Impending sense of something

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

All is calm.

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

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

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

I'm new here

Parker J. Palmer's and Arthur Zajonc's The heart of higher education: A call to renewal (transforming the academy through collegial conversations) (the centerpiece of the most recent faculty Learning Circle in which I took part, and about which I've posted somewhat recently) gave me more than its fair share of things to think about. Many of its insights offered theoretical, even spiritual, enlightenment regarding teaching, but other insights were more practical and practicable.

One of the more down-to-earth suggestions offered up by one of the coauthors' colleagues (Patricia Owen-Smith, Professor of Psychology and Women's Studies in the Oxford College of Emory University) was a means of encouraging contemplative practice in the classroom simply by playing music to being each class period. Owen-Smith (pp. 157-161 in the above work) describes how several years ago she began the practice of playing 7-9 minutes of music at the outset of each class. She encouraged her students to "go within, be still, and listen to the self." While she admitted the difficulty of tying this contemplative practice to improvements in students' achievement of cognitive learning goals (as if this would be the purpose of playing the music in the first place!), she reports that

over time, I learned that the music and meditative moments had an impact on many students. Some students began to ask for guidance with their contemplation and reflection....By midsemester several students per class would mention that they looked forward to this nine-minute period of music. Some students began to bring music from their own collections that they found inspirational and important. As we neared the end of the semester, the structure of the class had changed from a group of individuals reluctantly gathered together for study to a community of friends and partners who were creating a space of introspection, quiet, and respect for the process of study and the development of self.

Once or twice a week for the past few weeks I've begun a similar practice in one of my classes. It began with my reading of an excerpt from Rilke's letters to Franz Kappus (a reading which moved one student so much she had to leave the room), and continued with a passage from Dick Leith's history of the English language. At one student's suggestion one morning we watched a scene from Harold and Maude, and the next day I read one of my own recently-written poems ("Ode to Ned Maddrell," which I penned for the last-living speaker of the Manx language). To open off on Monday (at the suggestion of another student) we'll take in one of Gil Scott-Heron's last videos.

This practice has developed naturally, in an easygoing fashion, and perhaps unsurprisingly it's happened in the most natural and easygoing section of any of my courses this semester. From Day One nearly every person in that section of Precalculus has worked well together, helping each other out and asking for help when help is needed. This natural ease with which we've worked together all term has made it hard for me to tell whether the contemplative practice has had a real effect on the esprit de corps of the class...

...but I'm not going to take any chances. The practice has definitely helped me to make connections between the intellectual and the personal, between the scientific and the humanistic. It's helped me to remain focused on what really matters, and it's helped to remind both me and my students that we do well to look for mathematics' usefulness...and to look for its beauty. Much like well-chosen low-stakes writing activities, this practice is well worth the few minutes of class time that it takes.

Beginning right away I'm going to introduce this practice in all of my classes. It can't hurt.

What effect can it have in a class like my current Abstract Algebra class, a room full of stressed-out work-wearied students representing, honestly, probably the greatest range of mathematical ability I've seen in one section since I began teaching at UNCA over six years ago? I admit here and now that I find myself frustrated at how I've managed this course. For some time now I've felt the need to slow its pace to accommodate the most modestly quick learners, but without sacrificing the true nature of the subject, a nature laden with often-abstract proofs. I don't know how much slowing the pace down has helped: several students are still struggling, and as much as I hate to leave them behind (I hate hate hate people who teach to the top ten percent), I simply must move on. Moreover, I sense that the slowness has led to frustration on the part of some of the class's quicker students, and I can feel cliquishness setting in...not a nice way to end the semester.

Maybe it'd be wise at this point to remind everyone that we all have a right to say "I'm new here": we're all free to make mistakes from time to time. After all, no one's lived this life before, and the future hits us all at the same time. But no matter how far you go, you can always turn around.

What say, folks? What are we going to make of our last five weeks or so together?

Dead Zone

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

What to do at such times?

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

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

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

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

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

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

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

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

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

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

"Teaching"

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

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

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

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

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

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

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

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

Author! Author!

