break! I told her to expect me to try to get her to go to this conference in May. More on that as events warrant...)

Bile and bluster

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

In retrospect, perhaps I was a bit harsh.

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

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

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

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

And:

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

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

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

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

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

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

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

Now?

I.

Am.

Tired.

Well...

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

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

Except maybe Matt Ryan.

But that's a different story.

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

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

Stay tuned!

Long, long, long

As I'd suspected would be the case, the past week and a half or so has been absotively, posilutely insane, and I've hardly had a chance to keep on top of each day's work as it's come due. Between travel and teaching, City Council meetings, grading, grading, and a little bit more grading, Learning Circles, research, and reappointment shit, it's been a rough one. Ergo, no posts for a bit now. Many apologies, etc. Today's the first day I've not felt torn in several directions at once; once or twice this morning I actually had a chance to sit at my desk and ponder my next task before instinctively setting to it.

So what's up?

The first draft of the 280 students' Professional Proof Analyses, in which I asked them to apply our course rubric for superlative mathematical writing to three proofs of the same theorem, each written by three different authors, were fantastic. The students raised excellent points, made perspicacious observations, dug deeply into the "Four Cs" of the rubric (Correctness, Completeness, Clarity, and Composition) and applied these principles consistently and clearly. By one means or another most of them accounted for variable audiences, the evolution of exposition through time, the difficulties entailed in the analysis of a single proof extracted from within the context of the entire textbook, the subtle epistemological differences between putting a proof before a proposition's statement and the converse configuration, and so forth. And these were just the drafts! I was able to honestly say as I handed them back that those papers were among the strongest mathematical writing I've yet seen in any of the classes I've ever taught. How much of this is due to the students' inherent skill in constructing well-thought-out essays, how much to the clarity with which the assignment was designed and implemented, and how much to the fact that I feel I'm a much better teacher of mathematical writing than I've ever been before, is hard to say. I like to think it's a combination of all of the above.

I'll be polling the class more formally on Monday after the final drafts are handed in (this will be one of the assignments collected for the purposes of the Writing Assessment study, incidentally), but preliminary estimates show that the proof of the second part of the Fundamental Theorem of Calculus appearing in Hass, Weir, and Thomas's latest edition came out on top, beating Stewart's 2nd edition out in terms of completeness and composition, both of them beating out Abraham Schwartz's 1967 treatise that made use of outdated and relatively unfamiliar notation and awkward terminology. There was some dissent on this point, though: a few of the class's strongest students argued that in terms of completeness and composition, Schwartz's proof had the others beat. I believe the students' sense of completeness might come from Schwartz's explicit construction of Riemann sums; the other authors hide most of the messy details inside references to other theorems and corollaries contained elsewhere in the text, and so might feel a bit more scanty than Schwartz.

Class itself has had its ups and downs during the past couple of weeks. Last Friday I was in no mood to talk about permutations, so after a few minutes going over suggested corrections on the most recently-graded homework, I gave an impromptu lecture on Russell's paradox, proper classes, the infinity of infinities, and the cardinality of the reals. The topics are engaging, the students asked fantastic, insightful questions, and we all had a good time: that's how I wish every class could be. Quincy suggested that perhaps I should try to get my Chair to allow me to teach a "Random Seminar," in which topics are drawn from a fishbowl at the room's center, and teacher and students together spend a few weeks digging into the topic so chosen, convening as needed to fill each other in on the details. Sounds like fun, but it would be require an incredible amount of work on the parts of both the teacher and the students, and it would take some tweaking before it would fly.

The past week saw us slog through the remainder of our work on combinatorics, leaving us ready to tackle relations. On Wednesday equivalence relations proved a bit dodgy for the students, so I took some time yesterday to make up an additional handout that dealt more concretely with equivalence relations, asking the students to construct explicit examples of relations with certain properties, on small sets. I took several of the students aside after class, one at a time, and asked them if they felt the worksheet helped ground their understanding, and the consensus was that yes, it did. I'm glad.

Calculus, meanwhile, has been a hoot, but with the conference I attended all of last weekend, I've felt out-of-whack with regard to that class. I wasn't able to grade last Friday's homework over the weekend, as I nearly always do, so I didn't get it back to them until Wednesday this week, and that's made me feel as though I'm a bit behind. (Likely, the students couldn't give a rat's patoot.) I do feel more on top of things now that I've had a chance to catch up...just in time for this weekend's homework. Huzzah!

