Arrrrrr...avast!

The "pirate" section of Topology is on. Beginning on Tuesday, February 2nd, I'm going to try holding an intermittent "unofficial" (or "pirate," named for the manner in which we'll have to hijack a vacant classroom somewhere on campus) section of Topology, splitting the class in two in order to attempt to provide a more cozy and intimate learning environment to the students in the class. It looks like I'm going to have roughly 8 or 9 in the unofficial section and the remaining 15 or 16 in the section meeting at the official time.

We'll see how it goes.

In other Topology-related news, we've had two installments of homework presentations (operating in lieu of homework committees this semester), and the second flowed much more smoothly than the first. Students took turns explicating four of the five homework problems which were due today, and they all did a fine job. I was particularly impressed by the way in which the proofs they offered were for the most part merely sketches. It's best that the student presenters leave lacunae to be filled by their fellow scholars in their individual write-ups: to provide a complete and error-free proof would be counterproductive.

What I'd like to know at this point, from the students in Topology with whom I've worked in earlier classes (280, Abstract Algebra, etc.), is: which system do you prefer so far, homework committees or homework presentations? Obviously they both serve different purposes (one offers substantial opportunity to peer-review while the other is an instance collaborative learning in the extreme), but it might very well be asked which serves its purpose more effectively.

Which one is the most useful to you? Which one helps you learn better?

We'll see.

I'll leave by noting my frustration at Mother Nature for dumping several inches of snow all over what should have been the writing workshop I've been co-organizing, scheduled to take place in Black Mountain this afternoon. Snowpocalypse II caused my writing colleague Betty Lou and me to push the workshop back to next Friday.

Two weeks in the can

We just wrapped up Week Two here at UNC Asheville, and because of the MLK holiday I've yet to teach a full week of classes this term. In fact, I'll be skipping out on my Friday Calc II class this coming week, so it won't be until Week Four that I manage to teach a full week.

Things are...well, I think they're settling in.

I still feel unsettled, though. Missing the second half of the first week because of the Joint Meetings really threw me off. Nevertheless, the classes are starting to come together, the enrollments are stabilizing, and we're getting work done.

My Calc II students did quite well on their first miniproject. (It was the perennial favorite, "Confectionary Conundrum," in which they are asked to estimate the number of gumballs it would take to full the candy machine I keep on my office desk. They're meant to use Riemann sums, but there are always a few folks who come up with other clever ways of getting an estimate.) And I've got great attendance so far, which is somewhat troubling, since there are 36 people enrolled in one section and 37 in the other, while the room is designed to seat about 32.

Meanwhile my Topology students have weathered their first round of homework presentations. The presentations (of which there've been two so far, delivered by four students) were a bit different, as is to be expected, from the homework committee presentations in past classes. The presentations were less coy and more direct; this is unsurprising, given that in the homework presentations in this class the students are asked to sketch draft solutions for their peers, whereas the purpose of the committee presentations in past classes was not to solve the problem, but rather to highlight common errors and pitfalls, and indicate any clever methods one or more students may have discovered.

I'm hoping that as the homework sets become more difficult, the students will put more thought into their presentations, planning ways to offer their friends helpful hints without giving away the full story. Being able to sketch a proof is as important a skill as being able to write it out in detail.

I'm also hoping that students will start to take advantage of the unlimited revision and resubmission policy I've got for the course.

I think one of the things that's kept me from settling into a groove is that much of the work I've done so far this semester is administrative rather than pedagogical. We've got a lot going on in our search for a new faculty member, and I've been scrambling around putting out fires for the Writing Intensive Subcommittee (dealing with student petitions, setting a new meeting schedule, uploading course proposals, planning next week's workshop, puzzling out assessment) and the ILS Oversight Committee (another new meeting schedule, dealing with the Academic Policies Committee, and putting out a bajillion little fires regarding Clusters). And I'm trying to organize the growing list of students who are planning on going to the MAA's Southeast Sectional meeting at Elon University in March. (Twenty students have committed to going now, and I've got my sights set on nine or ten more.)

Ugh.

Okay, I just realized that I've said nothing interesting or insightful in this post, so I'll end it here and wait until I've got something more meaningful to share before I write again.

Quick questions

I've got some random ruminations to share, in the wake of this past Saturday's lovely WAC/CAC (that's Writing Across the Curriculum and Communication Across the Curriculum, for those not in the know) Symposium at UNC Greensboro.

Please discuss, in the comments section, anonymously, if you'd like:

1. Why are faculty so fearful of assessment? Is it because of the work involved? Is it that they're just not sure how best to do it, or are they afraid of what they'll find if they do it right?

2. For those of you out there who are into math (faculty especially): when in life was it that you first fell in love with math? What is it about mathematics that drew you in, that got you hooked?

3. Are my homework committees working as well as they ought to? Is there a different model I might adopt that would make them more effective? (For what it's worth, I think they work more effectively in 280 than they do in the 400-level courses; while the 280 students don't often "get the right answer" from their committee-dwelling peers, they're exposed to multiple points of view on the same problem, which is ultimately far more important than getting the right answer anyway.)

4. For my students: just how busy are you? I get the sense that this particular Spring 2009 semester has been one of the most stressful on record, for faculty and students alike, and I suppose a lot of it has to do with the economic situation the world's found itself in, and the commensurate busyness it's forced on us all as we struggle to make ends meet by asking 1 plus 1 to be 3.

No due date, no page limit. Lemme have it.

Committee report report

I've just picked a perilous path through the eye of the perfect storm of grading, fourteen and a half hours of toiling and troubling over computational exercises and low-stakes writing on derivatives from my Calc I kids, first-order logic problems and truth tables (not to mention first-time LaTeX exercises!) from my Foundations students, and a passel of problems on ring theory from my Abstract II class.

Fortunately the cold I came down with at the end of this last week made sure I didn't want to be anywhere else but home this weekend.

I've been meaning for some time now to say a bit about how the homework committees are going this semester, since so far I've been proud of their smooth functioning. (For the people reading this on my first-ever cross-post on the Young Mathematicians' Network, you can read a number of my older posts about using homework committees here. Briefly, it's a peer-review technique I use to encourage students to (1) begin their homework more promptly, (2) engage in self-authorship, (3) grow accustomed with the frequent multiplicity of correct solutions, and (4) develop their teamwork skills. Students volunteer to serve on committees tasked with reviewing and offering feedback on their peers' drafts of solutions to particular homework problems. After reviewing all submitted drafts they lead a brief class discussion on the problem they considered and return the drafts to their respective authors, who then have time to revise their work before submitting a final draft one class period later.)

There have been four reports in each of Foundations and Abstract II, and despite their relative inexperience with the genre, the former students' committee reports have been stronger than those of their Abstract II counterparts: they've skillfully avoided simply answering the problem placed before them (this was a major problem the first semester I asked students to serve on homework committees), they've intentionally made use of the course's writing stylesheet (the now-infamous Four Cs), and they've done a marvelous job of indicating common pitfalls, clever solutions, and helpful hints.

When the first homework set was handed in there was a little misunderstanding concerning exactly who received which draft of which problem, and as a consequence I was given a glimpse of the feedback the students were offering to one another on the drafts submitted to the committees. It was heartening: the few comments I saw were meaningful, respectful, and helpful without being too much so.