Every person is the author of her own adventures.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Out of the wilderness

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

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

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

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

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

Into the woods

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

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

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

Back

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

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

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

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

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

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

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

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

Fun Times

I'm smack-dab in the middle of the freest time of year for me, that three- our four-week-long end of May, during which time I'm generally "relaxing" after a successful semester, working like mad to take care of the various reports due from me in the next few weeks, and busily prepping for the REU that starts two weeks from tomorrow.

This year my wife and I were able to get away for our first "real vacation" (defined as "a trip involving neither work nor family visits") in several years, a five-day cruise to the Bahamas. Of course, being who I am I managed to make the most of it by bringing along some highly inappropriate pleasure reading, the thirtieth anniversary edition of Paulo Freire's Pedagogy of the oppressed, the central text for the summer Learning Circle I'm taking part in in June and July. (The only reading material I saw which may have vied with it for the "Most Ironical Reading on a Cruise Ship" award was one woman's copy of Orwell's 1984; I think if she'd have been reading Huxley instead, she might have had me beat.)

I'll likely have more to say in the coming weeks about Freire's philosophy as it applies to math education at the university level, but I wanted to put forward in this post an idea for a new ongoing writing project I hope to implement in at least one of the two classes I'm organizing in the fall (Precalculus and Abstract Algrebra I).

Oddly enough, I got the idea from the cruise company. Every evening around dinner time we were treated to a delightful turn-down service, featuring complimentary mints, expertly folded towel animals, viz.:


and a copy of the Fun Times, the cruise ship's guide to all of organized activities that would be going on on the ship the next day. It was little more than a newsletter, three or four pages in length, just the sort of periodical I think a class (or two) full of students, working together, could crank out at least once a week, if properly prepared to do so.

So here's the idea: ask students to put together an ongoing "newsletter" for their class, The Algebra Times, or The Precalc Picayune, if you will. Its content would be flexible, and what went into it from week to week would be left to the discretion of the students (I'd want to have as little to do with its creation as possible). Perhaps, for example, the precalculus newsletter could include

• study tips,
• hints for tricky homework problems,
  
• advertising for study groups,
• applications of course material to areas outside of class,
• games and puzzles,
• "letters to the editor,"
• recommendations for class activities,
• recaps of recent class activities,
  
• personal reflections on math in general,
  
• etc.

Whatever. It'd be up to them. I'd stay out of the day-to-day operations. Maybe I'm being overly sanguine, but I can imagine a handful of particularly eager students taking on editorial and managerial responsibilities (there are a few in every class). To ensure participation by the class as a whole I'd require every student to contribute to the newsletter at least once, twice, thrice, something like that, during the course of the semester. (Jointly-written contributions would count.) It would be difficult to "grade" contributions (I might shy away from this entirely, keeping it a low-stakes exercise), although I'd provide feedback to authors confidentially.

How's this sound? Colleagues: have you tried something like this before? Students: would this be something you'd be all upon?

Wrapping up

The end is near! (I realize I've been saying this a lot lately, but it's more and more true each day.) We've got just under a week of classes left, with only one more meeting of MATH 480 and three each of Calc II and 280.

My last "regular" meeting of MATH 179 was this morning. We dedicated the day to a free-wheeling discussion of the past semester, focusing on the three central purposes of the class: (1) learn how math is experienced from place to place and across time, (2) learn a bit about a liberal education in general and UNCA in particular, and (3) get some practice in writing academically in a specific discipline, and for specific purposes.

Students pointed out some of the course's strengths: it helped fine-tune their writing skills, gave them a broader view of mathematics, and offered a forum where they could talk about random school-related matters with students in similar situations without fear of looking like fools. They pointed out some rough spots, too: the first text we used (by Ascher) was awful, and the 9:25 time slot wasn't the most conducive to student excitement. A couple of students indicated that I should teach the course again, and I suspect that the next time around it'll go much more smoothly. The students' take-home final exam, due in a week or so, asks them to reflect more intentionally on the three points I mentioned above; I'm sure I'll get from it an even greater sense of the course's strengths and weaknesses.

The MATH 280 students get their last exam tomorrow, too, and like the last it'll be a collaborative one. I was tremendously gratified by the students' reception of the last exam: the learning environment the students created and cultivated as they worked on the exam was a rich and fertile one. I have no doubt that many of the students, particularly those who may struggle more mightily than their quicker peers, learned more from working with one another than they would have from any other form of assessment. (Incidentally, I've been keeping track of collaboration networks on the collaborative exams I've given over the past couple of semesters, and I hope to begin tracking the data more carefully to try to analyze correlations between indices of collaboration like number of collaborators and measures of success like initial exam scores.)

