Vox populi

The students have spoken!

Some of them, anyway. I thought I'd post the feedback I've received so far on this last Friday's class. The core issue is class structure: soccer ball or no? A few students brought up residual issues, but this single one remained at the center.

Saith one:

I find the best form of government is a benevolent dictatorship. Think about that premise and I think you'll be able to maintain an iron fist over the class without compromising the spirited environment.

In response to this point of view said I in an e-mail: "Point taken, kindly. I'll put my iron fist in a velvet glove and see how things go."

This student's take was echoed by a colleague:

I definitely felt like my thoughts were being forced to remain in my head, during Friday's class. I do agree that maybe some of those thoughts might be better off there, but what's most disappointing about that feeling is that it felt like much of the passion and the fun that existed in the first two class periods was sucked out of the experience.

I think that the soccer ball should be eradicated from the world of Graph Theory 473. What should be put in it's place is some self awareness, and some consideration on the part of those who are speaking, a few rules giving the 'presenters' more authority while they are in front of the class, and possibly a comment from the prof. when things start to get a little out of hand.

Man, I just went back and read your suggestions and ideas for next class. [See the excerpted e-mail from my previous post.] I didn't realize that you had already written the same ones that I did. Oops. Oh well, that's how I feel.

One student waxed a bit more philosophical:

Even though it was painful, I am glad we had the class we did on Friday. For me, two big ideas came out of what occurred during class. The first is, that in setting up time to ensure that we build a solid foundation for what we are learning, I believe that we are avoiding some long term pitfalls that might only have come up in the last weeks of the class. I feel like what really came out of the horse that we beat to death from problem sheet #2 was that carefulness leads to the deeper meaning that we are trying to glean. I know that there are Algebra and Calculus ideas that I learned only well enough for testing purposes and now wish that I understood more principally. I like the idea the idea of the review problem and the carefulness in answering problems.

Second, I think some good things came out of the 'structure discussion.' While I do not like the soccer ball, I do like the metaphorical one. I don't think that things were out of hand before Friday, in fact I love this class. I do think that we might not have had this discussion until things were out of hand though, and that would have been more painful. I like the 'sitting in a circle' idea. I believe that half of what we were concerned about will be fixed by that one addition. Also, because our Friday talk was not based in failure but in improvement, I think it gave us an early opportunity to be a little more conscious about how we do want to shape the class. I would guess that most of us gave that a little more thought after the discussion. So I guess the second 'big idea' for me was that awareness that came out of Friday. How could that hurt us!

Ultimately, some kind of order is necessary, as acknowledged by the folks above, and by our last commenter:

My thought is that there definitely needs to be some type of system, because the first few days it did get kind of crazy and was hard to really understand people's thoughts. But I also think we are all college students and should be able to respect one another enough to listen when they talk and then put in our thoughts (without interrupting). And people should be able to ask questions!

The plan from here: let's abandon that awful soccer ball, let's grant the presenter the authority to open or close the discussion, let's allow for open conversation while discussion is "on," with the understanding that that requires respect for one others' points of view and rights to have a say as well, and let's let me have another go at better moderating the discussion should the need for moderation arise (see my "velvet-coated iron fist").

Friday was painful for me, too. As the second commentator above pointed out, it felt a lot less fun, and when it comes to research (let's face it, folks, we're doin' research here) fun shouldn't be undervalued. I want this class to be fun and engaging, and I think we can manage that without the help of the soccer ball.

I'm glad we've had this conversation, I think it's helped us all to understand better what it takes to make up a healthy learning environment, and I'm glad it's happened in Week 2 instead of Week 11, allowing us twelve more weeks of organized, respectful, blissful interaction!

I'll check in again tomorrow and let you know how things go down.

Chaos

I've got mixed feelings about today.

Calc II felt all right: the first section was a little sleepy, but the second was more engaged and seemed to enjoy the sugar fix provided by the shortcakes used to illustrate the method of cylindrical shells.

Graph Theory?

Hmmm...

After Wednesday's class got a bit rowdy, with lots of cross-talk, overdubbing, interruptions, and just plain ol' mayhem, we decided that maybe we ought to try out a means of directing the discussion. I suggested the possibility of getting a small plush object to toss around: she/he with the ball was the one who got to speak. It seemed a bit puerile, but I didn't want discussion to get out of hand, lest people start zoning out, not understanding what's going on, what's being said, what's being proven.

So Theodoric brought in a plushy soccer ball, and we tried it out.

The atmosphere was...

...well, to me it seemed a bit dead. I'm not sure the deadness was completely unwelcome, but I don't want to kill off the natural enthusiasm that folks are having for the course.

I think some people had difficulty getting the attention of whoever it was who had the ball at any given time, others felt like they weren't going to stoop to "playing the game" of getting the ball before speaking and so said nothing...for the most part we stuck to the plan and didn't speak until given the ball...but it felt stilted, juvenile.

