Problematic

The last couple of days (as the coming couple of weeks will also) have seen preparation for my courses this coming semester. While Calc II is something I could do with my eyes closed, I'm spending a bit more time in getting ready for Graph Theory.

I spent a few hours yesterday afternoon putting together the first problem sheet for that class.. It asks for a nice mix of examples and proofs, all interspersed throughout a series of definitions that flesh out the basics of graphs. I'm quite pleased with it. It's got a little more than one question per person currently registered for the course, so I hope that everyone will have a chance to get involved through this first set of exercises. I don't think it should take us more than the first week or so to get through it. I'm going to get started on the second sheet soon, but it won't be made public immediately; I have a hunch I'll want to make adjustments once I see how the first sheet goes over. I may, after all, find that I'm aiming too high, or too low, or expecting too much or too little...the students may want to take the class in a totally different direction. We'll see how it goes.

After tinkering with various ideas for that class's exam structure, I've decided to just play it by ear and ask the students to help me design the exams when the time comes: they'll help in the construction of the tests, and in their organization.

Tidbits: I'll also be asking each of the students to read and present on at least one mathematics research paper, and I'm toying with the idea of giving a small amount of extra credit for learning and using LaTeX in the preparation of written homework problems.

Besides this, nothing much to report, teaching-wise.

Further bulletins as events warrant.

Monday review of books

I've finished two books in the last twenty-four hours, including one that I capped off in a single three-hour sitting this afternoon: Jonathan Kozol's The night is dark and I am from from home (Simon and Schuster Inc., 1990), and Kurt Vonnegut's Mother Night (Dell 1999). Although neither directly relates to teaching at the university level, the former deals with Education with a capital 'E,' and in reading the latter on the heels of the first, I couldn't help but be impressed by dramatic differences between the two, as well as some deep parallels.

Both books I read just now by accident: Kozol's book I came across at an ongoing book sale in the basement of the Swannanoa Branch of the Buncombe County Library system this past Tuesday, and just yesterday afternoon when I accompanied Maggie in to keep her company during her short holiday weekend work Vonnegut's novel was on the top of the bin we pulled from the Pack Library's return kiosk. The title was fresh in mind, recently recommended by one of my students; it was one of the small few of Vonnegut's novels I'd not read before.

Odd that I should have read them so close in time to one another, so different the views they express, on the face of it.

Kozol's book paints a bold picture in absolutist strokes, uncompromising, often laced with ugliness and spite. The world he portrays is one in which unassailable good does battle daily with the purest of evil. The battle takes place throughout our society, our author assures us, but not surprisingly his focus rests on the nation's public schools. His thesis: "the first goal and primary function of the U.S. public school is not to educate good people, but good citizens. It is the function which we call -- in enemy nations -- 'state indoctrination' " (p. 1). To support this thesis he erects argument after argument, exposing for the reader the "straightforward lies" told by the educational establishment, laying bare the artfully hidden connections between the adversity felt by the nation's poor and the plenty enjoyed by its wealthy, and indicating the means by which teachers and administrators encourage the children in their charge to substitute passive concern for meaningful action. Certain themes recur, indicators of our nation's evil: My Lai (the book was written during the period from 1969 to 1975), Kent State, socioeconomic discrepancies in infant mortality rates, income opportunities, and access to medical care. Every aspect of our educational system, we're convinced, goes to supporting the perpetrators of these crimes.

Kozol's criticism is harsh, his tone cold and merciless. In fact, many of his accusations are simply unjust, and many of his comparisons inapt, insulting, and hurtful. Fortunately an older, wiser, Jonathan Kozol recognizes this fact, and for the 1990 edition of the book I just finished reading has written a set of critical annotations offering a rebuttal to the words of his younger, more hot-headed, self. "It was a time [1969-1975] when people who grew up to love their nation felt a sense of shock and shame. If, like myself, they had also staked their early years to try to bring about some serious social change, they were also likely to feel bitterness and rage. This much background may explain, though it does not excuse, the passages within this book in which I seem to speak about America not as a good but flawed society but as a land of unabated cruelty and evil" (pp. 18-19).