The last 280 exam asks students to provide a few more "traditional" proofs, but, like its counterpart in MATH 179, it also asks the students to reflect on the course and identify specific topics they found interesting and write at length about those. My hope is that the exam will help me to discern which areas students are most interested in, and about which they feel most "iffy." This'll help me plan the course more appropriately in the future.

Speaking of course-planning...I've already got some ideas in mind (involving multiplication tables) for first-day activities for MATH 461 (Abstract Algebra I) this coming fall. I'm excited! I know it's several months off yet, but I'm looking forward to it already.

Before I go, I have to give a shout-out, to my colleague Kelli and to our first-year junior Kamryn, for spearheading the successful founding of a UNC Asheville chapter of the Association for Women in Mathematics (AWM): wonderful first meeting! I look forward to seeing what activities the students can put together in the coming months.

P.S. -- I sent the first draft of my manuscript to the publisher today. Much happiness.

Good day, sunshine

It's been a great week. Now that I'm over the semester's hump (for me the period of complete and utter insane busyness ended about a week ago), I'm relaxed enough to enjoy the day-to-day interaction with my students.

There's nothing like a Thursday for that kind of deal, and today was a wonderful one: I had several awesome meetings with students. A few just wanted me to vet homework solutions (way to go, Karlie! Your answer was spot on! Gorgeous!), others had meetings with me to talk about their upcoming presentations in 280 or 462 (Pascal's fractal and Ramsey numbers! This stuff is awesome, and your presentations are gonna rock!), and another met with me so we could talk turkey about undergraduate research into poset representation of graph coloring games. (Tish, you are so going to clean up on this one...)

I've been in a good mood all week. It doesn't hurt that we're doing awesome stuff in all three of my classes: Riemann sums in Calc I, sizes of infinity in 280, and Galois groups in 462...what's not to like?

It's cheesed me off a little bit that it's taken me roughly 13 weeks to really get into the groove with this semester...and now it's almost over.

Well, I've nearly managed to survive yet another term, and I don't mind saying this one's been brutal. The four-prep schedule (one new, one a rerun but with a new text, and a third, also a rerun, but with a substantial new component) has been a bear: in some weeks it took all that I had just to take care of the bare essentials for each of my classes. I hope my weariness has not shown itself too starkly.

Ah, well. Water under the bridge, I trust.

For now, let's enjoy the sunshine and the last week or so we've got together. This goes especially to my graduating seniors, with some of whom I've enjoyed as many as six classes over the past four years. Nadia and Sylvester top that list...I'm going to miss you guys! It won't be the same department without either of you around.

Happy holidays

Underneath this year's Christmas tree: a three-to-seven percent budget cut campuswide.

Yay.

First in line for the chopping block: adjunct salaries (and by extension ordinary faculty release time) or operating budget?

Our mission: to teach more students in bigger classes with tighter budgets and fewer resources, while we forgo our own professional development and opportunities for relaxation and disengagement from our all-consuming careers.

And by the way, happy new year.

In other news, the "just-in-time" workshop (that's actually how we billed it) on developing and implementing writing-intensive courses that Lulabelle and I ran yesterday afternoon came off very well. We had four very interested individuals take part, and I felt our conversations were elevated and meaningful. It exceeded my expectations, and I think it was worth my time.

I realized a day or two ago that I've not yet filled out my own report card for the semester, as I've done at the end of each of the past few terms. Let's remedy that.

Abstract Algebra I?: A-. This is the second time I've taught the course anywhere, and the first time at UNC Asheville. I think my relative lack of experience showed now and then: occasionally I miscalculated my students' abilities, and every now and then I fumbled a definition or a description, and as well as I I know the material (probably because of how well I know the material!) I still found myself struggling to impart intuition for it here and there.

Nevertheless, the committees functioned more or less smoothly, the students' presentations were the best I've yet seen from an upper-division class, the energy in the both sections was frequently palpable, and 12 out of 29 of the students are continuing on with me to the course's second semester. Any one of these facts indicates some sort of success.