Having to choose between lively and occasionally cacophonous debate on important mathematical topics and stultifying silence, I'll take the debate, even if it means a little chaos every now and then. As I put it to the students in an e-mail exhorting them to write to me and let me know how they felt (their comments will be posted here as they trickle in):

My own two bits, for what it's worth: we've got a room full of 16ish smart, eager people, and I know that there are time when we've all got something to say. I want to keep the class lively and the discussion excited, but I also don't want it to descend into utter chaos. I'm not doing my job well if I let the class devolve into a kindergarten class. That said, if people are overwhelmingly for it, I'll be open to trying to use the soccer ball again (thanks for bringing it, Theodoric, by the way), but my feeling is that (a) you're all mature enough to not interrupt one another and to not crack jokes when other people are trying to explain something, and (b) I can do a better job at moderating discussion should it need moderation. I'd like to come in on Monday and try to go without the soccer ball, we'll let the person at the board lead the discussion (opening it up once he/she is through presenting), and if things begin to get rowdy, I'll exercise my authority and rein it in.

We'll see what they have to say.

Mathematically, we finished off a single problem today, proving that a subgraph of a simple graph is also simple. Our proof, built up in bits and pieces, was ultimately a careful one. One person starting things off with an intuitive explanation, a second tag-teamed with a more solid justification, and a third stepped in to nail it down with some clear notation. The result was a pretty clean proof, and I'm glad we took the time to make it rigorous. Remember, folks: I'd like you to be able to understand these theorems, but you should also be able to prove them.

That's all for now. I'll post student comments on the Great Soccer Ball Fiacso of 2008 as those comments come in.

Promises, promises

At last!

Many apologies for the delay in getting around to posting on this topic, after all of the promises I’ve made. It’s taken a while to gather all the necessary permissions from my students (who shall remain anonymous in all of the following comments), the semester’s end brought the usual concomitant tumult, meeting after meeting with administration about this matter and that added extra delay, and the last couple of weeks have been devoted both to finishing up a number of loose ends and to enjoying a relatively unstructured, somewhat work-free languor.

I did want to share my thoughts (and my students’) on the Newton v. Leibniz project, since at least one of the two reenactments proved to be a very memorable learning experience for a number of the class’s students. Though there was certainly room for improvement, I feel that my management of the project was quite effective, and many of the students put a great deal of effort into making the activity a successful one.

Below I’ll highlight some of the successes of the project: what did the students take from it? What did they learn about math, about math history, about themselves and the way in which math plays a role in their lives? What did they take from the classroom reenactment itself? On the other hand, I’ll point out some of the project’s flaws, and ways various students have suggested the project be modified the next time it’s assigned.

First, I might say a few words about the character of the reenactments.

The first section’s trial reenactment was satisfactory, but comparatively lifeless, characterized by a lack of preparedness on the part of some of the trial’s principal parties (the teams comprising Newton and Leibniz themselves, along with their counsel, and the teams comprising the scientists’ friends and colleagues). Two of the, shall we agree to say, less modest students in the section had taken on the roles of Newton and Leibniz, and it was clear that they did not always take these roles seriously. (This levity was a source of distraction for at least two of these students' colleagues.)

The second section’s reenactment was positively splendid: it was lively, heated, authentic, and
characterized not only by preparedness and seriousness, but also sincere concern for the discovery of "truth." Nearly every member of each of the principal teams prepared meticulously, and this level of preparation was clearly on exhibit in the resulting trial, in which students engaged in excited debate. Those cast as friends of one party or another fit seamlessly into their parts, attorneys on either side deliberated feverishly in reaction to the other side’s statements and questions, objections tore through the air like shrapnel. The result was an electrifying activity, and a pleasurable one: several students expressed a wish that the trial could be extended to the following class period. Said one: "the execution of the trial was a blast and I would have been willing to stay an hour after class just to continue it."

What was it that held the first section back?

"I have to admit, I was disappointed by the trial," said one student. "I understand that it’s important to have fun with projects, but there’s a different between having fun and not taking it seriously. I feel like, in general, the groups of colleagues and the historical experts came to class prepared to present information, they collaborated, and took the project seriously. I was dismayed to find that the primary groups…were relatively unprepared, and spent a great deal of class time joking with one another."

One of this students’ colleagues in the class agreed: "The arguments and the overall project could have been given a heightened sense of importance…Coming from my personal thoughts and experiences working on this project, I thought it had the potential to develop into a really intense trial, maybe not to the caliber of the To Kill a Mockingbird trial scene, but pretty close. I thought that everyone loved to argue which would ignite a heated trial."

I later asked this student for advice on making the experience more worthwhile. He gave examples of a mock trial in which he’d taken part in high school, indicating techniques his high school teacher had used to make the most of her project. He explained how in studying Beowulf his English class had put Grendel on trial, and the teacher herself played an active part in the trial by assuming the role of Grendel. "She knew her stuff about it, being an English teach," the student indicated in an e-mail to me later, "this makes it hard on the team who is unprepared." I agreed that in this way the teacher can "call out" those who are underprepared. The disadvantage of this set-up is that the pivotal role of Grendel (comparable to that of either Leibniz or Newton in our class’s rendition) is then taken away from a student, who might gain much from being cast in that part. The relative merits of either side could be debated. I very much appreciate this idea, and will definitely consider it when running the activity again.