We've been talking about related rates, an ever-vexing topic that never fails to confound student understanding at first. The last couple of days we've been talking about exponential and logarithmic models, including the semilogarithmic model for network growth that my colleague and I came up with this summer. (I hold out hopes that by infusing my teaching with my research and vice versa I might catch a few students early in the game and entice them into considering a Math major...it might be working: Tallulah seems open to the idea of undertaking a little undergraduate research soon.)

The most exciting events concerning Calc I have to do with the Newton v. Leibniz project. Role proposals were due on Wednesday, and every one of them made it across my desk before zero hour. The proposals were...entertaining. Some were quite formal, serious pieces of persuasive writing, offering solid arguments for why I should make one appointment over another. Others were simply silly. I had fun reading them. In assigning roles I attempted to balance the strength of the individual proposals with the cumulative "happiness" of the class as measured by the number of teams receiving their respective top choice. In Section 1 I was able to grant five of the eight teams their top choice for roles, while only three of the eight teams got their number one choice in Section 3. There were a few long faces in that section, I know a couple of the teams had high hopes for their first bid and had only half-heartedly lobbied for their second. On the other hand, there was genuine excitement on some people's parts in both classes. I think enough students are going to get something truly meaningful out of this project that it'll be worth the trouble it's taking to organize it.

What else? The conference in Charleston was a conference. I learned a bit, met some new people, managed to insinuate myself a bit more deeply into the graph theory and combinatorics community. (Today I was offered a chance to referee some of the papers for the proceedings, to appear next year in the Journal of Combinatorial Mathematics and Combinatorial Computing. Exciting stuff!) I came back with several interesting problems to think about, including one that I pitched to one of our brightest junior majors almost as soon as I got home. He picked up on the general idea almost immediately and is already working on the problem I gave him.

Ummm...what else? Hmmm...yesterday I finished my reappointment binder and got it in to my Chair a week before it's due. I'm happy with it. I tried to cut my "candidate's statement" down a bit, but I really wasn't sure how to. I'm certain I included more documentation than was necessary: at roughly 75 pages, it's probably about three times as long as it needs to be, but I don't do anything halfway (or a third of the way, for that matter).

What else? Hmmmm...

...I'm sorry, I'm really tired right now, and should probably get to bed.

I'll try to post again tomorrow, as I really do have many thoughts I'd like to commit to paper (well...to...whatever passes for paper this millennium) before they escape me for all time: Fabian's astute observation that a many-authored proof might more quickly than a solo effort reach a sound and stable equilibrium, my conversation with Tallulah about a student's authority to assess the rightness and wrongness of a mathematical computation, and so forth. But sleep would do me well tonight, if I'm to survive the onslaught of the Super Saturday kiddies tomorrow morning. I'll likely have several stalwart students by my side to help me out, but no such Saturday goes by without my renewed appreciation for the role played by our nation's middle school teachers.

Until tomorrow, then!

Random thoughts

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

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

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

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

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

This week in college mathematics

Where to begin?

The week got off to a rough start with an uncharacteristically stern lecture on my part to my Calc I students. (Musta worked: their homework for this past week, graded this morning, was far more complete and correct. Well done, y'all!) They came back that night for a pleasant review session in preparation for a relatively tough test I'd dish out to them on Thursday. I always enjoy review sessions: the students are awake, receptive, responsive. I wish students would bring the same eagerness and vigor to class as they do to those evening sessions. They finished off the exam with an overall class average of 75.9%; one student nailed it with a clean 100%, and there were several other As. With revisions (I told them to think of the in-class exam as a first draft), they've got the chance to bring the class average up to 84%, and I promised a 3% cherry on top of that if they manage to make it within a couple percentage points of that goal.

I spent a good deal of time (most of it while running) thinking about how I'm going to put Calc I together the next time it falls to me to teach it. I've already created almost all of the resources I'd need to make the course decidedly more student-centered: the plan would be to pare down and tweak the existing class notes to the point where they could be used as effective worksheets for the students to complete outside of class and present to one another in class, much as I'm currently doing with 280. I'd break further away from the text than I have already by eliminating all textbook homework problems (they'd instead be "recommended" as practice problems, alongside illustrative examples from the relevant sections of the text) and replacing them with problem-centered applications worksheets (similar to the first team project I've just passed out to my current classes) which would be handed out and collected on a weekly basis. Completing these worksheets would require students to master all of the concepts discussed in class during the previous week, and would force the students to integrate content with application and to produce realistic technical writing. Exams and quizzes (both individual and team) would continue as at present. Classes would be focussed on student presentations and student-led discussions. With the exception of a handful of appropriate weekly projects, I've got most of the materials made up, the transition wouldn't be too hard for me. I'm ready; it all comes down to one issue.