The quality of these students' committee work greatly exceeds that of the students in the first Foundations course in which I assigned committee problems. Back in "the day" the kids'd frequently just get up in front of the class and solve the given problem for their friends, resulting in a couple dozen nearly identical eventual submissions in which the students would faithfully render every jot and tittle (whether correct or not) of their colleagues' proffered solution. During the past couple of semesters I've very deliberately pointed out that it is not the job of the committee to perform this disservice.

Word's gotten out.

So, props.

In other news, I'm happy to see that I've received a few (not even anonymous!) replies from my students regarding Gil Strang's free text, Calculus. I might be misreading the message here, but it seems that though people are not yet ready to chuck the text through the nearest open window, it might not hurt to mix a few of my own exercises in with the relatively recondite ones offered at the end of Strang's sections?

I believe this is exactly what I shall begin doing.

Thank you all for your feedback, I appreciate your attention to improving our classroom environment! Together we'll make it to the semester's end.

For real this time

What a strange semester this has been.

It's almost over now. We're already two days into Finals Week, I've got half of my grades submitted, and faculty wrap parties are popping up like badgers in a Weebls video.

I told Griselda the other night that it's been one of those running-to-stand-still semesters.

Maybe it was my relative unfamiliarity with both of my classes (first time ever for Precalc, first time here for Abstract) that kept me on edge all term long.

I just don't feel like it's done yet.

Somehow I feel like it never began.

Have you ever watched a movie that, twenty minutes in, didn't even seem to have started yet?

I spent almost two hours this morning in a meeting with Casanova and Lulabelle as we attempted to hammer out the dings and dents that marred the pre- and post-surveys Lulabelle and I and several others used in last year's writing assessment project. The goal is to ready a new pair of instruments to use in various classes next term. You guessed it, my 280 students will get to be guinea pigs again.

At Lulabelle's suggestion, we cut the pre-test in half by removing discipline-specific items. "I can think of two really good reasons for doing that," I said. "First, our ultimate goal is design an instrument that is discipline-independent anyway, and second, I'd find it interesting to see what sort of general writing gains students perceive having received quality writing instruction in the disciplines."

Harrumphs of agreement.

We added a few items, changed the wording in a few others ("I hate the phrase 'that professors like,' " lamented Casanova), and tweaked the demographic questions, particularly those regarding gender, so that they became less othering and isolating. "And we don't need to know about their AP and IB experience," we agreed. Out it went.

The new survey should be about half as long and much less cumbersome. Moreover, if we run it on Moodle (the online course management software UNC Asheville uses), the data collection will be a snap.

I'm still waiting to get some of the data from last year's study. I'd really like to know how a faculty member's own field of study affected her application of the course rubric to classes in various disciplines; I conjecture that the students' scored mean is directly related and the students' scored variance inversely related to the "distance" between the discipline of the rater and that of the "ratee." That is, as a mathematician I'm more likely to judge physics papers more strictly, and to assign a broader array of scores to such papers, than would a poet.

And no, I'm not sure just how this conjecture could be quantified completely.

But does it need to be?

Tomorrow I plan on collating all of the writing products I gathered from this past summer's REU students. The primary question I'd like to answer in assessing their writing is: "can one clearly discern the student's trajectory from novice math writer to accomplished article author over the course of an eight-week program in which developing writing proficiency is a stated learning goal?"

And hey, in case you didn't know it, I've been teaching math a bit lately, too!

My colleagues and I all agree that this semester's Senior Seminar talks were of consistently high quality. Not one of the eight talks was weak (usually there'll be one or two stinkers). It's too early to tell how much of this improvement is due to the more rigid scaffolding my colleague provided the students in the form of in-class "practice" talks and writing assignments. I'll be introducing yet more structure to this writing component in next semester's installment. (I know a couple of my mentees from this past semester would have been happy with more clearly articulated guidelines for their writing.)

Shit, there I go, on about writing again. Sorry.

My Abstract Algebra students' presentations were also significantly stronger than those of most students I've had in upper-division courses in the past. Their success was a result of thorough preparation on their parts: I forced them to get their acts together early and start right away in choosing topics, finding references, and assembling thei talks. In the end I felt that only three of the 15 talks were fairly weak ones (C-quality or worse), and even these had silver linings. I was particularly impressed with the talks on point groups in chemistry, free groups and group presentations, and ordered groups.

Several days ago one of my Abstract students gave me some good constructive criticism on the committee system, about to enter its fourth semester next term. "There'd be times when I'd get my draft back from the committee and it would say 'great job!' and 'perfect!' So I'd hand it in, and I'd get it back from you and it'd have half of its points missing. That was really disheartening."

Yeah. "I understand what you mean," I said.

I told him that I'd take care to emphasize that the burden of verifying the validity of a particular proof or computation lies on the committee's members: it's their job to check the facts. That's a hefty responsibility, but one that I firmly believe they should expect to assume (how else are they to make of themselves mathematicians?). I told him that I'd take care to emphasize that I would be there as a "resident expert" should there be any dispute over the validity of a particular student's response. May my word be advice and not edict.

I'm retaining committees for both 280 and 462 (Abstract II) next term, and I'll probably introduce a modified system for projects in Calc I as well. 280 students will see a greater number of "dialogue" problems in which students are asked to construct mock expository exchanges between one another with the aim of better understanding tricky logical points or proofs: students have said consistently that these exercises are particularly effective ones.

For 480 I'll once again devote a day to abstract-writing, a day to peer-editing of students' papers, and a day to designing a multimedia presentation using PowerPoint, SliTeX, or Beamer. Tomorrow I'll take a little while to hammer out the 480 schedule, which is going to be tight: with 19 (!) students registered for the course, at least seven (!) days will be devoted to student talks alone; with the three days indicated above, we'll only have time for four or five faculty "model" talks, and students will probably have to start presenting before Spring Break, or very near to it.

Ouch.

The most drastic change to my m.o. next semester will be my introduction of LaTeX to the 280 students. If they're going to learn how to write mathematically, then, by gum, they're going to learn how to write mathematically.

After introducing LaTeX through the handouts and worksheets I've already developed for the REU, I'll introduce a "TeX this!" assignment like the one the REU students worked on for about an hour this past summer. Then I'll start infusing their homework with LaTeX requirements. I won't require them to TeX all of their homework: just as certain problems will be designated as committee problems, others will be designated as LaTeX problems whose solutions must be TeXed. (They'll receive a nominal amount of extra credit for TeXing the other problems.) The percentage of LaTeX problems will increase in each successive assignment, from one in four in the first few assignments to maybe two-thirds by the semester's end. I doubt the students will resist: LaTeX is easy to obtain, free to use, and damned fun once you get used to it. In my experience once students get in the habit of TeXing their work they never look back, and even the most typographically challenging documents are seen as amusing obstacles to be overcome.

Besides, I have no doubt that I'll be making my colleagues' (and my own!) grading substantially easier down the line. Imagine how jumbled and disjointed a student's non-linear mathematical thinking might appear, with a disorganized blob of exposition tacked onto the side and inserted with an arrow, and a stray sentence placed at the page's bottom beside an anchoring asterisk that leads the reader to this appendix from the main body of the text. In typesetting her work, the math student is forced to linearize her thoughts, to compose them well, to say what she wants to say in the right order, all in one place. Typing, a tool for writing, is by extension a tool for thinking.