Others were discomfited by the trial experience itself, whether or not they felt it was a success: "overall, I like the idea of a mock trial, but I found this experience to be really stressful." After listing a litany of inconveniences and discomforts experienced during the lead-up to the trial, the student admits that "despite all of this, I kind of enjoyed it." One of his fellow students agrees, clearly a lot more comfortable with the activity itself: "the actual trial was the part I found most enjoyable, because I got to get outside of the person that I normally am and really attack the other team. Being an attorney ended up being more fun than I thought it would be."

What did they take from the experience, this first section?

A number of students indicated that they learned more about Leibniz than they did about Newton, given the relative obscurity of the former mathematician. Several students mentioned learning a good deal of mathematics and the history to go along with it. A few insights were more purely epistemological, dealing with the nature of mathematical knowledge and its acquisition than with the math itself: where does math come from? How does it arise? Who can lay claim to its discovery?

One student became more aware of the "social" aspect of mathematical discovery: "from this project I have learned that in math everything is a collaborative effort with [one’s] colleagues or predecessors, nothing is every truly invented on [one’s] own." Another student agreed with this assessment and reflected on the "human" provenance of mathematics: "if anything else, from this project I have taken away the important idea that a mysterious Calculus book wasn’t laid in someone’s lap. It took many years to discover and perfect." And no angels were they, the folks who cobbled calculus together. In the words of yet another student, "these [Newton’s] colleagues were not so innocent in their daily lives, and their moral drawbacks were made public, so as their future representatives we had to make Newton’s colleagues look trustworthy and knowledgeable. In all honesty I never realized such dramatic lives were led by the founders of mathematics and the sciences."

Others took from the trial more pragmatic insights, ones that might be incorporated usefully into other settings: "it was interesting to learn and expand my knowledge of Newton and Leibniz but what stuck out even more was the small details," says one student. "The seeming less significant aspect of the project was the most interesting to me. The most beneficial was the fact that this assignment involved, writing, mathematics, the overall use of language, communication, cooperation with group members. This was interesting because it was universal and very helpful in other subject areas…Thanks for incorporating more than just math into this project."

How did the second section experience the same assignment?

Overall, excitedly, and positively. I don’t believe I received a single negative comment from the second section. Not a one. The nearest thing to a negative comment came from a rather reticent student who knew herself well enough to anticipate her strengths and weaknesses: "this is not the kind of project that I like to express myself with since I’m more of a creative writing person. The poem that’s coming up will be better for me to express myself." (Indeed, she would do that; hers is one of the poems appearing in the second set of poems, in the previous post.)

On the other hand, most were thrilled by the trial experience: "with the three-second attention span of a goldfish, it really amazed me how focused I became on this project," said one. She continues: "I enjoyed working on this with my teammates, I had fun digging up facts during research, and, more than anything else, I loved picking apart the other [team’s] arguments. So many things about this project interested me, but one aspect has really stayed with me. I was inspired and charmed by the level of dedication that came out of the students, including myself, during this project…what amazed me was the dedication of the students during this project. Weeks of planning and research culminated into a wondrous thing."

Her colleagues agreed: "I think the whole approach was creative and stimulating," said one. Several indicated disbelief at first, but later conversion. Many made revelations, about math and its history, and about themselves, much like those uncovered by the first section’s students. Here’s a representative testimonial:

"When the Newton v. Leibniz project was assigned I was, in all honesty, unmotived and in no way looking forward to the trial date. The assignment somehow seemed to be kind of irrelevant and distracting to what we were actually learning in class. After the actual trial on Monday though, I cannot stress how wrong I had been…Math had seemed like one of those things that has just always been around. I never thought twice about the history behind Math because I had never had a need to do so. By doing the [trial] however, one seemingly obvious thing was made evident: someone somewhere had to actually derive these concepts and be the first to think of them. Before, I had just thought of math as math. I never took into account that there is more to it than that. Math isn’t just math; math is a conglomeration of ideas and concepts over time. A major part of math is in its history and to truly understand math, one cannot ignore its past…Math is a collection of evolved ideas over time, whose history is almost as important as its functions."

And a second:

"When I enrolled in calculus in order to fulfill my biology requirement, I expected that the semester would be full of long, grueling problems which would bring back painful memories of math classes long past. And so it has (although with a lot less stress than I had anticipated). But if you had told me at the beginning of the semester that I would be doing a project later in the semester that I would actually enjoy, I’d figure you were crazy. But I did enjoy Newton vs. Leibniz, probably for two reasons: one, there were no math computations involved; and two, I felt I could relate it to what I am interested [in] more than anything else we’ve done this semester."