Class size.

To realistically expect freshpeople to speak up and participate in class to the extent that this course scheme would require, I'd need to establish a relatively small and tight-knit community of learners; this semester has reminded me just how difficult that task is when working with a class of 30 students.

Further bulletins as events warrant.

And then there's 280, with another round of committee reports. They did a bang-up job on Wednesday, raising a number of crucial issues, including appropriate choice of notation, simple vs. short in the context of proofs, writing for a given audience, and the fact that there may be more than one way to skin a mathematical cat. So far I've been impressed with how well the committees have appeared to work. Beyond the great conversations we've had in class, I've no doubt that homework has been made immeasurably stronger as a result of input from the committee members. That's my take on things, and I encourage any of my students to give me their side of the story: how are things going on behind the scenes, folks?

All in all, I'd characterize this semester's 280 class as a friendlier place to work than last semester's was. Not only have the students had little problem in communicating with each other, they've shown willingness to speak up and let me know when I'm full of it, too. This past Wednesday a handful of them objected, quite openly and strenuously, to the way in which I'd worded one of the examples in a worksheet, and sure enough, a subtle semantic oversight I'd made in designing the sheet last Spring came to the fore, and I was forced to change it before we reconvened on Friday. Bravo!

We'll be continuing with set theory for the next couple of meetings, and by the end of the week should be making our way into the realm of relations. I'm eager to see how they handle the first take-home exam, due this coming Wednesday.

Finally, I ought to mention that the first meeting of this semester's Learning Circle, on self-authorship, went down this Monday. I have a good feeling about this group, we had a great discussion concerning the basic idea of self-authorship, and how it fits into our various philosophies. In particular, I mentioned that I appreciate (among other effects) the way in which the concept of self-authorship effectively displaces "content ownership"; my colleague Thibault from the Drama Department concurred and added that he's happy to say farewell to the term "development," a word that simply connotes passivity, as though students just happen to turn into learners, magically, mysteriously. (One of the contributors to Meszaros's volume addresses this mistaken view of development.) I'll likely have more to say on these issues as we continue to meet.

Well, it's after midnight, the Badgers have just beaten back the Hawkeyes (on, Wisconsin!), and it's time for me to head for bed. Until next time...

Author! Author!

Nearly three weeks in, and still going strong!

The past week has been a good one, and not just for the vacationlet in Virginia Beach, where after the half-marathon Maggie and I and friend/ex-student Mariposa (now teaching middle school in Fredericksburg, VA) hit a local pizzeria called Pi-zzeria, whose theme is the letter pi and from whom I bought a wickedly cool shirt with a pi on the front. I've brought a few fun activities into all of my classes, and I'm particularly happy with some new ideas I've incorporated into Calc I.

There, last week, we pieced together a mathematical jigsaw puzzle, an exercise I thought up on my way into campus that morning. Here's the recipe:

1. Print out a somewhat familiar picture (I used Mona Lisa for one section, and a detail from the ceiling of the Sistine Chapel for the other).
2. Subdivide another sheet of paper into a number of rectangles, grouped in fours, equal in total number to the number of students in your class.
3. In each rectangle so created, write an unreduced expression involving exponents and logarithms, in such a manner that the group of four rectangles contained in any given "region" of the paper holds equal values.
4. Photocopy the picture onto the backside of the grid you've just created.
5. Cut the rectangles apart from one another and shuffle 'em up.
6. Distribute them to the students, and let 'em assemble the picture by first piecing together the local regions with similar values, taping these together, and then fitting these regions into one another.

Of course, the exercise can be modified to provide a review activity for just about any concept you can imagine: compositions of functions, derivatives, integrals, you name it.

The first section took about 9 minutes and change to put their picture together, while the second section came in around 9 minutes.