That, and the student feels all the more accomplished fo having mastered a pretty snazzy typesetting tool.

I'm really looking foward to next semester.

I think I say that before every semester begins, don't I?

I just love what I do.

I hope it shows.

Okay, I'm off for now. More to come, more to come. Always more to come.

Until then, comments are appreciated. Take care!

The promised post

So, yeah...how's about them homework committees?

They're doing well. I don't think I've been challenging them enough. For instance, members of two of the six (three per section) committees who offered up reports in yesterday's classes professed that they really didn't know what to say, as the problems they'd dealt with were so straightforward. Eventually both of these teams, who worked on the same problem in different sections, delivered some variation of "It's an if-and-only-if proof; make sure you prove both directions."

One or two of the committees slipped momentarily into "show mode," essentially solving for their classmates the problem they'd been assigned. "Please remember," I gave them a gentle reminder afterward, "that it's not necessary, nor is it even best, for you to solve the problem you've been given." While one ought to offer some feedback and direction by indicating trouble spots and possible avenues to a successful proof, one does one's fellow scholars a disservice by simply solving the problem for them.

I also reminded the class that it's not up to me to weigh in the veracity of a given mathematical proposition; it's just as much up to them. Mathematics, a passel of mutually accepted axioms and rules of logic and inference, is the the product of centuries of give and take, of argument and counterargument, of disputation of every sort. Mathematics, like any other human-made system, is a social construction, and its application is open to any who master its agreed-upon conventions.

I'm liking the committees' work, and the committee system runs more smoothly in each successive class in which I make use of it. (I'm sure this in part because I'm getting better at managing and mentoring the committees, and in part because my more recent committee classes contain veterans of my earlier committee classes, so many of the students know the drill by now.) But I've definitely got to push them harder.

No, this last round didn't offer much of a hill to climb. In order to provide a greater challenge, I'm selecting some of the harder problems from the upcoming homework set for the committee problems.

Meanwhile, at other end of the Karpen Hall basement, Precalc is oozing along. We're still several sections behind where we "should be," but I'm growing more and more comfortable with that, especially after having a brief talk with my Chair this morning about my concerns. "Not having taught the class, I'm not as sure what I should be expecting them to know, and I'm not as sure of the pace I'm taking," I admitted. "It's getting better, and I'm starting to get a sense as to their abilities, but I know I'm going more slowly than I should be." He agreed with me, though, that there are certain things they need to see, and other things they can do without. As long as they're exposed to a wide variety of functions and get the chance to apply various algebraic techniques to those functions, they'll be fine.

And that they're getting.

Today's class was a tiny one, 10 of 32 people absent. (What's up, folks? Did Fall Break start early?) Class went very smoothly, though, and I noticed today how easy it now is to coax them from their seats and get them at the board doing problems. Although there are a few showboats in the class (they know who they are) who stride to the board with alacrity, even the more wallflowerish of the bunch have grown more confident in getting up in front of the class to chalk out a few figures.

All I have to say to my fellow teachers is this: if you're going to use group work, board work, inquiry-based methods, problem-based methods...any sort of classroom technique that'll take the students out of their comfort zones for even a moment...do it early, do it often. Start 'em on the first day, and do it regularly.

Indeed, this was the advice I and several others gave to Frodo, a new high school math teacher whom I met last night at a dinner meant to bring UNC Asheville's math and science faculty together with the region's high school math and science teachers. He'd related a tale of a problem-based activity he'd done in which he'd engineered the work groups by placing a strong student, a mediocre student, and a weak student in each group. The results were all right, but not what he'd hoped for. Everyone at the table assured him he'd made the right move (while the weaker students benefit from having a peer guide them through a solution, the stronger students benefit from having to provide the guidance in the first place, thereby improving their ability to communicate mathematical ideas) and encouraged him to keep at it, and to introduce such activities to the students as early in the course as possible.

I've decided that I'm going to introduce committee work to lower-level courses beginning with next semester's Calc I class.

Oh, yeah, I promised to say a bit about the Precalckers' poetic achievements! This past weekend I spent about six or seven hours reading through the 31 rough drafts I received (all but one! y'all rock!). They range from whimsical and funny to dark and brooding (one was simply titled "Dread"). Some were humorous, some wry, some philosophical. The tendency was for the poems to be more narratively personal than the Calc I students' were, and whereas the Calc I students often chose to incorporate mathematics into the structure of their poems, the Precalc students more often chose to place math squarely within the content of the poems. (This may simply be because the Calc I students have a deeper and more sophisticated understanding of mathematics in general.)

There were very few poems that I would consider weak. In fact, I would say the "low end" was significantly higher than the corresponding low for the Calc I classes last fall. On the flip side, there weren't any that approached the caliber of the Calc I students' best offerings (I'm thinking of Farrah's "Motivation for a sweet tooth," Lisette's "Mathbeth," and the anonymously penned haikus from last year's class).

This past Monday night I held an optional "poetry reading," and sadly only three students showed up. "Don't tell me we're the only ones!"Belinda moaned when she and I arrived on the scene simultaneously, finding our classroom an empty cave. "We'll give it a few minutes." A few minutes passed, and no one else had come, so Belinda and I just talked about poetry and math for a bit longer. Then Tootsie and Omar arrived, almost at once, and we got underway in earnest. Each of the three students read their poems, and we spent about ten minutes on each, offering comments and insights. Often the conversation wandered off on poorly-lit philosophical roads (Gödel's Incompleteness Theorem, the "universality" of math, ethnomathematics, cultural scientific relativism, and so forth), but I think we all learned a lot.

I feel that these poetry exercises truly help the students to engage mathematics in an entirely different way than that they're used to, and I'm hoping they're getting something meaningful out of it. Students, feel free to chime in with a comment or two! And know that I'll soon be handing out "surveys" that'll help me to understand what you got out of this exercise, and what you put into it.

I'm looking forward to hearing more from a few students in particular. For instance, not long before handing in her rough draft at the end of last week, Gwendolyn told me that she'd had a hard time finding something to write about at first, and it sounds like she spun her wheels for the first week and a half after the assignment was handed out. But then, she informed me, she'd been inspired during class last Wednesday when I'd mentioned the ways in which math could be viewed as a metaphor. The poem came quickly then.

I'd really like to get at what it was she was thinking as she put her pen to the page!

Okay, for now I must go. Tomorrow night I hope to put together Chapter 3 in my series of CWPA-inspired essays, so please stay tuned.

Deep thoughts and more dilettantish dalliances

Hey, All! It's been a long time since I blogged here about math pedagogy, so I wanted put out a post that lets you to know that I do indeed still reflect on my teaching, and that I am keeping close watch on my classes as they develop this semester.

I thought long and hard about teaching math this morning as I walked into campus. Specifically, I thought about our use of software in Precalc, and I've come to the conclusion that I'm not very happy with it.

Let me start out by saying that I'm glad that I've had the chance to use the software this semester in teaching the course: it's been an opportunity to gain proficiency with a particular pedagogical technology I'd not used much before. Moreover, I recognize the usefulness of some aspects of the software for some students: the software's ability to generate problem after problem of samples fitting particular problem molds is useful for students who learn well by example and iteration. Nevertheless, while before I couldn't say with certainty that I would prefer to not use computer-graded homework in teaching introductory math courses, I feel that having made use of the software in my course I can credibly affirm that statement.