How about...

Precalculus?: A- there, too. I didn't have a bad run of it, considering this was my first time teaching it, ever. I don't mind saying I was scared about my perfomance in the class every so often, and lost sleep over it at least once (just a couple of weeks ago I spent a whole sleepless night worried about "coverage"). Ultimately, though, I think I pulled through: the projects were well-received, the classes animated and engaging enough to distort the students' sense of time (a frequent comment was that the 50 minutes seemed to fly by), and a substantial number of of the students felt comfortable enough to perform at the board with very little cajoling by the semester's end: by then I'd managed to get a room full of relative mathematical novices to overcome their collective trepidation and lead each other to better understanding.

My greatest failing was, not surprisingly, "coverage": I could have used another week. Hell, I could have used another two weeks.

As early as three weeks in I began to suspect that I'd eventually run out of time: at that point I was only at the end of the first chapter of the text, knowing full well that I'd ought to get through the first seven chapters. By the time we'd capped off the second chapter I knew that, even though our pace had picked up substantially, there was no way we were going to make it that far.

Reprecussions?

Minimal, I think. I know 7 of the 29 students will be continuing with me to Calc I next semester, and as all of these students are quite solid and as I know how it is that I conduct the first couple of weeks of Calc I (i.e., with a good deal of review of the skills from precalculus), I don't fear for these students' learning next term.

It'll be okay.

Oh, I guess I could also rate my performance as a Senior Seminar mentor: A. That grade, however, belongs to all of the students with whom I worked, too. It was their willingness, as well as my own, to meet on their own time and go over their own and each others' talks well before they were given that helped us polish them up and make them the wonderful talks that the were. Moreover, their written work was fantastic. Thanks, y'all, for making my job such an easy one. May I ever have such intelligent, diligent, and cooperative students working with me in this course!

Okay, it's late, and tomorrow is another day. Until then, adieu.

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!

What's there to be said?

The cold winter months of 2005 were stressful ones.

My three-year gig in Illinois was coming to a close.

In the previous autumn I'd sent out several dozen job applications, to schools ranging from tiny Alfred University in upstate New York to the Harvard of the West, Stanford University, where maybe just maybe I could secure a second postdoc. Came December, and I started hearing tiny peeps: Cal State Chico had requested a phone interview (which I granted; I thought it went reasonably well), a colleague at Louisville had indicated that he'd seen my application and that his department might be in touch with me in the coming weeks, and a few other schools indicated they'd like to tawk. "Are you going to the Joint Meetings?"

I hadn't planned to. "Unless you're giving a talk in a special session, don't bother going," my Ph.D. adviser had told me three years before when I'd first hit the job market. "The only schools that interview there are the little ones, and you should be gunning for a postdoc." Like a good father, he wanted more for me than he'd had himself.

That was then, this was now.

Within the span of a few short days Seattle University, Carleton College, and UNC Asheville, all on my Top Ten list, contacted me and let me know they'd like to see me in Atlanta.

What the hell, why not? With only a week to go before the meeting, Maggie and I once again secured the services of Gisela, the matronly German retiree who was our regular dogsitter, packed a couple of suitcases, and with the first flurries of 2005 falling gently on the roof of our '97 Camry we began the 10-hour trip to Georgia's capital. (That it's only 10 hours from Urbana to Atlanta is somewhat surprising; it seems that it ought to take longer than that.)

I felt that my meetings with the three schools named above went well (clearly one of them went particularly well!), and although we were only there for half of the conference (I never actually registered!) we had a good time while we were there, catching a few interesting talks, seeing old friends, and putting together the first of what have now been four Annual Semiofficial Vanderbilt Mathematics Department Dinners.

I felt good about the conference, and was in good spirits during the northward drive.

Soon, though, bitterness set in.

Ned, my officemate of two years, was a highly talented number theorist who by the end of his first year at UIUC had already had an article in Inventiones and who was popping out papers at an impressive rate. I'm no slacker, but it was no secret that I put as much stock in my teaching as a I did in my research, so my publication rate couldn't compare with Ned's. Moreover, while offering fascinating and fun problems that are accessible even to talented undergraduates (thus making the field an ideal one for the focus of an REU), geometric group theory is not a particularly "sexy" subfield of mathematics, and its practitioners are not in unduly high demand. Number theory it ain't.