That was last Thursday. Then they had their first team quiz on Friday, and everyone did very well (between the two sections only a couple of teams missed a perfect score, and even those got 4/5). For the quiz I gave them a problem which likely would have been rather hard for my Calc I students from last semester, and these kids just ate it up. I can tell I'm going to have to challenge these folks with some tougher open-ended problems. From what I could tell, most of the teams collaborated smoothly, too: as I walked around the room, I heard a good deal of explaining, cooperating, clarifying. I don't think there are any truly indomitable personalities in either section. (I do have to say, though, that one student, Tallulah, did mention that she was a bit disgusted with the nattering negativity coming from a pair of her peers in class the other day. I hope this was just a blip on the radar, not to be repeated. I'm doing all I can to create a classroom environment in which people can feel free to pose possible solutions to the problems we discuss, even if they're not entirely sure of their answers; careless critiquing of those brave enough to venture such solutions is hardly appropriate. I don't know of whom Tallulah was speaking, but if you're reading this and you recognize your own behavior, shame, shame!)

What else? Yesterday towards the end of class I asked each student to provide me with a pair of topics discussed so far in class, one of which she or he understands thoroughly and a second on which she or he feels fuzzy. I took some time last night to match each person up with someone else from the class, pairing people off who expressed the same uncertainties in understanding: two folks who felt iffy on inverse functions might have gotten grouped together, or two who reported feeling lost with logarithms. For next Friday I'm asking the pairs of people so matched to work together to construct a dialogue in which they help one another through their mutual difficulties with the topic with which they both expressed confusion. My hope is that in addition to understanding the relevant mathematical concept more clearly, they'll all uncover something about their own learning styles as they examine what it is they're unsure about. Moreover, hey, it's a great way to get them to do a little writing. (Boy, I am the WAC nerd, aren't I? Speaking of which, I've still gotta finish up an abstract for Austin...)

Finally, before and after class yesterday I approached the students who had done particularly well on the most recent homework sets and asked each if he or she wouldn't mind sending me a short e-mail indicating the way he or she completes the homework assignments, in the hopes that I can glean from the ensuing comments some helpful hints I might compile in a handout to give to these students' colleagues who are having more difficulty with the work. I hope they might answer questions like: what do you do when you do your homework? Do you work alone, together with friends, in the Math Lab? Do you make use of a solutions manual? How do you use it, if you do? Is there a way you approach certain problems, a particular way of viewing them? Do you have any specific techniques you recommend, tips for your peers? I didn't ask these questions specifically, hoping to receive unprompted and unfiltered responses. So far I've heard from Magdalena and Xavierina (whose homework, by the way, is some of the most beautiful I've ever seen: it's clearly written, organized, well-documented with an appropriate amount of work shown, and almost entirely correct; Xavierina, if you're reading this, kudos!), and I've had hallway conversations with a few of the others who promise to send me their comments soon.

Meanwhile, there's 280. I'm still having a bit of ball in that class. As well as last Spring's 280 was received, I feel better still about this most recent installment. I can't put my finger on it, but I feel there's a healthier dynamic in this group of students than there was last semester. It almost feels as though the class is significantly smaller, even though there are only two fewer students now than a few months back. It's cozier, comfier, somehow.

The first committee report was made last week (Monday, I believe?), and the three members of that first deliberative body seemed to work well together. At least, I heard no complaints. Their report was a brief one, doing little more than illustrate a couple of the superior responses the committee received (by the way, participation was salutarily high, with about 2/3 of the class submitting solutions). I think the students might have felt a little uncomfortable about indicating others' errors in front of the class (even anonymously), so they avoided outright criticism, but I hope future committees (two more reports tomorrow!) will feel it's okay to indicate common pitfalls, especially if many people fell into them.

Voluntary committee involvement has been strong, I've had no trouble getting people to offer themselves up, and both committees received submissions from over half of the class yesterday.

I have 280 components like the homework committees at the front of my mind as I make my way through the latest in a long line of teaching-related reads, Peggy S. Meszaros's (ed.) Self-authorship: advancing students' intellectual growth, Jossey-Bass, San Francisco, 2007, the focal text of yet another university learning circle I'm taking part in this semester. So far though I've not found the book thoroughly engaging, it's served to reconfirm much of what in the past few years I've come to know and believe about progressive pedagogy at the university level.

In the opening essay, "The journey of self-authorship: why is it necessary?," Meszaros takes the definition of self-authorship offered up by the now-canonized Marcia Baxter Magolda: "the capacity to internally define [one's] own beliefs, identity, and relationships" (p. 10, from Baxter Magolda, Making their own way: narratives for transforming higher education to promote self-development, Stylus, Sterling, VA, 2001). (Justifiably this concern takes center stage in many of today's progessive college classrooms: time after time we hear that what students in today's universities most need to learn is indeed simply how to learn.)