Without going into detail, let me lodge three objections (relatively briefly! I'll flesh these out later once I've had a chance to further reflect on them, most likely once this semester's behind me) to the software:

1. Its use foregrounds the medium at the expense of the message: in asking the students to master the software's often unnatural and indeed often byzantine commands for entering mathematical notation with a Flash-driven interface, the I feel that all too often the computations the students need to perform to generate a correct answer are overshadowed by the mechanical manipulations they must undertake to enter the answer into the computer.

2. The software places the students at a distance from the instructor, to an extent that effective bridging of that distance vitiates the need for the software in the first place. To wit, even with the ability to zoom in on a single student's solution to a single specific homework problem, the opacity of the software's interface does not allow the instructor to penetrate beyond the student's final response. Should this response be wrong, there's generally no way to deduce from it just what it is the student did incorrectly without asking the student to submit her or his handwritten notes. Of course, I am all for students' working out solutions by hand...I continually exhort them to do their work on paper and use the computer only to enter their solutions...yet if ultimately I have to dig up their handwritten notes in order to tell what it is they're doing right and wrong, what's the point in having the software in the first place? I might as well simply ask that they submit their homework directly to me, let me grade it, see it, be in contact with it, and reestablish the missing and much-missed bond between the students' understanding and my own, without the electronic intermediary.

3. Finally, and most fundamentally, I feel that the software system by its nature reinforces the common and erroneous perception of mathematics as a rigid, timeless, universal enterprise. As the computer is trained to expect only a very particular form of answer to each problem it provides, the student may come away with the mistaken notions that math is a field in which there is a single correct answer, in which there is no gray but only black and white, that process is unimportant if the product is ambiguous, that every instance of a certain calculation requires a single form of solution. All of these claims are preposterously wrong and further the view of math as far-removed from ordinary ways of thinking, as something undertaken only by pointy-heads who've mastered arcane rules of mathematical computation and communication. Human-graded homework, on the other hand, far more sensitive to idiosyncratic-but-correct responses, to slight variations in notation and style, to math's true nature as fluid, era-dependent, humanly-crafted enterprise, allows students to succeed by responding in various ways that reflect their own particular learning styles. The unmediated bond between student and teacher facilitates the latter's ability to convey the perception of mathematics as a ground in which critical thinking can be taught, and the former's ability to construct a personal mathematics all her own.

These are deeply-rooted philosophical objections about which I hope to say more later. I'd be interested in hearing others' take on this matter, I don't claim to have the final word!

On a wholly different note, I've finished my first "exotic" (G,φ)-gram, although I must admit that I'm unsatisfied with one of the steps I took in its construction. I'll admit up front that I wrote another Mathematica notebook to help crunch the noncommutative multiplications that go into the poem's analysis.

The poem is a (D₄,φ)-gram, D₄ the dihedral group of order 8. The homomorphism φ that governs the poem is one Mathematica chose at random; this is the part I'm not happy with, as I'd rather choose a φ that's "meaningful" in some way, that relates each letter to an element of D₄ in a "useful" way. But here's the rub: what choice of φ would work best? My thought was to assign to each letter the longest element of its stabilizer subgroup (relative to the presentation of D₄ in which a represents the reflection in the vertical and b the reflection in the SW-NE line)...but it turns out that practically every letter then goes to an element of the abelian subgroup {1,a,bab,abab}. (The only one that doesn't is "Q", with its funky NW-SE symmetry, which goes to aba...and how often is "Q" used?) Thus if one reflects (or in fact in any way permutes!) the letters of a given line, the value of that line under this particular φ is unchanged.

Boring!

Even worse, any choice of φ(α) from Stab(α) will lead to the same problem! Thus my opting for a randomly generated homomorphism.

My hope was to write a poem in which the value of each line is the group-theoretic inverse of the same line written backwards (letter-by-letter, not word-by-word). The poem below achieves this, although it's an admittedly simple poem. However, it was surprisingly easy to construct (owing probably to the smallness of the governing group), so expanding on this theme would likely not be hard.

So here's the poem, with the homomorphism following it:


Inverse

What kind of mirror symmetry
must a piece possess
for its value to invert itself
when we trade east for west?


Let me give the homomorphism by a listing of the preimages:

φ^-1(1) = {C,L,P,Q,X,Z}
φ^-1(a) = { }
φ^-1(b) = {A,F,O,R,S}
φ^-1(ab) = {I,U,V}
φ^-1(ba) = {G,Y}
φ^-1(aba) = {H,M,W}
φ^-1(bab) = {B,D,J,K,T}
φ^-1(abab) = {E}

Thus only 7 elements ended up in the center of the group, and only 4 of these ("C","P","L", and "E") appear in the poem.

Anyway, I'm having fun. My Mathematica code will enable me to work with dihedral groups of any order, so I may try out a more complicated example later if I have time. Now though, I've got to meet with Sylvester in a few; he and I are continuing research on caterpillar labelings this semester, and we're meeting to debrief after his presentation in the Senior Seminar yesterday afternoon.

Let me close with a note to my Abstract students: keep up the good work! Your committee presentations are already at a very high level. You're doing a great job of highlighting common difficulties and errors, and in giving credit to particularly insightful methods your peers apply. I like that you're all making note of the fact that there's generally more than one way to prove a given proposition. You're also demonstrating a good understanding of the "metamathematical" aspects of mathematical writing. In particular, I appreciate the attention you're all paying to the "Four Cs" criteria, and I hope those criteria are helping you to learn to discern good math writing from bad.

And now, adieu! Thank you for reading.

Post #200

It's been a long and tiring week, but a pretty good one.

We're now about three weeks into the new semester, and I'm feeling good about it so far: my classes are fun, I've set a good pace in both, my students are engaged and willing to work.

The past week or so I've felt on edge about something, though, as though I've not been able to find a groove. It wasn't until yesterday morning that I put my finger on what it was that was keeping me from settling in: homework deadlines.

In every class I've taught since I started teaching here three years ago, all homework, projects, papers, projects of any kind, have always been due at 5:00 p.m. on Friday evenings. This singular deadline meant that I didn't have to think about when a particular assignment was due, it meant I could flip from class to class and be able to say in the wink of an eye when students owed me something. It was easier on the students, too, for they knew the answer they'd get if they asked me when something was due to my desk.

It's the course software for Precalc that's been throwing me off: freed from having ("getting"? I'm missing it!) to hand-grade the students' homework, I've been able to glibly toss about due dates without any restriction imposed by my grading schedule: Monday at midnight? No problem. Wednesday at noon? Fine, again...and somehow that glibness carried over to my assignation of Abstract deadlines, too, so their homework has been due willy-nilly throughout the week.

No more.

The nerve-wracking chaos ended yesterday when I informed all of my classes that I'd be reverting to the Friday at 5:00 deadline. It's a seemingly minor change, but we'll see if it doesn't put me at ease!

Meanwhile, as I mentioned above, all's going quite well. I'm well aware that we're making our way through Precalc a little more slowly than we should be, but I imagine once we get through the algebra review I'll pick up the pace a little bit; I feel that it's important to establish a firm foundation of algebraic skills before moving into a place where we'll have to apply them. I've just passed out the first written assignment, asking the students to minimize the cost of constructing a box with a fixed volume and certain dimensional constraints, and to put all necessary data and computations in the form of a written report to a container manufacturer. I'm eager to see how wel they handle this project. I'm imagining that it'll be a snap for some, and a mountainous challenge for others.