In his critical notes Kozol often decries his own words as "exaggeration," "abhorrent," and "reckless overstatements" (pp. 240-243).

I have to say that I agree.

While I agree also with the general thrust of many of his arguments, his tone is so continually demeaning that by the end of the book I felt a disgust which surely must have lost him many allies. Ultimately this is the problem with the book, the way it's written: though I too believe that the American educational system is deeply, deeply flawed (for many of the reasons Kozol highlights in this book, and for others reasons that didn't obtain when it was first written), I believe that through the sort of hectoring he does with The night is dark, he is likely to lose the support of those most ready and able to answer his call for change.

The chapter that struck closest to my mark, of course, was that titled "Colleges and Universities." Here Kozol sounds much like an acquaintance of mine in Illinois, who insisted on referring to the University of Illinois as "that little university down the street" on his spite-spewing radio program, all the while both he and his wife drew their paychecks from that same university. Here Kozol bars no holds: "the university is built on blood and nourished by injustice," and functions "both to sedate the lives and to protect the conscience of the university population, to insulate the college common and the paneled dining quarters of the college faculty and deans from either knowledge, memory or recognition of the pain, the description and the devastation of those tens of thousands who live just beyong their reach and recognition" (pp. 214-215). The claim that university faculty have difficult jobs to perform he calls the "Fiction of Hard Work" (p.215); the seersucker-wearing, lime-sherbet-slurping ivory-tower intellectuals with whom Kozol populates his college campuses (he has a thing for lime sherbet, mentioning it at least three times) are said to lead nearly effortless work lives. This would come as a surprise to me and to my colleagues (I can name several) who work 70-plus-hour weeks on a regular basis. I'm not claiming that we dig ditches for a living, I'm simply claiming that it takes a great deal of time and, yes, intellectual effort, to create meaningful coursework, serviceable assessment measures, significant learning experiences, and insightful evaluations and subsequent revision of our own work; to work with one another in planning and implementing truly impactful activities that involve not just the university but also the surrounding community we are chartered to serve; to craft original ideas and to incorporate those of others into our own, and to spread these ideas far and wide; and yes, to challenge our students to do more than say, but to do as well; to do more than care, but to act as well.

I found that chapter insulting, as did Kozol himself, in retrospect: "too many words like 'evil,' 'brutal,' 'fraud' demean these pages" (p. 251).

At the end of the day, I feel that Kozol's downfall is his insistence that in order to wash away the brand of "hypocrite" we must not simply put in a few hours each week at the homeless shelter, tutor math in our neighborhood elementary school, or work weekends for Habitat for Humanity; we must instead disavow all connection with the bloodthirsty machine that is the U.S. government and its corporate allies, we must pull our names from the university class rolls or quit our university posts, we must throw away all but the clothes on our backs and the staffs in our hands, shaking the sand from our sandals at any house where we are made unwelcome, making of ourselves martyrs for the cause of the have-nots, when often such martyrdom removes us from the positions in which we are most able to affect meaningful change. While I agree that too many people lead relatively comfortable lives blissfully unaware of the sometimes-causal connection between their own comfort and others' agony, I find ludicrous, for instance, Kozol's claim that he is hypocritical who, with access to quality medical care for his family, does not eschew that care available to him but instead brings his family to one of the understaffed and undersupplied clinics that service the nation's poor (pp. 204-205). Kozol's insistence on absolutist actions like this out-and-out boycott is the stuff of relatively immature extremism.