On the facing page in Meszaros's essay, we find the following snippet: "Becoming the authors of their own lives involved reshaping what they believed (epistemology), their sense of self (intrapersonal), and their relationships with others (interpersonal)" (p. 11). As I read this, I jotted some notes in the margin regarding the role played by a discovery-centered approach to proofs and proof-writing in helping to affect changes of all of these sorts:

1. By being encouraged both to construct their own proofs and to thoughtfully critique others', the students gain a deeper understanding of the nature of mathematical knowledge, particularly of the fact that it doesn't inhere in any one person, no matter how intelligent that person is. Knowledge ceases to be "out there, somewhere," but rather "in here."
2. By allowing students to take command of both the proof-writing and the proof-reading (in a literal sense) processes, as I'm attempting to do in our class by establishing the homework committees, I challenge the students to take on the role of the mathematical authority: mathematically speaking, anyone who can grab hold of the governing rules of math logic can stand in judgment of the correctness of a given proof. No longer are the students simply vessels for knowledge not yet bestowed; they are the bestowers themselves, they are the experts. They are participants in the mathematical process, not merely spectators.
3. By cooperating and collaborating in the proof-writing and proof-reading processes, the students come to appreciate that mathematics is a social enterprise, that it is conveyed in a transmittable medium, that it is a part of our shared heritage, ultimately constructed by human beings working in concert with one another.

How successful will this class prove (no pun intended) in easing my students down the road to self-authorship?

I don't know.

Do my colleagues think as deeply about these issues as I do?

I don't know.

I hope so.

I'd really like to see my department develop a more coherent pedagogical philosophy.

But that's another story.

And it's late.

I'm going for now. I'll let you know how tomorrow's committee reports go.

Mirror, mirror, on the wall...

Another day past.

I felt skittish in my Calc I classes today, and awkward. I felt elsewhere, out-of-place, out-of-sync.

MATH 280 made up for it, though. I was definitely at home in Karpen 033 this afternoon.

This afternoon, I started thinking about what I'd like to talk about at a Writing Across the Curriculum conference.

What do I have to say?

What am I qualified to say?

Hey, even if I have to say so myself, I think I'm pretty damned good at teaching math students how to write math...but is that enough? I don't know how hardcore into the scholarship of teaching and learning I'm expected to be in order to "have something to say."

I'd like to talk about my rubric-building exercises: how does one set out to teach math students to teach themselves what to look for most in quantifying quality in math writing? How does one teach them that, given a few ground rules and a little practice, they are as qualified as I am to render an assessment of a proof's goodness?

Good enough?

Hmmmm...there's a kernel of irony here, isn't there?: maybe I've just got to teach myself that I am as qualified as anyone else is to render an assessment on the goodness of my own writing-instruction methods, at least in the context of my own classroom.

Is it that easy, or is that just a bunch of relativistic hooey?

Ah, fugeddaboutit.

I'm going to go rustle up something to eat.

To be continued, for sure.

Conversations

Had a little doorway chat with Karl (longtime Math Lab student employee, now graduated) this afternoon about the universality (or lack of it) of mathematics: to what extent is math just waiting around for us to discover it, and to what extent is math itself an artifact of human invention, the residue that's left by the human mind as it makes its imprint on all that it encompasses? The whole conversation started when I was showing him a book of logarithm tables I picked up at a garage sale or flea market somewhere a long time ago, and I wavered indecisively between the words "invented" and "discovered" when searching for the right word to describe the initial human engagement with logarithms.

"Since you said 'invented' first," said Karl, "I can tell which camp you're in." This led to a discussion of whether mathematics can truly be universal, a position neither of us defends. Karl mentioned recent research (see this link for more info) into the language of a certain Amazonian people suggesting limits to traditional Chomskian analysis, and I let him know about Anthony F. Aveni's Uncommon sense: understanding nature's truths across time and culture (University Press of Colorado, Boulder, 2006), an interesting book I worked my way through this summer. Aveni discusses the scientific undertakings of the members of various ancient and modern societies and provides accounts of culture-specific scientific knowledge that might seem patently alien to practitioners of science as defined by the Western European Enlightenment tradition. I'll definitely be looking through that text again when I start to put together my thoughts on the history of math technology course I hope to run.