Hardly a day went by during January and February without our office answering machine recording a new interview request for Ned. Every day I'd trudge into campus through the snow, pull aside the heavy front door to Altgeld Hall, wind my way up the stairs to our office on the west wing overlooking the Math Library, and the first thing I'd see as I entered our office would be the blinking light on that goddamned answering machine.

"Hi, this message is for Ned Cochran...this is Joe Schmo at the University of Upper Iowa, and we were wondering if..."

Damn it.

The stress began to mount. I was a wreck. I never seriously thought that I'd be jobless, but what if I ended up having a to take a one-year appointment at a school with no research activity and a 5-5 teaching load, at $5,000 less per year than I'd been paid at Illinois? The market that year was better than the past had seen, but offered nothing close to milk and honey.

To help us kill the time our friend Kurt, an avid Trekkie, began lending us his Deep Space Nine DVDs, season by season. It took a few weeks to make it through the first set of disks, but as the pressure grew and I saw less and less of Ned (there were weeks when he had on-site interviews back-to-back-to-back and I felt like his secretary, piling sticky notes on his desk: "Prof. Gregson at Lower Quebec Superior College requests that...") Maggie and I quickened our pace. We finished Season Four in two nights.

"Can I come over and get Season Five?"

"Sure, dude. Rough week?"

Several of 'em. During which I felt impotent and useless. Somehow I managed to get a lot done: I continued to co-host our weekly radio show, I began finishing up the manuscript I'd soon be sending to my publisher, I started what would soon be two more papers of which I'm still quite proud, and I successfully put together my first (and so far only) grad-level course. How I managed all of this I'll never know, as I felt daily and nightly paralyzed by anomic and debasing stress.

As regular readers of this blog (I pity you, poor souls!) must know, this story has a happy ending. Once the clog consisting of the superstar scholars like Ned worked its way through the pipes, the second-tier folks like me (a better researcher than most "teaching" faculty and a better teacher than all but the rarest researchers, but not absolutely aces in either) had our go: the calls began to come. Within the span of two weeks at the end of February and the beginning of March I had six interview requests, from one of which grew my current position.

Happily. Ever. After.

Present contentment calms the waters of the past. Lacking the scarifying effect physical violence might leave on our bodies, psychic violence often can be forgotten. Trying times seem positively historical until those times are relived once again.

The past two months have seen me gripped with a paralytic torpor the likes of which I'd not seen in three and a half years.

I blame the economy.

And the Republican party.

And the general helter-skelter state of world affairs.

The last two months I've spent in almost biminutely refreshing sites like FiveThirtyEight.com (who's ahead?) and HuffingtonPost.com (what're they saying?) and TalkingPointsMemo.com (hasn't he fucking won yet?!?), sitting zombified in front of my computer, obsessing on the electoral effect of out-of-work black Pennsylvania lesbians in the automotive industry.

And it wasn't just me. For two months just now, all was atwitter. Water cooler conversations were echoic, little more than he-said/she-said murmurs, gurgling trickles of rehashed political rhetoric. We all walked around dazedly and without conviction, as though all we did was rehearsal for the real deal, some grand pageant to be enacted a few months hence. We lacked motivation, and we were nervous. We fidgeted. Fingernails were made ragged and gnawed, eyes were rendered sleepless and puffy. We went to the edge, over it, even, and then we dangled there, gazing downward, pleading skyward, feeling our grip slacken as we sank...

Then we slept. If only for one night, we slept.

I slept for four short hours, drunk on a bottle of my lovely friends' very lovely red wine and on the excitement of a 21-hour day on which, as the Chief Election Judge for my electoral precinct, I truly feel I helped to usher in a new era.

We slept, and we're waking up now. I'm waking up.

What's gone on as I've slumbered?

The semester's nearly gone now. There are three class periods remaining in Precalc, and only two in each of my Abstract Algebra sections.

I've misjudged the amount of time I'd have for the former class, and I find myself cramming too much trig into the few hours that remain. I'm discarding all but the bare essentials, as I've got only a tiny duffel bag to serve for an oceanic passage. I feel rushed and uncomfortable.