And a third:

"Upon the receiving the details of the assignment, I would be lying if I say that I didn’t let out a few groans. First of all, it was a group project. Group projects have seemed to be the epitome of evil to me for the past several years of my life. I kept flashing back to previous group projects in high school where it seemed that I pulled most of the weight of the project, whether or not it was intentional…Second of all, the assignment seemed a bit silly. I had only briefly heard about Leibniz and I felt as if there was no point to argue the Leibniz side of the dispute. Obviously, I was a bit wrong…From this project, I have changed. I have changed my outlook on the idea of group projects. Everyone in my group was amazing, and they all did their part. I realized that I needed to stop being so paranoid and pessimistic [about] others in my groups. I actually like group projects a little bit, now, and it is nice to work and talk with others who I would have otherwise possibly never talked to."

Though this last student had a change of heart about working on a project as one member of a team, others had different experiences: "this was one of the most interesting projects I have ever participated in and not in a bad way. At the beginning of the project I was very [skeptical] and did not know what to [expect]. Even though I had a rough start, in the end I thoroughly enjoyed myself. I learned a lot about myself and what role I play in groups. This may be cliche, however, I truly did learn about myself and realized that I do not like working in a group at all, unless I can pick the members in my group and that does not always turn out great. I realize this is a big problem because for the most of my remaining life I will be expected to work in groups."

As in the first section, some students had much to say about how the project had lent them insight into the process through which mathematical knowledge is constructed: "it’s [mathematical discovery is] a constant building of ideas on top of each other, almost like a pyramid with each layer built being built upon a previous layer. Newton of Leibniz could not have done their work without the work of mathematicians such as Isaac Barrow, Barrow couldn’t have done his work without the influence of scientists before him." As this student indicates, this "influence" causes difficulties when it comes time to ascribe credit: "The idea of creating new concepts also made my group and myself question what it means to actually invent something…Whoever created it first gets the credit, right? Well, then I started thinking, the more complex something is the more parts there are, the more parts there are the more other people are involved in the process."

Assigning credit is made more difficult by the fact that real people were involved, a fact brought to the fore by the students’ research during the project. "We often think of scientists as being ‘morally superior’ to politicians in that they are interested in obtaining the facts," says one student, who continues, "while I am sure this is true to an extent, scientists want fame and success just as much as politicians want power…In researching Newton vs. Leibniz, I was reminded that scientists are only human." This student was able to relate his experience in his section’s trial to his leisure reading: "It’s reinvigorated my interest in reading the last two books of the Baroque Cycle [a series of historical fiction novels], the last of which specifically deals with the calculus controversy. I can’t wait to see how Neal Stephenson portrays Newton: embittered, jealous, or sincere in that an injustice was done? In either case, I know it will be an entertaining read."

Others were able to make similar connections to their interests outside of class. Our friend with the three-second "goldfish" attention span hopes to recycle the courtroom concept for her own classes once she begins her career as a teacher: "as impressed as I was with the level of participation during the trial, I began to formulate a way I could use the idea of a trial in my own classroom. After reading a little of Change of Basis, I realized that this was not a strict model and could be applied across different disciplines. It was reassuring to know that while the process and student involvement stayed with me, it also stayed with my professor before me."

Perhaps the most ecstatic comments came from a student who had an epiphany concerning his own approach to mastery of mathematics: "for me it is much easier to understand mathematics when I say what I am writing down, in verbal form in my head," he explains. "I have always done this and before I had no idea why. Now that I have reasoning and have explored the subject, I have a better understanding of how to teach myself and be a good student of Calculus and the broader field of Mathematics. As a person that does not make breakthroughs like that on a regular basis, I was floored to learn more of how I learn. All due to this project!…If I were you I would keep assigning this project to Calc I classes."

Never fear, I believe I shall.

While the first section’s experience of the reenactment of Newton v. Leibniz offered mixed results, I was overwhelmed by the energy and alacrity with which the second section embraced the project, and gratified by their positive feedback. In all honesty the project was a bitch of a row to hoe, but it’s clear to me that the harvest was a rich enough one to warrant sowing these seeds again. Made wiser by the suggestions my students have offered me, and simply by the experience itself, I’m sure the next installment will be more successful than the last. I think I can safely say I’ve put together a significant learning experience with this trial.

And with this post, I’ve fulfilled a promise I’ve now made for several weeks.

I hope to post again soon with comments on my upcoming courses (choose-your-own-adventure testing in Graph Theory), on my teaching-related Winter Break leisure reading (Jonathan Kozol’s vitriolic The night is dark and I am far from home), and so forth.

Now, however, the night is indeed dark, and I should soon away to bed.

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!

Numerical nightmares and Newtonian nattering

Could it be that I became a mathematician in order to exorcise personal demons?

At dinner tonight, I was telling Maggie about an unpleasant dream I had last night.

As in many of my similarly unpleasant dreams, I had lost myself in the heart of a giant hotel, one replete with cascading staircases, ornate chandeliers, and atria with glass ceilings towering to dizzying heights. I'd lost myself, I couldn't find the room I was seeking, didn't know if it even existed. Somehow I sensed I was late for something. Hours, days, even weeks late.

"Were you at a conference?" Maggie asked. I suspect so, and I said this.

"It's weird how many 'conference anxiety' dreams I have," I commented.