Rewind several hours: as I walked into campus this morning I thought about our discussion on the topic of "Good Proof/Bad Proof" in 280 yesterday. "Damn," I thought, "that was a nice conversation." I really felt that we got right at the meat of the matter (or whatever vegetarian substitute one would like to put in its stead), and the students themselves were quick to point out, unprompted, what it is that makes a given proof a weak one or a strong one: does it use notation correctly? Consistently? Does it prove the claimed statement in full generality? Does it use correct grammar and punctuation, use complete sentences? Does it "lead the reader" conversationally through the thought processes of the prover? All of these questions get at the issues of clarity, correctness, completeness, and cohesion, my "Four Cs" of assessing the quality of a proof. Above all else, the exercise helped them develop (oh, that meaning-laden term!) "ownership" of the process of mathematical discovery: they have the same right that I do to question the validity of a proof, to test the hypotheses of a theorem. Math's truth does not inhere in a single individual no matter how much experience that individual possesses, and even the greenest of mathematical parvenus, equipped with the right tools and techniques, may approach, with healthy skepticism, a given mathematical statement with the confidence of a professor emeritus. I think that yesterday's exercise helped folks see that, and I hope that it gave them the confidence they'll require to feel free to explore the problems we'll face the rest of the semester.

I'm really glad we took time out for that activity.

Notes to self, part 2

We're just a few days away from beginning the new semester (Monday, August 20th: do you have your calendars marked?). In between periods of vegetative depressurization from the newly-ended REU and continuing research in probabilstic graph theory, I've been spending a bit of time during the past couple of weeks putting together various activities for MATH 280 and MATH 191. Much of my planning is outlined nearly illegibly in the margins of my copy of John C. Bean's Engaging ideas: the professor's guide to integrating writing, critical thinking, and active learning in the classroom (Jossey-Bass, San Francisco, 2001), but I really need to compile it all in one place. Ergo...

NOTES TO SELF

(An open exercise in academic free writing)

During the coming semester I will be putting greater emphasis on "decentering" activities that promote cognitive dissonance, unorthodox points of view, and healthy skepticism. In presenting students with counterintuitive mathematical ideas, I can imbue them with a sense of surprise, wonder, and curiosity. In asking them to develop the ability to see a problem from all perspectives (literally and metaphorically), I ask them to become stronger problem-solvers. In encouraging them to question unproven assumptions, I not only charge them to be more careful in their calculations; I also open up to them unexplored fields of inquiry. Where would geometry (and by extension, much of modern mathematics, not to mention the physical sciences) be had a number of brilliant minds not questioned the validity of Euclid's Parallel Postulate?

Now, how to do all of this?

I hope to introduce the students to the idea that mathematical discovery, along with its recording and transmission, is a dialectical process that involves the researcher in conversation with herself and with others. The act of discovery is almost never a burst of light illuminating the void, but rather is born from the steady nurture of an ever-growing spark. Discovery begins with the posing of a question, the wrestling with a problem. One's first thoughts on a problem are generally chaotic and messy, and harken back to solutions to analogous problems and inchoate modifications of earlier ideas. After long hours of talking with others and lying awake at night staring at the ceiling, and after countless pages of notes have been scribbled, studied, and redacted, more complete ideas take shape. Bean (p. 20) speaks on the nature of the written manifestation of this process: "the elegance and structure of thesis-governed writing -- as a finished product -- evolves from a lengthy and messy process of drafting and redrafting."

Below are a number of the exercises I plan on implementing in some fashion during the coming semester:

• Response writing to Polya; possible guiding questions: "Is Polya's proposed process relevant in a modern problem-solving course such as MATH 280? Take a position on this question, and defend your point of view." "Have you ever applied Polya's process, knowingly or unknowingly, to solve a problem posed to you? Explain carefully." "Do you feel that intuitionism is a defensible mathematical philosophy?" "Use Goldbach's Conjecture to illustrate the difference between constructivist mathematics and nonconstructivist mathematics."
• Decentering exercises focusing on puzzling phenomena such as various sizes of infinity, space-filling curves, fractal dimensions, et cetera.
• For the 191 folks, to get them to take a position in a short thesis-governed paper: "Suppose you need to differentiate a function of the form f(x)/g(x). Do you prefer to apply the Quotient Rule, or would you rather rewrite the function as a f(x)(g(x))^(-1) before applying the Chain and Product Rules? Explain the reasoning for your preference."
• For the 280 folks, to accustom them to "mathematizing" messy problems and developing intuition (skills I feel are overlooked in even the more discovery-learning oriented proofs courses): "Consider the game of Nelinurk (see tonypa.pri.ee/start.html). Play the game for a half-hour or so to get used to the rules and the flow of the game. Once you feel comfortable playing, see if you can describe the game mathematically and develop a strategy for optimal play. Explain your strategy as clearly and as completely as you can. (It may help to develop your own terminology and notation as you write.)"
• More "intuition-building" activities for 280 students: estimation exercises? Incomplete proofs? ("How big?...", "How many?...", "Give the outline for a proof of...") As I said above, I feel that the nurturing of mathematical intuition that's done in most proofs courses is woefully outweighed by the emphasis placed on learning how to do formal proofs. (And no, I don't think it needs to be put off until a "problems course" like our 381; good intuition makes for clearer, more succinct proof-writing, and clearly written proofs feed a healthy intuition like Wheaties feed Mary Lou Retton.)
• Taking a page from my own playbook, five or six years ago at Vanderbilt: have 'em write a few "poems inspired by mathematics." It can't hurt, and it might be just what the more humanities-minded students need to get their creative juices flowing.
• Have a "show and tell" day on which I ask everyone (myself included) to bring in all of the notes, scribbles, emendations, and so forth that went into the final draft of a given project. (I might simply ask them to save all of their homework drafts?)
• Class-opening and class-ending one-minute essays: "where do we need to go today?", and "where did we end up in our travels?"
• Mock trials: in 191, the obvious, Newton v. Leibniz. In 280, perhaps Brouwer v. Hilbert? Each side is taken up by roughly half of the class, certain individuals chosen to act as the given personages, with others as their supporting staff (i.e., legal counsel). For the 191 debates, I could even have the two sections square off in a "finals" round, one section taking the side of Newton, the other that of Leibniz.
• Analogy games (cf. Bean, p. 111): ask the 280 folks to complete the following and elaborate upon it: "Writing proofs is like ________ ." For the 191 classes: "If differentiation were an Olympic event, it would be most like ________ ."
• Precise proof-summarizing and theorem stating: require students to write a proof summary or a theorem statement using a precisely defined number of words, giving maximal credit only for using exactly that many words. Example: "explain the Axiom of Choice in precisely 25 words."
• To give the students a taste of "original research," I can hand the 280 folks the data I obtained this past summer on consecutive inverses modulo p and ask them to describe the patterns they see. I can do the same for the Calc I kids, giving them the graphs of the sequences of expected degrees for various of the random tree construction algorithms, asking them to supply likely models (and concomitant analysis) for the shapes they see.

Hmmm...that's all for now. More to come, I'm sure.

Soldier, sailor, tinker, tailor, ploughboy...

Who are you?

Let's say that the instructor waltzes in and announces that you're going to be working in groups. You can't call on your best friend in the class to help you; the instructor's choosing the groups for you, and the way you're all split up appears to be random. Oh great, you're stuck with Jessica. You heard about her. Giselle you don't know, except for the fact that her cell phone's gone off in class three times so far this semester. And then there's Dante. You've never heard him say a word. You're given five minutes at the end of class to meet with your new group members, to get to know each other a little, to exchange contact information. You've got a week and a half to put together the project just assigned, and you want to get to work on it as soon as possible.

As early as your first meeting, two days later, you notice certain interpersonal dynamics. You're focused and on-task (or at least you try to be), while Giselle is not. She gets up every five minutes to get a snack from the vending machine or call her best friend on her cell. Meanwhile Dante has started to work on the project, but he's off in his own world, performing computations that you don't understand and that he seems unwilling to explain to you. That leaves you and Jessica, and you find her to be (quite frankly) dumb as a box o' rocks. Indeed, almost every other sentence out of her mouth is "I don't know."

"Well, did you understand this one?"

"I don't know."

"What did Prof. Buxfizz say about this method?"

"I don't know."

"What in the hell is taking Giselle so long this time?"

"I don't know."

What good could come of working with her? You finally decide to peer over Dante's shoulder as he works away at the project's first problem. At least maybe you can learn a little by looking on.

Do any of these habits sound familiar? Chances are quite good that you've observed one or more of these personalities in group work you've done in class. Maybe you're Dante, maybe you're Giselle. Maybe you're the poor overtasked Jessica, or maybe you really are the monkey in the middle whose role I've given to you as our fictional observer.