In final assessment, again, I find myself agreeing with Kozol in much that he says, but his message is lost in the brutal static of his medium. He loses much by demonizing not only his "enemies" (whoever they are; in sketching them his job is poorly done. As near as I can tell, his attitude is a Bush-like "yer either fer us or ag'in us" in which a nebulous Marxian elite plays the role of bogeyman), but also his allies. The world he paints is black and white, right and wrong, good and evil, and he makes it very clear on which side of each respective terminator he stands.

Perhaps not surprisingly, Vonnegut's world is a different one. The novel's anti-hero, Howard W. Campbell, Jr., served the Americans as a spy by adopting the role of a high-ranking Nazi official charged with promulgating anti-American propaganda. Throughout the novel he tells us of the pragmatism that kept him alive during the war and after it: to himself he ascribes neither guilt, nor a sense of loss, nor a loathing of death, nor heartbroken rage, nor a unlovability, nor a sense of the cruelty of God (pp. 231-232). He steals, lies, deceives, schemes, and somehow even when caught manages to purchase freedom by one means or another. His world is not black and white, but gray, a full-blown cloud of obscuring mist. In order to perform the greatest acts of espionage, he was forced to author and deliver the most hateful of anti-Semitic screed. Which outweighs the other, the sinner or the saint? Impelled by whatever situation he finds himself in, our hero eventually loses all sense of black or white and opts for suicide over freedom when freedom might lend him another opportunity to pick sides.

Still there are parallels between Kozol's work and Vonnegut's. Most notably, both admit that in order for an ordinary human to live with herself, she must adopt a certain degree of hypocrisy. Early on, Kozol quotes Tolstoy's The Kingdom of God is within you: "the men of the ruling classes -- the honest, good, clever men among them -- cannot help but suffer from these internal contradictions...We cannot pretend that we do not see the policeman who walks in front of the windows with a loaded revolver, defending us, while we eat our savoury dinner or view a new performance...We certainly know that if we shall finish eating our dinner, or seeing the latest drama, or having our fun at a ball, at the Christmas tree, at the skating, at the races, or at the chase, we do so only thanks to the bullet in the policeman's revolver and in the soldier's gun" (quoted on p. 62). Kozol limns a similar modern American myth: "the lives of children of the white and well-to-do within a land like ours exist upon a plateau of relaxed and innocent intent: one which turns at times, if we so wish, to passages of benefaction, at other times to academic labors, string quartets or summer garden parties, yet one which is at all times disaffiliated from the exploitation that it rests on, uncontaminated by the blood that nourishes the soil" (p. 63).

Compare this with a passage from p. 223 ff. of Vonnegut:

I have never seen a more sublime demonstration of the totalitarian mind, a mind which may be likened unto a system of gears whose teeth have been filed off at random...The dismaying thing about the classic totalitarian mind is that any given gear, though mutilated, will have at its circumference unbroken sequences of teeth that are immaculately maintained, that are exquisitely machined...The missing teeth, of course, are simple, obvious truths, truths available and comprehensible even to ten-year-olds, in most cases. The willful filing off of gear teeth, the willful doing without certain obvious pieces of information -- That was how a household as contradictory as one composed of Jones, Father Keeley, Vice-Bundesfuehrer Krapptauer, and the Black Fuehrer could exist in relative harmony --That was how my father-in-law could contain in one mind an indifference toward slave women and love for a blue vase --That was how Rudolf Hoess, Commandant of Auschwitz, could alternate over the loudspeakers of Auschwitz great music and calls for corpse-carriers --That was how Nazi Germany could sense no important differences between civilization and hydrophobia...

For Kozol, school is the file with which the teeth are removed.

Vonnegut also seconds Kozol's observation that in totalitarian societies there is safety in imitation and danger in originality (that, in fact, true originality can be punishable by death): "'What harm is there in writing what's already been written? Real originality is a capital crime, often calling for cruel and unusual punishment in advance of the coup de grace' " (Vonnegut, pp. 206-207). Vonnegut stops short of labeling our own society a totalitarian one, as Kozol does throughout his work.