Overall I've got mixed feelings about the way my first experience in teaching Precalculus has turned out.

As I've said in earlier posts, I'm glad I've been given the opportunity to teach the course, and I've learned boatloads from so doing. For instance, I've learned to take nothing for granted, even concepts that have become so automatic to me that I feel as though I understood them upon springing forth from my mother's womb. I've gained much more valuable practice in the art of using in-class team activities in lower-level mathematics courses. And I've had a chance to interact with students the likes of which I don't often meet in my other, higher-level, courses; namely mathphobes who really don't give a rat's ass about mathematics and who, once the semester has ended, have no intention of ever looking at another number so long as they live.

On the flip side, I've fallen in the mud a few times. I've muddled clumsily through explanations of things I've never had to explain before, I've faced attendance problems that dwarf any I've seen in classes I've taught in the past, and I've reached the semester's end feeling saddled with a slight sense of resentment. Do I resent them? I don't know. Do they resent me? I don't know. I could certainly be mistaken (I almost always have an end-of-semester freak-out about my students' perception of me as a teacher: "are my evals going to suck?"), but I feel that there's a chill, a distance, between me and them.

I've worked as hard as I've known how to in order to make this class a success...but have I done all that I could have done?

I don't know.

Maybe I just have to accept that most of the folks who took this course came in with very low expectations, and that if I've helped them to go even an inch beyond those expectations, I've succeeded.

Student presentations began in the second section of Abstract Algebra today. Derrick and Miguel started us off with a bang, giving two unintentionally interlaced talks on fundamental groups and free groups. Their talks were technical, clear, well-timed, and well-organized. I hope the next couple of days will bring more presentations of this quality.

Twelve of this semester's twenty-nine Abstract I students are continuing on to Abstract II with me in the spring. I'm looking forward to it already. (I might also mention that several of my Precalc students are on board for my Calc I class, too, and my 280 course is filled by wonderful blasts from the past...it's going to be a great semester!)

What else has happened while I've slumbered?

Hell, what hasn't?

Just as in the winter of 2005, life's gone on even as I've numbly watched it pass. There's a lot to be said, but I'll say it later.

For now, it's late, and I've got another long day tomorrow.

Torpor

Tired.

We're all tired.

I can feel it, I can see it.

Absences are up in all of my classes (not a day goes by without a handful of students missing from Precalc, Godzilla movie marathons or no), and my students are all sick with the second major wave of colds to sweep across campus this semester. My colleagues and I wander around the department with glazed-over looks and blunder bumblingly through committee meetings.

There's more to it than a change of seasons or an onslaught of midterms. There's more to it even than the dip in the Dow and corporate profits.

I admitted to a friend the other day that I feel as though I'm suffering from an existential paralysis, and I don't think that I'm alone in that feeling. The world is holding its breath, and we're all standing here blue-faced, wondering what's going to happen next.

The moon's crawled closer to the sun each morning this week as I've made my daily trek to campus. Dawn breaks around seven or so, with blots of orange cutting through a crisp crepuscular fog. I like this time of year: the mornings are fresh, and though it's cool it's not yet cold, and I arrive at my office invigorated.

I have to admit that motivating my morning Abstract class has been a challenge for me this semester. The students are by nature quieter than those in the afternoon section, and it's often hard to rouse them. Nevertheless, they're a strong bunch, and they're quick to learn. Their committee presentations were wonderful this morning; not one of the three committees fell into the "show 'n' tell" trap of solving the problem for the rest of the class. One committee made explicit reference to the Four Cs rubric, breaking their problem down along the axes laid out by the rubric.

"Honestly, are many of you finding the Four Cs a useful tool?" I asked. "I put it out there as a tool to use, but I certainly don't want it to simply be an exercise in demagoguery. If there's anything I can do to make it better or improve upon it for future classes, let me know."

There was murmured agreement. "I use it to help critique others' work when I'm on a committee," Norbert said. "Since I'm planning on being a teacher, it really helps me to be able to grade others' work. Having some in front of me to structure my comments makes it that much easier." Others agreed, and though it sounds like no one's using the Four Cs explicitly when they're writing their own work, they're often keeping it in the backs of their minds.