Abstract's chugging along; we've had two sets of committee reports so far. Yesterday's reports were very strong, they've already begun to break away from the "here's the answer" mode of reportage, in which they don't really comment on their peers' work so much as solve the problem for the class. "That's not really helping anyone," I've reminded them. I've exhorted them to provide helpful, respectful, and specific feedback: "good!" is as useless as "wrong!".

I was particularly gratified by one of the students' reports yesterday, in which she indicated the importance of clarity and composition, emphasizing how even if someone had the right answer it was often difficult to discern this if the proof's wording were awkward, if its structure were nonexistent. It was clear that she'd read the "Four Cs" style sheet, if nothing else.

Well, I'm off...the weekend lies before me, and I've got relatively little grading to do (a few factoring problems and a team quiz from my Precalckers...shouldn't take more than a couple of hours during some football game later), so I might actually get some down-time tonight.

Happy 200th post!

Same proof, different theorem

It's been a heartening day since I last checked in.

I'm happy to say that today's installment of 280 proved a useful one, as far as I can tell.

A few weeks ago I spent a bit of time designing a suitable peer review component for the third and final exam for the course. Since time permitted neither exam revisions (as I'd allowed for the first exams) nor a by-now-typical committee-based peer review of one of the exam problems, I decided to allow those who completed a draft of a particular problem (namely, the first on the exam) to take part in an in-class peer-review activity in which participants were divided into groups of three at random, allowed to discuss their approach to the problem within these small groups, and finally given the chance to share their groups' discussion with the reconvened class at large.

Without fail, everyone completed a draft of the indicated problem, and group discussion was lively and apropos. After ten minutes, we met again as a class, and several of the most eager folks in the class took turns presenting their solutions to bits and pieces of the problem.

At one point Quincy scrawled on the board both a certain proposition and a "proof" of this assertion. Although the proof was a flawless justification of the proposition we really sought, the proposition as stated was incorrect. One of his peers pointed out the error, and with a slight modification, the theorem read correctly.

"Same proof, different theorem," I said. "I need a t-shirt that says that."

I'm impressed with how willing these folks have become to get up to the board and perform math in front of their peers: even Dewey, a relatively reticent soul, spoke up once or twice today when he believed his friends to be in error. And Fiorello didn't skip a beat before taking the board to slam down a nearly perfectly composed proof of one equivalence relation's transitivity.

I'm going to miss this class, it really has been one of my favorite so far at UNCA. I've learned as much from them as they've learned from me.

Today I've had several other things to be happy about pedagogically, professionally, philosophically.

This afternoon I had a brief tête-a-tête with my 280 student Keiko, who over the past couple of weeks has made tremendous strides in coping with equivalence relations. It's clear to me that she's truly understanding them, not just going through the motions. Though she's still making little errors here and there, the mistakes are typographical and not logical. I'll take an armload of typos over a single conceptual slip-up any day.

In an e-mail from Barrymore, one of my first-section Calcsters, I got some of the most useful teaching ideas I've ever received from a student. A veteran of several mock trials in high school, he offered me some advice on how to make the trial experience a more useful one, a more intense one, a more authentic one. His advice centers on introducing the instructor as an actor in the drama, perhaps as the defendant, or perhaps the plaintiff. As students are called on to challenge not their peers (who may be more or less knowledgeable about the subject at hand, depending on their level of preparedness) but rather their assumed-proficient professor, the care with which they must construct their arguments is concomitantly heightened, and the stakes are upped.

Barrymore therefore suggested having faculty play the roles of Newton and Leibniz, while students are asked to play the lawyers, witnesses, and colleagues.

I'm not sure how I feel about this advice. It's solid, to be sure...I'm only wondering if the benefits described above would be outweighed by the loss of the students' opportunity to play the leading roles.

Barrymore also suggested that the various experts should be asked to meet with the respective legal teams in order to perform "depositions" of sorts, to make sure both sides agree on a consistent set of evidence. I like this plan.

The specificity with which Barrymore was able to offer advice showed that he has really thought about this project. I respect his judgment and will certainly consider his input when I put this project together again.

Just half an hour ago I got an e-mail from Bethesda, eight pages into her final paper on the issue of computer proofs she's writing for our independent study on the history of math technology. She's frazzled. She feels uncomfortable making claims about the proof of the Four-Color Theorem, the proof of which she can barely understand (especially its migraine-inducing implicitness). She's questioning the validity of computer proofs, questioning what it truly means to be able to prove something in the first place. In one long-running paragraph she spat out a dozen or so insightful observations whose perspicacity made me want to weep with joy.

I'm going to have to think for a bit before offering a robust reply.

It's days like this that make me glad that I do what I do.

I'm off to bed now. It's nearly eleven, I've been up since four this morning, and I spent nearly thirteen hours on campus getting "caught up" after a "break" during which I worked for nearly a day altogether.

What's wrong with me that I work so hard and yet love that work so much?

One parting note: I've received the go-ahead from several Calc I students to quote from their reflections. Excerpts to come soon!

Harder than it looks

Teaching is hard.

What makes it this way?

After 280 wrapped up yesterday, I was walking back to Robinson Hall with Quincy, and he was rehashing the experience he'd had less than an hour earlier with two others from the class, their committee report on the latest problem set. He was worried that what I had seen as a successful endeavor had gone horribly awry; it hadn't gone as he'd expected it to. It wasn't as smoothly executed, perhaps, and he hadn't been able to clearly get across (pardon my split infinitive) the "subtle nuance" he was trying to point out in the proof they were critiquing.

"It's really hard to lead someone to say what you'd really like to hear her say, without telling her to say it," I assured him. "That's called teaching." From a pedagogical standpoint, I felt that their committee report had been a solid one.

Most heartening to me is the fact that the students have begun to break away from the "here's the right answer" style of committee report with which they began the semester. "We want this to be a discussion, so if you have anything to say, just come out and say it," Nicolette exhorted her peers yesterday. (Like Quincy, I think she feared that their team just wasn't saying whatever it was they'd have to say to get the others out of their seats.) They were aiming at a different model for their presentation: rather than simply hand out the "correct" proof of the indicated proposition, they intended instead to guide their peers to a proper understanding of the weaknesses of the proofs they saw, and of the necessary elements of a valid proof. The second team had the same goal, and they strove towards it with different steps. In lieu of doling out a proper proof, they instead gave an outline of the elements such a proof would need, indicating where they felt people might have tripped up most commonly. Neither presentation was fully explicit, both focused on the process instead of the product, both adopted a more mature attitude regarding the course content than most earlier discussions had.

In case you can't tell, I'm pleased!

I cornered one of the other students in the Math Lab after class and asked her how she would compare this semester's installment of the course with last semester's. (She'd been enrolled in the course in the spring until health issues forced her out about 2/3 of the way through.) She's enjoying it much more this time around, and she feels like she's learning more effectively. The format, she says, is much stronger (what about it? This was unclear. Is it the idea of the homework committees? The structure of the worksheets? The revisions? I'm not sure...I'll have to probe further), this particular group of students is more open to the idea of learning in this way.