What could it mean? Is this indicative of some fundamental insecurity in the quality of my research? Or does it belie a more general sense of confusion or befuddlement, some sense of unease at my place in life?

Tonight's conversation reminded me that I'd mentioned a childhood anxiety dream to Quimby's wife over dinner last night. She'd mentioned that she'd had a dream in which she was asked to count out some exceedingly large sum of money, and just as she was finishing, she lost count. When I was a child, I then explained, I had a recurring dream in which I'd been asked to count to astronomically huge figures, and the simple act of counting so high terrified me. "I woke up in a sweat," I swore.

Not falling, for me. No death by drowning, hanging, shooting, or fire. No ordinary nightmare would do.

For me it had to be counting.

The most common theme truly was an astronomical one: I'd be hauled out into the midst of a far-off asteroid belt, where I'd be left, presumably with some sort of life support system, and having found myself a cozy place to sit on one of the larger planetoids, I'd set about counting.

Millions gave way to billions, and billions to trillions and so on...fast forward, and before I knew it my dream-me was sitting there, enumerating the asteroids with fictitious numbers I'd invented for the sole purpose of completing my never-ending task.

Jung would have something to say about some sort of Sisyphean archetype.

I would often awake from these dreams too scared to lie back down for another hour. I'd often have to walk it off, pacing the kitchen floor in the sharp blue light of the microwave's LCDed digits.

I don't remember how young I was when I first had these dreams...twelve? Ten? Surely no younger than that.

Number.

Maybe Number has always had something to say to me, and I've spent the last few decades learning how to say something back.

Next week I'll be asking my Calc I students to call upon metaphor, synecdoche, all kinds of poetic imagery, to explore their feelings about mathematics. I hope that they'll take care to have fun in writing a poem about math, but I hope also they'll take the exercise seriously enough to discover something truly meaningful about their own feelings towards mathematics.

What does Number say to them? What can they say back?

Meanwhile, I really ought to say something about today's trials.

They were great.

I gotta admit, what they pulled off today was a tough task: the active participants were asked to think on their feet, to come up nearly instantly with viable explanations for actions that happened thousands of miles away hundreds of years before they were born. They had to continually adapt to newly-discovered evidence, they had to keep tabs not only on their own lines of attack but also on their opponents'. They had to shift chimerically from one train of thought to another, without skipping a logical beat. And they had to do it all in front of a couple dozen peers and their instructor.

Both trials saw their fair share of ers and uhs and uncomfortable pauses (as I put it in a congratulatory e-mail I sent to both classes), but given the leviathan load with which they were tasked, a fair amount of hesitant deliberation was understandable.

They did well.

The exchange got downright vitriolic at times. In my second section, especially, the litigants threw themselves into their roles and engaged in vigorous and rigorous debate. Once or twice during that section I was sure it would come to fisticuffs. Throughout the affair, hastily scribbled notes were passed from colleagues to legal teams, and there was ceaseless whispering between the debaters on either side. There were numerous and strenuous objections. I tried to be as fair as possible in allowing both sides some, but not too much, leeway; I hope this is how my actions were perceived. I attempted to quash speculation and focus on fact, but given the fineness of the line between the two, it was a difficult rope to walk.

Beatrice, who along with Farrah had spent a good deal of time during the trial passing notes to Leibniz's legal team, mentioned after class that in hindsight both of them wished they'd proposed to take a more active role than that of Leibniz's colleagues. "We didn't realize until today that we'd've had fun in that role," she said. Cassio, as Leibniz, did an admirable job in defending himself, and he was ably helped by Xavierina, his lead counsel. On the other side, Tallulah was chief counsel for Newton, played, astonishingly enough, by Newton, who was rather reticent and let Tallulah and Ambrose do most of the talking.

I talked with both Tallulah and Cassio a bit after class. Both thoroughly enjoyed the experience. "At first I thought, 'how are we going to stretch this out to fifty minutes?' " said Cassio. "Now I wish we'd had more time." Others agreed. "Why couldn't we continue on Wednesday?" a few folks asked.

So little time, alas!, so little time.

I thank you, one and all, for putting your time and effort into this endeavor, to make it the success that it was today. What will you have to say about the experience in your reflective essays? I can only guess at this time.

Super Saturday IV

I'm not sure why I got up at 6:30 this morning...at least I got a head start on the grading. I'm halfway through the Calc HW at this point, but 280'll call this afternoon.

At this moment I'm sitting in Zageir 227, 45 minutes remaining before our fourth Super Saturday session begins. I've got nothing to do for a bit, so I thought I'd take a moment to check in here.

How'd the week wind up?

In 280 we talked about injectivity and surjectivity, concepts that are substantially more puzzling in the abstract than they are in the familiar context of real-valued functions. We had a bear of a time working through a proof of the equivalence of the two most frequent characterizations of injectivity. There was something about the subtlety of the contradiction that was tripping up even the best and the brightest in the class. The confusion likely had to do with the convoluted structure of the statement we were trying to prove. Roughly speaking, the statement looks like (A ⇒ B) ⇔ (C ⇒ D), for various statements A, B, C, and D, so at one point we're dealing with a hypothesis that looks like (A ⇒ B) ∧ C. Say huh?