The afternoon section, bigger, louder, raucous and rambunctious, offered up shorter committee reports too. They gave fewer details, and seemed a bit more timid in their responses. I wonder how much their trepidation has to do with the size of the class? I wish they'd be more bold, more descriptive. It's hard, it takes practice, but oh, how the practice is repaid!

Midway through this class the torpor returned, and I felt horribly tired. "Are you okay?" Nadia asked as class ended and people began to stow their books and take their leaves.

"I'm just really tired," I admitted.

"You don't seem tired when you're teaching," she told me. I took it as a compliment, whether it was meant that way or not. She's one of the wonderful students in our program who makes my job as pleasurable as it is.

After nearly dozing off during a frankly awful talk in the senior seminar (students: to say nothing about the poorly-organized slides and the too-detailed computations, never ever ever run over time in your talk; it's the singe worst thing you can do and while they'll forgive nearly any other sin audiences will rarely forgive you for those extra minutes), I had an a truly electrifying run home, and for the last few hours I've felt refreshed.

Just now as I was lying on the couch I gave some thought to the third chapter of my stalled series on writing pedagogy, and I promise it will come soon.

For now, bed, with a promise of less tiredness tomorrow.

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.

Achieving self-awareness

I handed back the first exams of the semester in my Precalc class today.

By and large they did quite well. There were handfuls each of As and Fs, slightly smaller handfuls of Bs and Ds, and a passel of Cs. It was a nice trimodal distribution, with the middle mode dominating strongly at a mean of roughly 76.2%. The students struggled most mightily with anything involving absolute values, an assessment with which they wholeheartedly agreed when I made that observation at the beginning of class this afternoon. Long division of polynomials? No sweat: they nailed that. Radicals? Not anyone's favorite, but they came out okay on those. Absolute values? Fuhgedaboutit.

I'm not sure what it is about absolute values that proves so tricky to these beginning mathematicians. Admittedly their manipulation involves a good deal of practice and carefulness...but the same could be said of other concepts over which the students have shown a good deal more mastery.

I'm just not sure.

One thing I'm finding out about myself as I teach this class is just how much about mathematics I typically take for granted. I've blogged elsewhere (strangely enough, in this post about absolute values!) about the mathematical concepts I take as given; I'm only just now, in the midst of our first round of testing, realizing the tenuousness of some of my assumptions about students of mathematics.

I have to remember that though these Precalc students are every bit as intelligent as my Calc I and Calc II students, and though they're often exceptional students who are remarkably dedicated to their studies and who put every measure of effort into their educations, they're simply not as experienced mathematically as the students I find in my calc classes. I'm used to starting off the semester with students who come preprogrammed with knowledge of absolute values, of the algebra of polynomials and rational functions, of radical manipulations. And if a student makes it to my class without such knowledge, I've had the right to turn them around and send them down the hall to...

...precisely the class I'm now teaching.

It's new for me, and I know I'm messing up here and there, simply because of its newness.

I'm probably making assumptions I shouldn't make, about the knowledge you might or might not have, about the skills you do or don't possess. How many of you have ever graphed a function before? Probably most of you, but I probably shouldn't make that assumption. How many of you are bored to tears in having to spend an hour and a half going over simple examples of functions and the many forms they may take? Probably most of you...but I'm guessing not all.

I'm learning, folks. I'm learning what I need to say, what I need to do. And I need your help. If any of my Precalc students are reading this, please feel free to drop me a line, or even comment on this post (anonymously, if you'd prefer), and let me know: am I saying enough, or am I saying too much? Am I making any unsafe assumptions?, making an ass out of...you know the drill.

Let me know.

Today's one-sentence essay at the class's close asked the students to define "function," in their own words. Additionally, I asked each of them to indicate whether or not he or she liked poetry. Yes, I'm gearing up to assign another poetry exercise. I'm trying to decide how I'd like to modify this iteration of the assignment: ought I ask them to confine their theme to the topic of "function" in some way? Or should I merely let them traipse about the whole of the realm of mathematics, as I did my Calc I students last Fall?

I'm also toying with an ongoing "Pet Function" project, in which each student will be held responsible for the caretaking of a particular function, whose life cycle they'll sketch as we reach various points in the semester.