I suppose we all, those of us well enough off to be writing or reading these words right now, must pull a bit of ethical legerdemain in order to sleep well in a world as rife with inequality as is our own, and I suppose we all must mask our true intentions from time to time in order to preserve the integrity of the society in which we live and work.

How much is too much?

And how, if the balance is upset, do we reset it?

I'm going to have to think some more about these books, and how they relate to what I do, with what tasks they charge me.

Until I next check in, I welcome my readers' insights: have any of you read these books? What do you think?

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.

Round two

I've got one more meeting in an hour and a half, and commencement tomorrow...but then everything...classes, projects, meetings, grading...EVERYTHING...is over for the semester.

Give me a week, and I'll be bored out of my ever-loving skull, itchin' to get back into the classroom.

The last couple of days have been a whirlwind of excitement (grant-related activity, proposal-writing, this meeting and that, toying with various research ideas), and I'm looking forward to starting a few weeks of mercifully highly unstructured busyness.

For the moment I've got nothing to say, really, but I thought I'd post the second of what will be three rounds of student-authored math-themed poetry.

I should say up front that I've received permission from all of the students from whom permission was sought to post their writings; the only two poems I'm not including in this post are rich concrete constructions in which the pagination plays a very meaningful role. I'll have to see if I can render the effect in Blogger, I might have to settle for obtaining the original MS Word files from the students themselves and posting links to those.

For now, I give you

Mathematical Poetry, volume 2

Let's begin with a simple haiku, courtesy of Mark. Mark indicates that he felt justified in forgoing the tradition in American haiku of making mention of a season, since his poem deals intimately with the theme of Nature:

Nature is Chaos, by Mark

Nature is Chaos,
Math brings order to Chaos,
Math Quantifies life

***

Cory's poem deals with the sweet taste of victory, written after Leibniz had been exculpated in his section's reenactment of Newton v. Leibniz; Cory was one of several of his classmates who described the experience as an immensely positive one:

Newton v. Leibniz, by Cory

It started with a proposition
Which led my team to the path less traveled
Whitney, Cory, Ben, and Justin...Leibniz and his colleagues
We were doomed from the start
Newton this, Newton that, all in favor say Newton
With the invention of calculus at hand
We had to bring it home for the one and only Leibniz
As we ravaged the library and surfed the web
Little by little it pieced together
Closer and closer victory was ours
As underdogs we had no choice but to be prepared like no other has ever prepared
Days grew long
Nights became dawn
And before we knew it the hourglass was empty
The epic trial of Newton v. Leibniz was here
In only minutes we could see that we were better prepared
Minutes passed and minutes left before the deciding factors
As the trial came to an end Leibniz and his colleagues left the room with their heads held high
And a sense of pride lingered in their step
BAM
It was here
The verdict
Dead quiet
Leibniz found not guilty
Therefore he published first
And was crowned the king of calculus
It came with no surprise to Leibniz and his colleagues
WE WERE PREPARED FOR GLORY
AND NOTHING LESS!

***

The following is the only e-based poem I received, strangely enough. I love the way the author has made use of the fact that e's first several digits alternate between 1s/2s and 8s/9s: this alternation leads to a series of quick exclamations followed by more prolix elaboration.

e, by Anonymous

Great Calculus.
.
Can put to use in everyday situations.
Crazy.
Challenging yet the only subject that is constant.
Mind boggling.
Sometimes you want to pull your hair out.
Satisfying.
Confusing/makes perfect sense at the same time.
Intricate simplicity.
If used can solve many problems we face.