In Calc I we rapped about the way a function's graph is determined by its derivatives. On Monday I'm going to have them do a few exercises as a class to drive home the meaning of properties like concavity and inflection, and how they relate to increasing and decreasing.

Oh, by the way: they made substantial recovery on their exam revisions, raising the class average to roughly 77%, up from the woeful 68% last week. Several students offered touching apologies or narratives on their submitted revisions. Magdalena, for instance, mentioned how she saw herself in the inner monologue I included in the post following last week's grading frenzy. "Oh my god, he's inside my head!" she said. She vows never to become so reliant on the solutions manual again.

They're good kids.

Newton v. Leibniz takes place a week from Monday; this coming Monday I'll get the rough drafts of their "arguments," many of which I will share with opposing teams so that each legal team may adequately prepare a defense having had access to the other's evidence and data. So far most of the teams seem to be functioning pretty well together, with a couple of exceptions. There's been a little tension stemming merely from personality differences present in one or two of the groups. In a perfect world, perhaps...

...you know, when I was a student, I didn't so much like working in groups. I didn't have to do it all that often, but when I did, I got it done with as soon as possible and moved on. I can't say now how I must have come across to my teammates. Bossy, maybe? Headstrong? I hope not. I know that I'm the sort of person who wants to get it done "right," even if that means that I have to take the entire project onto my own shoulders and do every last bit of the work myself. Better that than that the project turn out shoddily because one of my teammates had the audacity to not be as smart as I was.

I suppose I might have been bossy, without meaning to be.

And I wasn't always my sweet current self, either. Though sarcasm is my native language, over the years I've learned to tone it down with others whom I don't know so well. Sarcasm, spite, vitriol. My mother loves telling the story of how I frustratedly called one of my kindergarten colleagues "stupid" because she couldn't grasp the nuances of the game I was trying to play with her. I was insufferable as a young 'un, but I've gotten more mellow with age.

Why can't we all just get along?

More later...I'll let you know how my first session of the departmental Moses Learning Circle went yesterday afternoon (short version: I'm not so sure my colleagues are sold on the claim that mathematical literacy is a civil rights issue, or the claim that improved numeracy is a societal panacea), but for now I've got to get ready for the SS kiddies who'll be here in a few minutes (Tallulah and Deidre are already here, ready to lend a hand).

Avanti!

Continued!

Lunch went well!

I need to stop hanging around with Quimby, he keeps making more work for me.

Short version: I'm now in dialogue with some folks over in psychology regarding work they're doing to put together a grant to get support for a lab they'd like to put together. There'd be a substantial human element to their program, involving, for instance, student researchers. Quimby thought that my experience with the REU would come in handy to them, and that their need for researchers might dovetail nicely into the ongoing STEM-related NSF grant we're all busily working away at over here...it's all good.

After lunch I got waylaid in the Math Lab by student after student...Calc I folks cleaning up their exams (good job on those so far!), 280 students making good progress on the homework and the take-home exam...a good group.

Topic for a future post: how in the hell do you manage blips and burps in group dynamics when students are working together? This is a cause for much consternation. When personalities clash, well...

...to be continued, again...

The show must go on

So it is: the learning experiences one remembers best from years past are those in which one played the most active role as a student.

The Latin American Civilization and Culture course I took during my last quarter as an undergraduate, some...holy crap...um...eleven years ago, was taught by one of the finest teachers I've ever had, Dr. Miriam Bornstein-Gomez. (Truly the entire faculty of the University of Denver's Spanish Department is fantastic, and I was made happy this evening to find, upon looking up the department's website, that all but one of the professors I had during my years there are still teaching.) She herself was strong and strict (I feared her for the first few weeks of class, honestly) but supportive and approachable, and her class was exciting and fun. She challenged her students to seek out every ounce of their talent, every day. An advocate of active learning, Profesora Bornstein felt there was no better way for us to learn about the atrocities committed during Pizarro's conquest of the Inca empire than to let her class put Pizarro on trial for the crimes he'd committed.

As the sole male member of the class of a dozen or so (and one of the most proficient speakers), I suppose I was the natural choice to play Pizarro, and so I was cast. I was assigned a defense attorney, played admirably by a young woman named Shailini, whose name I still remember because, truth be told, I had a bit of a crush on her and had vainly asked her out earlier that semester. Opposing us in the classroom court would be a pair of prosecuting attorneys who would have the right to call on any one of a handful of witnesses, played by other students in the class. The remaining students fulfilled the role of the jury, while Profesora Bornstein presided as judge.

I remember some impressions of the proceedings, but few details. I seem to recall that I adopted a cocksure attitude and tried to shift the blame onto the conquered Incas. In my good but not completely fluent Spanish, my attitude probably came off more comic than cocky, and I recall Profesora Bornstein laughing several times during the trial.

Strangely enough, though I can recall with punctilious detail the layout and orientation of the classroom (it was a small room, tucked into a tight corner on the third floor of the General Classroom Building, one wall of windows looking out over the quad to the building's east), I can't remember what verdict was rendered.