That's the skinny in Precalc. The thème du jour in Abstract Algebra was a litany of technical lemmata that read like one of the toledoth passages out of Exodus: "and the existence of cycles begat a decomposition into disjoint cycles, and once these cycles were found, verily the other elements they moved not. Existence begat uniqueness, and so was proven the Fundamental Theorem of Arithmetic, symmetric group version."

The first section hung in there, but the weight of the afternoon's weariness and the length of the week's work and the dark depths of the unintuitive computations we'd had to wade through were too much for the yawning and blinking and note-passing second section. Fuck it, we ended a few minutes early and half a proof short.

Hang in there, folks! Next week we'll cap off the FTA for S_n, we'll meet the alternating groups, we'll start exploring properties of groups in their most abstract form. It's all good.

What else?

In addition to about an inch and a half of grading (Precalc projects, abstract homework) I've got to spend a couple of hours this weekend throwing together a couple more handouts for the Carolina Writing Program Administrators conference to which I'm heading off on Monday afternoon. I've already got a handout that breaks down our (my, Lulabelle's, and Casanova's) views on the "future of writing at UNCA." I just need to gather some qualitative and quantitative data on the previous assessment project...and maybe say something about faculty intentionality. I'll figure it out.

Oh, and! I just found out that I will indeed be co-organizing a short course on designing and implementing writing-intensive mathematics courses at this coming spring's MAA Southeastern Sectional Meeting in Nashville! I'm excited.

Okay, my pizza's coming out of the oven in about five minutes, so off I must be. Take care, Gentle Readers. I'll check in again soon.

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!

Weak Two

You ever have one of those days when you just can't seem to get traction?

Today began early, and pleasantly enough. I had a nice walk under cover of slate gray skies, the sun's rise obscured by a feathery blanket of cloud. It was muggy, but comforting.

I'm trying to figure out where my Abstract folks are right now: this morning had me meeting with several of them about the committee problems that were due in class today. A few of them seemed a little overwhelmed, but in class most seemed on the ball. My hunch is that at this point what we're doing doesn't seem relevant, since we're developing skills and techniques, not results that are important per se. Moreover, with the scant background we've developed so far, it's difficult to see how what we're doing fits together with the overall thrust of the course. Once we start doing congruence arithmetic (which we're poised to start discussing on Friday), things will start to come together. In particular, I'll be able to point at congruence classes and say "See? See?!!? Groups!"

From this morning's Abstract section (I'm sorry I was a bit sluggish, y'all...I had a too-short weekend, too!) I went back to the office and took care of a number of bureaucratic tasks...those damned things pop up like whack-a-mole moles at a second-rate funplex. I was busy enough to not even notice the clock inch towards 1:45.

I felt out of sorts in Precalc, too. I think what threw me off at first was my continued frustration over the software. I'm sorry, y'all, that registering on Educo is such a pain. I promise you, I promise you, that once you're all in the lane and plugging away on-line, things will go a bit more smoothly.

My stutter-step start to that class continued, and I felt a half-beat behind for the whole hour. I think part of it's due to my sub-par preparation...I realized about halfway into one of the exercises I'd prepared (describing the interval defined by |x+6|<3) that I hadn't completely prepared a description of the algebraic solution to the problem. That oversight's due in part to my unfamiliarity with teaching this material (again, and again and again: first time doing this, folks!), but part is also due to my abhorrence of "plug 'n' chug" explanations.

Me: "You'll wanna do it this way because if you remember, what the absolute value |a-b| represents is the distance between a and b..."

Not me: "Why do this computation? Because it works, that's why."

Ouch.

I need to find a middle ground.

Happily, I've noted a couple of students who are very attentive, diligent, and responsive to my efforts...they're clearly engaged in class and give great non-verbal feedback as I'm teaching. By watching them I can tell if I seem to be getting through. My thanks to you!

I have to say that I enjoyed my second section of Abstract today, overly large as the section may be. Although there were a few people who I sensed were having difficulties with Euclid's Algorithm, I think most were picking it up, judging from the alacrity with which they offered to share their computations with the class at large.

My grades for the day: Abstract: B+; Precalc: B-; Overall: B.

Bleh.

I'll be ready for Wednesday, folks. You can count on it. Thanks for your patience with me.

Now, off to bowl!