***

Ben's work, below, paints an idealized picture of the way the world might look through a mathematician's eyes. I'm not sure exactly what's meant by the cryptic closing lines.

mathematician, by Ben

late nights and classroom lights
chalk board dust and old book must
soda redbull coffee and tea
oh! the world the mathematician sees

to take heed - a desire planted so deep

to justly discover and describe
the fine natural truths held inside
deep blue seas, big old trees. its
size, shape, speed, that he sees

joy and fulfillment abound
beautiful! the answer is found
and my dear! the truth will set you free
oh! i long to see the world that he sees

but as his time ends, there is one regret -

his curiosity reigns supreme
and learning all is but a dream

some regret a dollar lost
or an unforeseen cost

but as it comes fro'
his time to go

things unknown
leave him
'lone

the world the mathematician sees
is the world the mathematician must leave

this is the pain the mathematician knows

***

I felt the next poem was one of the strongest submitted, I've mentioned it in a previous post as its author was working it out in its preliminary form. Its theme is one of the most personal I encountered as well. Stylistically, the author expresses an affinity for the chiastic form the lines took on the page, and, not surprisingly, professes to be a fan of alliteration!

Frustrated, by Anonymous

It used to come so easy.
Never being challenged or troubled
Always loving the beauty and complexity of it,
Now getting bogged down in the cumbersome intricacies,
Confused not knowing how to help myself,
Frustrated with the fucking functions,
Wanting to get back to the beauty,
Seeking guidance.

***

Elise's piece was a particular clever one. Wanting to work with the Fibonacci sequence, she soon came up with the idea of laying out a flower garden, in which the nth flower named not only occurs Fn times, but also possesses Fn petals in its floral structure!

I think this poem would make a lovely children's book, if we could find the right illustrator.

My Fibonacci Flower Garden, by Elise

In my garden I have many flowers
The bloom elegantly after May day showers.

Here, I must bring you on a tour
Unless you've seen it all before.

It first starts off with a single white calla lily
That a friend gave me named Billy

Then next is a single pitcher plant
This is home to some ants.

After that is two euphorbia flowers
Couldn't you just look at them for hours?

Then there are three trilliums
Don't tell, I picked them from a mausoleum!

Then come five columbines
I swear, they bloom all the time.

And eight bloodroots
Aren't they just so cute?

Next to those are thirteen black-eyed Susan
Susan shouldn't have been cruisin' for a bruisin'.

And twenty-one asters
They seem to grow faster.

Then last but not least, thirty-four field daisies
I hope I don't seem crazy.

Now that you have seen my garden
Please beg your pardon.

It is now time for you to go
It looks like I have to put on another show.

***

Nathan's pi-based poem used the number's decimal places to count off the syllables occurring in each line, rather than the words. The effect is to render each line tighter, more succinct. What results is a geometric tour involving everyone's favorite constant:

Wonderful Pi, by Nathan

Beautiful.
Ace.
Vast bottomless
Pi.
Without existence
Not a Circle would ever compute.
Golden.
True value never known
Yet still you exist.
Essential.
Impossible, no
Inside an orbicular world
Your escort, radius, assists
finding area and volume
Infinite. Great. Infinitely great.
You, my friend,
Are pi.

***

I love this next poem, an ode on an abbreviation I use frequently when grading: "w.b.c" stands for "wrong, but consistent," and arises whenever a student makes use of an incorrect calculation but does so in the correct manner, resulting in a solution which is wrong, but consistent with an incorrect intermediate value. "W.b.c" is good, because generally it means the student's got the right idea, but just made a minor boo-boo in her computations. Partial credit-wise, "w.b.c." might mean only a point or two off.

Clearly our anonymous author has been haunted by this abbreviation:

Untitled #1, by Anonymous

Progress.
I stare and stare at the door
Feeling the approach of a familiar stranger
(Like pointless anticipation
Of an unnamed and unknown event)

Lingering.
On the very brink of my thoughts
The solution totters
Awaiting enlightenment (as a form of escape)
From the chamber of my mind

Disappointment.
Wrong but consistent.
Partial credit.

***

Sarah's poem, much like Sam's (see the previous post!), poses a mathematical puzzle embedded in its lines. In addition to performing this feat of numerical legerdemain, she based the structure of her poem on the sequence of prime numbers, the only student to do so.