Nevertheless, the fact that I remember anything at all from that course, in a field that was not my major, more than a decade ago, is a testament to good teaching.

Today I distributed the first handout dealing with the next team project in Calculus, Newton v. Leibniz. Over the next five weeks, each team in each class will be asked to (1) "audition" for one of the roles in the trial (Newton and counsel? Leibniz and counsel? colleagues of either? historical or mathematical experts? jurors?), (2) write a brief or letter of support or document describing initial findings, as appropriate, before the trial, (3) fulfill an active role in the trial's proceedings, up to and including rendering a verdict, and (4) write a brief paper reflecting on the experience of the trial once everything else is done.

A few of the teams in the first section seemed to show some interest in the project already, exchanging knowing glances and nodding when I spoke of the "casting call" that'll start the project off. The second section's response was a bit more subdued, but I'm chalking this up, at least in part, to the fact that their teams have been shuffled around, whereas the first section's teams have remained the same. (The team evaluations for the first section were overwhelmingly positive ones, with a number of people saying explicitly that they'd like to stay in the same teams; there were no negative comments. Meanwhile there were a few teams in the second section that didn't function quite so smoothly for various reasons, so I felt a rearrangement was definitely in order there.)

I encouraged them to throw themselves into this project, to be creative, to have fun with it. As is true every semester, I've got a good number of talented students in both sections, and I'm looking forward to seeing what they can pull out of their hats.

Today I also distributed the handout describing the 280 folks' next written assignment, Professional Proof Analysis, in which the students are asked to apply our "Four Cs" rubric to three proofs of the same (hopefully somewhat familiar) mathematical result, the second part of the Fundamental Theorem of Calculus, written by three different sets of university textbook authors (Stewart, Schwartz, and Hass-Weir-Thomas). I'm interested to get their take on these proofs, if some will stand out from others in one category or another, if they really all seem the same. Whatever they come up with, I'm sure the results will be interesting. This is definitely one of the assignments I'll collect for the Writing Assessment Pilot.

Yeah, I'm excited.

I only hope the students are half as excited about these projects as I am.

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

Who are you?

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

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

"Well, did you understand this one?"

"I don't know."

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

"I don't know."

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

"I don't know."

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

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

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

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

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

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

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

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

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

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

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

My 100th post

How 'bout that? It's taken me a while to make that first hundred, as rarely as I've been posting this semester. It's happened that most of the time when I've thought, "huh, that's an interesting thought. I might could write about that," I've ended up being too busy to post it before forgetting about it again. (Me? Busy?)

I've come to realize that for the most part, bloggers are either

1. college freshpeople who have more time on their hands than they know what to do with, writing about why Green Day is the greatest band in history (hint: they're not), or

2. pseudo-intelligent ex-English majors working in the food-service industry, writing about the brilliant conversation on Sartre they shared with the checkout guy at the Piggly Wiggly, and who think that now that they're blogging everyone's gonna find out what sort of genius they possess and that they're sure as hell gonna land that six-figure book advance.

Considering these options, it's probably best that I don't often have time to blog.

Nevertheless, I do corner a few seconds here and there, and sometimes those few seconds come at a time when I happen to be thinking about my teaching, specifically or generally.

Like now.

I just spent an hour or so hanging out in the comments section of one of my favorite blogs (Waiter Rant). Recently he (the anonymous New York-based blogger going by the name "Waiter") devoted a couple of posts to "assholes": one post listed 50 signs that you might be an "asshole customer"; a second, 50 signs that "your server might be an asshole."

This makes me think of an exercise I recently read in Maryellen Weimer's Learner-centered teaching, a work I referenced a few posts back, and which has given me a number of neat ideas to try out in my own classes.

Saith Prof. Weimer: think about starting the semester off with a brainstorming activity in which your students finish open-ended sentences like "I find that I learn well in a classroom where..." or "I find it annoying when the professor...". Let them discuss the matter, arrive at a consensus. This exercise promotes reflection on the learning process and on creating environments conducive to learning, and can serve as a prelude to a "classroom contract" in which the instructor agrees to work to construct an environment where the students' admitted concerns are addressed, and in response, the instructor can offer up a short list of behaviors s/he finds annoying in students and ask that the students do their best to avoid said behaviors.

Both I and my sole colleague in this semester's Learning Circle (shout out, Darlene!) agreed that this activity would probably seem condescending in an upper division class, but it might be a useful one to pull on first-years at the semester's outset.

Why not try it now? I'll share with you a list of my own pedagogical pet peeves, and in response, I hope you can feel free to share yours with me. I'm not claiming that any of my current students are guilty of any particular charge, but you might just recognize yourself in one or two of them. If you do, I hope that you'll do what you can to rein it in. As you'll know if you've been in one of my classes, I'm an easy-going guy, and I'm not likely to tear you a new one if you occasionally step out of line, let your cell phone ring because you sincerely forgot to set it to vibrate, can't seem to stay awake because you were up all night cramming for your Organic midterm, come in unprepared every now and then...I'll let it go, because I know we all have days like that, and I'm not an ogre.