Perhaps what's working well for her is the more conscious focus on writing instruction, in particular writing as a discipline-specific endeavor. She indicated that partly as a consequence of our class she's seen her writing improve in all of her classes. (Her partner, who proofreads all of her written work for her [what a kind soul!], has noticed this as well " 'I only had to add a couple of commas for your last paper,' " my student reported her partner's words.) With the clearer writing has come a clearer understanding: her scores this time around are noticeably better than her scores in the spring. As I told her yesterday, I'm so happy I could hug her.

I am dying to find out what kind of responses we get on the exit surveys for the writing assessment project. This is an exciting study!

Yesterday too I had the first of two interviews with one of the Writing Center's new student consultants. I prepared myself by going over the interview questions I'd been sent in advance, and when Beulah came by, I was all ready...perhaps too ready...with my responses. I actually printed out the abstracted answers to her interview questions that I'd typed up for myself, and asked if she'd like a copy. She accepted them happily. "That's fewer notes I have to take!" she said.

"I hope it's okay that I give you those," I said. She assured me that would be fine.

Our subsequent interview was a brief one, but she asked some good questions. "In what math classes do you feel that writing is important?" I warned her that I was answering for myself and not for my colleagues up and down the hall, and I told her I felt that writing was of pivotal importance in any class, including any math class. It wasn't until I said it out loud that I realized how strongly I feel that way, and how rare that feeling might be among my colleagues. (By "colleagues" I mean my colleagues in the profession, not necessarily the other folks here on the Third Floor.) I cannot imagine teaching a class in which writing didn't figure into the curriculum in some way, whether it's a conscious focus of the class, as it is in 280, or whether it takes the form of a few simple papers on vaguely mathematical topics, as in Calc I.

What else do I need to say right now? I thought to make a dent in the list of topics about which I wanted to say a little, but it seems like I'm running to stand still. (This is a good thing, to be sure, but a frustrating one...I need to invent a way to increase the length of the day by 20% or so.)

With the help of my students, I'm busily amassing good ideas for activities to take place in existing classes, and for classes we could offer as a part of our curriculum:

1. Last week the idea of math-themed poetry arose from two completely unrelated sources. We batted the idea around a bit in both of my Calc I sections, and it came to the fore as a topic of discussion on listserv of the MAA Special Interest Group on Math and Art, of which I am a member. I've decided I'm going to make math and poetry the focus of the last of my Calc I projects for the semester, a short one that'll cool the students down after the leviathan efforts they'll have expended on the Newton v Leibniz project. I'm going to incorporate a bit more guidance and instruction into this project than I did the last time I asked students to construct mathematical poetry, a sad little project I put into action during my grad school days at Vanderbilt. (Totally unrelated note: Vanderbilt, at 5-3, has the same record right now as Florida. Go 'Dores!)
2. I'm liking more and more the idea Quincy pitched a couple of weeks back regarding a "Random Seminar," in which participants, students and faculty alike, did research into and subsequently constructed classroom exercises around mathematical topics pulled from a goldfish bowl placed at the center of the room. It could work. It would take some fine-tuning, but it could work.
3. Quincy pitched another good idea to me by e-mail. A bit less ambitious, this one involves a component exercise for the 280 course: each student has a turn in which she presents a particular nasty proof she's been struggling with to the rest of the class, receiving feedback, suggested revisions, and so forth. Sounds kinda Moore-ish to me. Maybe we'll get a chance to do this a little bit before this semester's over. (Quincy, you're gonna love next semester's graph theory course!)
4. Another course idea that'll take some work to clean up is an "Unmath Seminar": students are asked on the first day of class to make an alteration to some fundamental axiom of mathematics, somewhat akin to denial of Euclid's Fifth Postulate. From that point on, students are asked to construct an internally consistent mathematical system that obeys all laws that are consequences of the assumptions made at the outset. If inconsistencies arise or inordinate difficulties ensue, seminar participants would be allowed to return to the starting point or some other point intermediate in the construction in order to modify their assumptions to make the resulting system more amenable to analysis. This whole project would be hard, and would require a good deal of advance planning to make the exercise worthwhile. Moreover, the students taking the course would have to buy into the project completely to make it work.
5. On a less ambitious tack, at some point I'd like to offer special topics courses on lattice theory and set theory.

So much to do!

I've got lunch this afternoon with Quimby and a couple of our cognition folks over in the Psych Department. I have no idea what they're going to spring on me, but I've no doubt it'll prove to be interesting. I've yet to send an e-mail to my colleague in mass comm whom Quimby recommended to me as an interested partner, regarding my idea for a "communicating mathematics" course.

For now, I'm off to class...to be continued!

Random thoughts, volume 2

I had a few interesting conversations today, with colleagues and with students. I also found myself unfairly piqued during my second section of Calc I, and I feel an apology is in order to my students.

Let's start there: Quiz 4 came today, asking for a brief rundown on the two fundamental interpretations of the derivative. As I would soon forcefully point out to my students (post-quiz), it's not as important to me that they memorize the formula for the derivative, nor that they master every one of the rules for differentiation we will soon study, as it is that they understand what derivatives mean, and how it is that we can see them in nature, and put them together to help us understand natural phenomena. As I put it to them, Mathematica can do all the derivatives for us, and much more quickly than we ever could. What Mathematica can't do is study a natural process, recognize that there is an interrelationship between two or more dynamic quantities, chart those interactions over a long enough course of time to posit a model that describes the way they depend on one another, and use that model to put together the derivative that gets fed into Mathematica at the end of the line. Mathematica, in this sense, represents the mathematics of the past, when it wasn't yet the case that there was a handy formula that one could apply to find the derivative of a given function. That was then. This is now, and the future is yet to come. The mathematician of the future needs to know more than a mechanical rules for finding derivatives (as important as they are to be able to apply well); she needs to know how to use derivatives, how to interpret them.

Of course, when I said all this to them, it came out ne'er so fluently as it did just now, above. Figures, huh?

Performance on the quiz was...meh. It wasn't horrific (I've given harder quizzes), but it was by no means stellar. A couple of my best students cornered me after the second section and asked if they were going to be okay from this point on. Tallulah: "because I didn't do well on that quiz." "I doubt anyone did," I told her. "The folks in the first section pretty much biffed it. It was a hard quiz, largely because you're not used to being asked questions about concepts rather than computations." She's a fantastic student, she'll recover splendidly.

I realized even as it was happening that I was (unfairly) letting my frustration with my students' conceptual misunderstandings get the best of me for a few minutes during that second section's class. I threw markers and punched the board like I always do, but I did so with more vigor than is typical, partly to dissipate my frustration. "How dare they not get this? Damn it, what, they think they can coast in this class if they spit up a formula or two?!"

My righteous indignation subsided as the class went on, and I realized by the end that if they'd not focused on the concepts over the calculations, it was as much my fault, and my colleagues' faults, as it was the students', for not asking them to refocus their attention elsewhere in the first place. I've got to try harder at that myself. For some reason, I have to admit, Calc I has proven the most resistant of all courses to redesign along the lines of discovery and application-based learning. Only now am I beginning to understand what a truly problem-based Calc I class might look like, and I admit that this semester I'm falling far short of that mark.

I promise a less angry, less frustrated tone tomorrow, folks: you really are a great bunch of students, and I enjoy working with you very much. Let's make tomorrow's class a good one, huh? I'll bring some donuts tomorrow morning, and we'll start off with a couple of conceptual exercises to get our creative juices flowing. Sound like a plan?