Last week my Learning Circle colleague Darlene pointed out that when small groups convene, very predictable personalities manifest themselves. There are type-A leaders who take it upon themselves to see that everything's done right, often dominating the workload and shopping the simpler tasks out to the others. There are the absent slackers, who more often than not don't bother to show up. There are the silent types who are afraid to speak up, fearing they'll betray their ignorance and be laughed at. There are the dittoheads who go along with every answer uncritically, there are the speed-demons who just want to finish everything as quickly as possible, and there are the perfectionists who aren't happy until the seventeenth draft of the group's write-up has at last been produced in the optimal font-size.

What type are you? I've only recently (in the past couple of years) begun to appreciate that successful performance in group work really does require of one an awareness of the sort of persona one tends to take on in group get-togethers. (Likewise, it's not enough for me as a teacher to simply throw the groups together and say, "have at it!") To get a group up and running, you've got to do more than make sure there's a time available for everyone to meet: once all are assembled in one place, there's then the matter of getting everyone to contribute her or his fair measure, to the extent that each is able to contribute according to her or his talents.

What is your talent? What good do you typically contribute to a group endeavor? Can you ask yourself to contribute your share of your positive energy, and can you challenge yourself to minimize your adverse behaviors? Can you bring yourself to contribute something else that's usually left inside of you?

I mentioned in my last post that throughout my schooling I was always the "get it done" guy. I'd rather do all the work myself than let the slower folks in the group take control and botch it up. Of course, having now spent a long time on the "other side of the glass," I realize that this attitude probably rendered all group exercises practically useless for my teammates, but hey: I got what I needed out of it, and everyone got to share in the good grades. Win-win, right?

Now in group work I challenge myself to stay quiet, to not dominate. I contribute, but I wait for contribution from others. I make sure my piece is heard, but I do what I can to incorporate others' views with my own, and whenever I can I paraphrase, reiterate, recount, others' takes on things to make sure that I'm understanding them properly. I offer help when it's needed and do what I can to facilitate the others' learning. If I find myself in danger of dominating the conversation, I try to shut up.

What if you were Jessica? Could you challenge yourself to speak up? This must be hard! Though it's somewhat awkward for me to sit on my hands on not go as quickly as I know I could if working alone, I realize that it must be downright terrifying for a shy and unsure group member to risk the derision of her peers by admitting that she doesn't know what in the hell is going on. Last semester in MATH 365 there was one group in which three of the group's members were decidedly more self-assured than the fourth. This fourth frequently confided to me about how difficult it was to tell his friends to "slow down! I can't understand things as quickly as you all can."

And Giselle, what could she do? Perhaps her challenge at the outset would be simply to stay in the moment and keep her focus. And you, the nameless observer in the comedy above? Could you, perhaps, challenge yourself to be the one to bring the group together? Could you make it your place to call "time out" and reconvene the group to say, "all right, folks, we're just not on the same page on this one. Can we lay out a plan that'll work for everyone?"

I don't know. I don't think there's any one right answer. Every situation is different.

What do you think? I'm really curious to know what's on your mind.

Owner/Operators

Monday in class an excellent question came up: someone (I think it was Tomassino) asked if permutations behave like combinations in the following fashion: "is it true that P(n,k) is the same as P(n,n-k)?"

"I don't know," said. "Let's find out. A minute or so later, we'd completed the computations. Of course, it was little more than three or four lines of simple arithmetic, but the lesson learned (I hope!) was more than simply how to manipulate a few factorials. Rather, "I want you all to know that the authority to do mathematics, to ask questions and to solve them, to prove things, to come up with new theorems and new theories, does not inhere in me. It doesn't lie in your textbook, it doesn't lie in the 'experts,' whoever they are. The authority lies in the mathematics itself, and therefore in anyone who takes the time to learn the mathematics. It lies in the logically sound arguments and valid computations of which mathematics is built. Anyone who can learn the rules of logic and algebra and adhere to them correctly and consistently has authority to do mathematics, and so to ask questions, to answer them, to create new mathematical ideas. Anyone. The authority is in you, if you take the time."

As much as I despise the term (primarily for its blatant capitalist and patriarchalist overtones), "ownership of" the material, or better yet, "partnership with," the material, is an end towards which I hope I help my students strive.

The math ain't mine. It ain't the domain of the experts, the pointy-heads, the mathematical gurus that rest on high in chaired positions in Harvard and Berkeley. Hell, it ain't even theirs.

It's everyone's.