And I like my students. I really like you guys. I have to say that in the almost-decade I've been teaching at the college level, of the roughly 700-800 students I've had in my charge at one time or another, I've personally liked about 99.5% of them. There have been a small few who've rubbed me the wrong way, a couple here and there that've gotten my cheese for one reason or another, but at the end of the day, I can literally count on one hand the number of students I've had whom I just couldn't stand. Really. You wanna know how many? Two. For real. Just two, and neither at UNCA. One at Vanderbilt University (initials RG), and one at the University of Illinois (initials KC). That first was a real piece of work. Remind me to tell you about his golf game up in Kentucky sometime.

If you find yourself identifying with one of the annoyers in the list below, please remember that it's the annoying habit I despise, not the person performing it. Chances are really good that I like you, and I want to continue to work with you as best I can. Just cut the crap, and we'll get along fine.

With no further ado, let me present you with

8 Annoying Student Habits
(I honestly couldn't think of any more. See how easy-going I am?)

1. I'm annoyed by endless complaints about how long it takes one to do one's homework (in my class or someone else's). Complaining about it doesn't finish it, it doesn't make it any easier, and it's not going to earn points from your professor (me included). If I think an extension is warranted (and often one is), I'll figure that out for myself, I don't need your help. Note: freshpeople are most often guilty of this behavior, as they've generally got a pretty poor sense of how much homework is "appropriate." By the way, I'll almost guarantee you that I spend at least twice as much time (often much more) in thinking up, designing, writing, photocopying, posting, grading, commenting on, and returning any single assignment or exam than you do in completing it. (If you ever wanna know how long a particular assignment took me to process, I'd be glad to give you an estimate, it's probably longer than you think.) Please keep that in mind before lodging a complaint.

2. It annoys me when students ask in class about course information that's available on the website. This isn't a big issue, but it's an annoying one nonetheless. I keep a pretty well-stocked website (this too takes a lot of time to maintain properly); if something's not listed/available from the course website, chances are it's not all that important. So if you've missed a couple of days of class and you need to find out what homework was assigned while you were gone, please don't ask me to spend three minutes at the beginning of class tracking that information down for you.

3. In the same vein, if you miss a few class periods, please don't expect me to give you a "synopsis" of the classes you missed. If you had a valid excuse for being gone, I might very well be able to spare 10-15 minutes to brief you on what went down while you were away, but I'm much more likely to actually give you this time if you've taken time beforehand to prepare for this briefing by reading the material we covered in your absence ahead of time.

4. Please don't complain about having to work in a group. I don't care if you don't like to work in groups. You know what? Not all of us do. I include myself in that list. I've always been one of those folks who wants to do everything for himself because he's not quite sure anyone else is going to do it as well as he will. You know what else? At some point in life, you're going to have to work in groups. It's called "committee work," another term for "hell." The experience in group work you gain now, in the relatively low-stakes, comfortable, safe environment of your classroom, the better you'll be at it in the future.

5. I've never been a huge fan of going over homework problems in class if doing so is not an integral part of the course's design (as is the case in my current 280 and 368 courses), especially if the students are not the ones doing the "going over" (see previous parenthetical comment). Some profs like to devote a good chunk of time to going over homework problems, while I, most of the time, don't. Occasionally I'll find it worth the class's while to go over the odd problem, but I'd rather you not ask me at the beginning of every class, "can we go over Problem 346?"

6. In classes where the solutions manual is broadly available, it annoys me to no end when students submit homework which was clearly copied from the manual. The manual can be a useful tool, if used properly, but it's worse than useless if the only purpose it serves for one is as a crib sheet. In the end, it's usually the student's loss, for a few extra points on the homework will be more than counterbalanced by the smack in the face the hapless student'll get come exam time when the solutions manual is unavailable for consultation.

7. Obvious obliviousness on the students' parts annoys me. If you're not gonna mind what I'm sayin' at all, then go home. If you're going to be your group's fifth wheel, go home. If you just can't be bothered to stay awake, go home. If you'd rather sit back and check out the box scores (Spring 2006, Calc II, Section 1?) in the sports section than focus on what the rest of the class is doing, go home.

8. Hateful speech. I hope this goes without saying, but for Pete's sake, people: please don't be crackin' "jokes" or whippin' off "smart" remarks about others' color, gender, ethnicity, nationality, religion, sexual preference, disabilities, intelligence, and so forth, whether it's in general or specific terms. There's really no room for that kind of thing anywhere in this world, and there's sure as heck no room for it in my class.

***

That's it, for now. Honestly. That's all I can think of off the top of my head. I'm probably in the minority, but little things like inadvertent cell phone rings and discreet lunch-eating don't get to me much. I don't even mind class clowning, if it's not too rambunctious or mean-spirited. It's just the big things, really.

So how 'bout it, Studenten? What professorly habits annoy you? What things have your profs done in the past (no names needed!) that you really could have done without? I'm truly curious.