Good.

From the second section of Calc I, it was off across the quad, to the second of the semester's Writing Intensive meetings (the first was this past Monday) for me. As I anticipated, I'm enjoying working on this committee, conferencing with a group of peers who feel as strongly and as passionately as I do about writing-to-learn and writing-across-the-curriculum and writing in general. I'm starting to get a good sense for the way writing is integrated academically, campus-wide, rather than simply in my own courses and in those of my math colleagues. The bar is quite high; the quality of writing instruction university-wide is solid. Nevertheless, there is room for improvement, and I found myself in a heated exchange of hallelujahs with Lexington, the WI committee's acting chair, as we walked back to our shared building after the committee meeting.

We agreed that the university has made tremendous strides forward in terms of embracing writing-across-the-curriculum, undergraduate research initiatives, outcome-based curricula, discovery learning opportunities, and so forth...but that there's also a lot of work to be done before perfection is reached. "If we're going to advertise that we're using discovery learning," Lexington said, "we've got to start doing just that, and to do that we're going to have to get serious about giving people the resources they need to do that." We agreed that we need to try to drive class sizes down (I mentioned my conversation with my own Chair last week regarding getting my Calc I classes capped at a lower level), we need to offer kids the opportunity to engage in alternative classrooms early and often, we need to make a focussed, directed, campus-wide effort to provide these opportunities to students from the get-go.

After a brief stop at my office and a moment in the Math Lab to unstick the stuckness of a few of the Calc I kids in computing the derivative required of them in the team project, I was off across the quad again to 280. Davina caught me before class with a few concerns about her service on one of this week's homework committees. For one, she wasn't sure about what to write on a person's submission if he said something like, "I'm stupid, I can't figure this out," or something along those lines. "That's a hard one," I agreed. More substantially, she wasn't sure she was giving the right kind of feedback, and she felt like she was being hypercritical, telling people to reword this, change that, and so on. I suggested that she might try to balance positive and negative feedback, and to offer comments like, "I'm having trouble understanding this, could you make this more clear?" or "This is a really good insight, it really helped me to see this point more clearly!" I later reiterated some of these ideas to the whole class, and wrote on the board: "Recall that the purpose of the committee work is not to homogenize, but rather to help people to clarify their own individual ideas."

The subsequent committee reports were good ones. I was particularly impressed with Davina's discussion during the presentation she and DeWayne gave on the homework problem they'd been assigned. I admired the way she was willing and able to come out and say that her serving on the committee definitely helped her to better understand the concepts involved in the problem they'd reviewed: "seeing how other people did it made me see how I could make my own writing more clear and more concise." I'm glad she came out and said that, and I hope her sentiment is shared by the others. I'm certainly going to ask the students about their committee experiences explicitly when I pass out midsemester evals in a week or so.

Came then (after another half hour of set theory) the trek back to Robinson Hall, where I'd spend a few more hours before heading home. I finished grading the second section's Quiz 4s, on which they did marginally better than the first but still not wonderfully. I also got a chance to work with a number of the teams as they struggled through their projects (they're all doing quite well, from what I can tell), and I met up, one-on-one, with several of the 280 students, helping them to polish various drafts of homework problems. They're definitely developing an appreciation for more and more subtle nuances, meanings of stereotypical mathematical phrases ("thus...," "for every...," and so forth), and clarity, clarity, clarity in writing. (A funny, and very heartening note, if I may: at the close of today's committee reports, I reminded the students to keep an eye on the rubric I'd handed out last week as they worked on their math writing, and I asked them if they could remember the "four Cs." In nearly complete unison, they intoned: "correctness, completeness, clarity, and composition." I didn't have to say a goldarned thing. I was a happy man.)

While I was finishing off those Quiz 4s, Cuthbert, Chemistry colleague of Lexington and a big, big man in undergraduate research, came by to ask me if I wouldn't mind providing him with the titles of the projects the REU kiddies worked on this past summer: he was putting together material to take to a regional conference on UG research, indicators of the sorts of things that can be made to happen in a liberal arts science curriculum. I was happy to oblige, and not an hour later I'd send him a list of the topics. We then got to talking about the inclusion of research components in UG courses, and he asked if I'd done much of that. "Depending on how you defined research," I told him, "I do that in nearly every class I teach." I told him a bit about the course that spawned this blog (MATH 365, Linear Algebra I, taught last Fall semester), and he asked if he could have information about the projects those students worked on. I obliged him there, too, sending him the prompts for a few of the research projects (Monopoly, Traffic Patterns, and Come on, Feel the Noise).

We hit a few more points in our short conversation, enough to give me a sense of déjà vu, feeling as though I was reliving the conversation I'd had earlier with Lexington. We talked about the increasingly interdisciplinary nature of scientific study; the need for broader, deeper, more meaningful implementation of discovery learning in our courses; the need for more robust interdisciplinary course offerings than an occasional and disingenuous cross-listing: we need team-taught classes, classes that provide a true interface between one field and another, like one Cuthbert mentioned that took place at his previous institution, in which beginning chemistry students learned their ways around a lab while generating real, unprettified data that could be fed to statistics students who would put it through realistic rigorous analysis, the results of which analysis could be funneled back to the students who ran the lab experiments in the first place in order to help refine their techniques. We agreed that a course that combined the concepts of physical chemistry with those of linear algebra would be a tremendous boon to both the Chem and Math departments; I have no doubt that we'll be talking about this again soon.

Thence, from my office, to dinner at Sorrento's (offering the best Italian in town in a cozy out-of-the-way den halfway to Oteen that sadly never seems very busy), and then home. Once home, I finally had a chance to call Bedelia, my colleague late of Harvard, now of Lesley University, and catch up with her. As it was bound to be, our conversation turned to...teaching! Imagine that! We talked for a bit about a program she's like to get off the ground, a sort of summer prep course to get local (in her case, Bostonian) college-bound kids up to speed on the math skills they might need to succeed in a challenging entry-level math course, while providing them with some useful advice on making the transition to college more smoothly. I compared the program she was describing with UNCA's own SOAR program, and she seemed to agree with the comparison, only she emphasized the local nature of the plan she envisioned. Then we talked shop for a bit, I about my own six courses (if one counts the senior seminar and my three independent study students), she about her Quantitative Reasoning and Pre-Calc classes.

We're both having a good time.

Hey, I've rambled on long enough, it's time to stop. I'm going to make a smoothie, maybe watch an episode of Mr Bean or some other such nonsense, and call it a day. Thanks for reading, I hope you'll free to comment, I always appreciate your feedback!

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...

Carolina dreamin'

I had an odd teaching-related (so Maggie says) dream this morning.

I dreamed that I was house-sitting for Maggie's Aunt Susan (if you know Susan, this whole dream is inexplicably funnier). In her dream house she had a titanic tropical fish aquarium...we're talking floor-to-ceiling here, occupying the entirety of a room or two. As you might guess, she had several large exotic fish in this aquarium, and though I seem to recall one or two were sufficiently weird to try to remember what they looked like to describe them to Maggie after I woke up, I've since forgotten all about them except that they were rare and rather large.

At some point in the dream, a dog (not one of ours) jumped into the aquarium and in some fashion broke through the glass, causing all of the water inside to wash away, dissipating through the dream's drainage system without causing a lick of damage to the house's furnishings.

The fish, however, were gone. There ensued several dream-minutes of minor panic as I careered around the house, vainly seeking the fish that had disappeared. I was in a bit of a bind; I seemed to think that Susan's return was imminent (if you know Susan, you know it's probably best not to be on her bad side). After a thorough search I could find neither fin nor scale of even the largest fish, but I was saved from almost certain death by waking before Maggie's aunt returned.

I told Maggie about the dream later this morning. "It sounds like you're worried about failing in some responsibility," she said matter-of-factly.

"Thank you, Dr. Freud."

"Probably in your work or your teaching, you feel like you've let someone down."

Funny, after Friday's 280 class (see yesterday's post).

Thinking a little bit further about what went on in 280 (I promise, I'll stop over-thinking this once I've done writing this post!), I realize that I most definitely should have just shut up and let the committee report play itself out.

Oh well.

I'm going to muster my courage to do just that from now on. I'm pretty certain that this semester's class is on average older than last semester's, and as I've mentioned recently, more cohesive, more mature, more ready for the responsibility I'm entrusting them with. For my part, as well as I led the class last semester, I feel that I've done a superlative job so far this semester in getting the students ready for the responsibility I'm expecting them to fulfill. They can handle it.

I'm going to throw them a sop, too. Well, it's not really a sop, it's simply an opportunity to add another layer of revision in the construction of their written work, but in addition to giving them the chance to improve their writing, it'll give them the chance to improve their grades as well (and a little extrinsic motivation doesn't hurt, right? I just happen to get paid fairly handsomely for doing a job that I'm good at and that I love). To wit, I'm going to give them the chance to revise two problems per homework assignment, after I've given them feedback. Grades'll be handled as with the exams, into which I've already built in guidelines for revision.

I'll put it to the class tomorrow, but I don't foresee major objections.

For now, I'm off to enjoy a lazy Sunday evening. Tally-ho.

Friday

Boy howdy, was my brain ever in Friday mode yesterday. I was muffing this and that, one thing after another, and until late in the afternoon my to-do list was growing longer far faster than I could cut it back.

I've had a good weekend so far, though, so I guess things are evening out in the end.

So, yesterday, what of it?

The first misstep of the morning took the innocuous form of a forgotten stapler. I'd meant to make a clean start and begin bringing it with me to class Fridays so that my Calc I students could properly assemble their ever-unattached homework pages in some manner other than messily crimping the corners together in a sad and useless little lump. Of course, the stapler found itself left behind on the corner of my desk.

No biggie. 'S all good, 's all good.

Then I started handing out the Mathematica installation disks (finally). Six students in each section got a disk, and I calmly went through my little spiel about the installation process...completely omitting three crucial points: (1) they'll need passwords to register, (2) they won't get the passwords instantaneously, but rather those'll be sent to them by e-mail within a day or two, and (3) they must use their school e-mail addresses when registering, otherwise our license manager won't know any one of them from Moses on a pogo stick when it comes time to kick a password studentward. (I later remedied the oversight by sending the entire class an e-mail about the correct procedure.)

I soldiered on. Saturday, after all, was but a half-day away.

It was 280 that was truly frustrating for me, though I know it probably shouldn't have been.

The second committee report of the season (regarding the construction of a truth table purporting to demonstrate a certain tautology) went off without a hitch, and there was robust, respectful discussion on a number of salient points: "Do you have to include all relevant columns in the truth table, or can you omit a few if you can perform some of the operations (like simple negation) mentally?" (It was then determined that one ought to write not for one's own understanding, but for that of the reader, and error should fall on the side of liberality in column inclusion.) "Even though it's obvious if you're looking for it that the first column (containing statement P's truth values) and the last (containing the truth values of a logically equivalent compound statement R) are identical, so we're meant to conclude that R is true if and only if P is...but wouldn't it also be correct to answer the question 'what can you say about when R is true?' by comparing R's truth to some other column, if you didn't notice the equal columns?" (It was agreed that though technically another answer might be correct, indicating the tautology P <=> R would be a "more correct" response to the question.)

The third committee report (concerning a tricky proof by contraposition, asking for a verification that [not Q] implies [not P] in order to prove P implies Q) went a bit more roughly. Tamar and Cornelius led the charge, and all went well in negating the given statements P and Q. But things got a little dicey when it came time to prove the desired contrapositive. Perhaps due to the subtle nature of the negated statements (one was a conjunction requiring a DeMorgan Law), the team got [not Q] and [not P] turned around. I'm not sure they were completely comfortable with the proof they presented, though: Cornelius correctly indicated that he wasn't sure their proof would handle one of the three cases which might arise in [not P], but they weren't able to salvage the proof once it foundered.

At this point, I stepped in for a few minutes to try to patch together a proof of the corrected implication [not Q] => [not P], but my own hastily-assembled argument was a weak one. It was technically correct, but smacked of proof by contradiction, something we had yet to discuss (and in fact would begin discussing a few minutes later), and didn't explicitly use the DeMorgan Law I hinted at on the homework sheet.

By the end of the report, I think everyone (including me) was a little confused and wearied, and we had only twenty minutes to finish a direct proof from the previous class period and to begin tackling contradiction. We made our way through an outline of the proof technique, and now stand poised to construct our first contradictory proof, a feat we'll undertake on Monday.

After class had ended, Dewey came up to me and asked me to take a look at his solution to the contrapositive problem.

It was beautiful.

Aside from a small boo-boo in the negation of one of the statements, his proof was error-free, elegant, and made full use of the required DeMorgan Law. Best of all, it didn't have a whiff of contradiction about it.

Ah, c'est la verification.

So why was I "frustrated"?

Because I know what it's like to muff something in front of one's peers: it ain't pleasant, and I hoped that the folks on that third committee didn't feel overwhelmed by the problem they'd been given.

I also felt frustrated that at the time I'd felt obligated to step into the ring, when I probably should've just stayed the hell out. After all, the whole point behind the use of the committees is to (fittingly) commit its members to take authority over the task they've been delegated. I see my role as that of an ex-officio, advisorial member of each committee formed: though I might provide a little input behind the scenes if it's asked of me, it's not my place to usurp the committee's authority in the classroom. If I keep doing that, how can I expect them to grow more comfortable in wearing the crown? Until yesterday, I feel I've done a really good job in reining in my own reign, and my frustration is probably born from the fact that yesterday, improperly, I let slip my own authority.

In that regard, I definitely fucked up.

On the other hand, the experience provided an object lesson to everybody: we're all going to miss a few now and then, and as I've said many times before (and as many smart people have said before me), learning how to prove things and learning how to clearly record those proofs are iterative processes that generally make their progress (sometimes) painful fits and starts. The students all contributed their versions of the desired proof, the committee undertook the thankless (if you're reading this, thank you, all!) task of collating these versions and producing their own, incomplete version, I made a half-assed attempt at cleaning this up, and Dewey succeeded in showing me up with a nearly flawless feat of mathematical legerdemain.

And hell, isn't this sort of imperfection the essence of discovery learning?

Yeah, I'm learning, folks, I'm learning. Slowly, maybe, but I'm learning. I've certainly got a rough road to travel to my own mastery of that method. In the past several years I've come a long way down that road, but the had sun set on Friday before I could take another step.

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.