Magical number, by Sarah

Magical number
Add twenty-four
Then multiply me by two
Half of my last digit is one
Eventually there will be a number that is right for me
Multiply my very first digit by six then subtract it all from me
Average my two digits and you will find a golf term
Triple me to see consecutive numbers
I am only one digit
Count backward to
See me!

***

In unpoetical news, I've just come back from the final faculty meeting of the calendar year, at which our fearless Learning Circle leader delivered directly into my hands the text for the LC in which I will be participating: Learning partnerships: theory and models of practice to educate for self-authorship, by Marcia Baxter Magolda and Patricia M. King. It should prove an interesting follow-up to Meszaros's book from this past semester.

This is the book we should have read first, perhaps?

Allow me now to end this post and set to work reading the paper I've been asked to referee for the Journal of Combinatorial Mathematics and Combinatorial Computing. (Say that one three times really quickly...)

I sing the verses eclectic

I've just been granted leave to share a number of my students' poems with my readership! Much thanks to the students who have gotten back to me and let me put their work on display, both those of you who'd rather remain anonymous, and those who'd like their names plastered proudly on their work!

In no particular order, then, I give you...

Mathematical Poetry, volume 1

The first I pulled from the stack at random, a little ditty by Sam, features a hidden message (read every seventh letter starting in the right position, and you'll find it...I cheated and typed the text into Mathematica to get it to paste together every seven letters):

Though It's Early, I'll Be There Surely!, by Sam

My, the amazed small one
At hardish attention will surmount
A test ahead
By reading a bunch, in easy bed stead

At this you can be delighted
As we, farsighted
Show our faces
In our favorite old "Man, is it eight?" places

A math class adored
I named thy award
It's found in my 4th
My paper explains

***

This next one is a stroke of parodical genius...I hope Lisette didn't spend too much time working on this send-up. I have a hunch her background includes a measure or two of Shakespeare (and her nights a good deal of euphoria-inducing caffeine):

Mathbeth, by Lisette

Act lim_(x -> 1+) 2x + 2, Scene lim_(x -> infinity) x^(1/x)

Student: By the scratching of my pencil
The answer slowly this way comes.
Open, mind,
the catacombs
Of transcendental sums.
But hark! Look you how the utensil moves!
(Enter Pencil Stage Right)
Pencil: How now, caffeine-riddled youth
What is it you do?
Student: A quest without an end.
Pencil: Horrors! Not another, fiend!
Student: Yet by your yellow wood I swear
'Twas not by my own choice!
Pencil: However the task was first derived,
To a veteran of your caravans,
Make good this oath with eye and voice:
Look to the ink of stalwart pens
In your aimless waste if parchment.
For never will your proofs amend
Those errors in your quotient.
Day upon day you dulled my lead
As ere you chased the numbers 'round.
So many times I thought, perhaps
You'd finally reached your limit,
Then watched my world shake upside-down
To briskly hide your mishaps.
I grew quite bald from misadventures
With wild domains and vicious powers
No more! I say again, No more!
My lead is soft, my wood is fragile.
Find some youth with a liquid core
And a shiny plastic shell.
We of wooden constitution
Have failed our last equation.
(Exeunt Pencil)
Student: So I, alone
But for my mug of coffee black,
Must weary waste the night away
With a knavish rogue called BIC.
(Flourish, Exeunt)

***

Farrah's oeuvre was one of the finest of the pi-based poems I received. The imagery (as much olfactory as visual) that it conjures up is magnificent, and the ironical twist at the end is superb. In her words, "I chose the topic of this poem because I was extremely hungry...then my thoughts on food and frustration led me to think about Pi which is delicious food that make my apartment smell like pastry and fruit and it is a pretty neat number...My favorite thing about the poem is the shape the poem made when centered on the page."

I hope the Blogger's pagination will render a similar effect, Farrah!:

Motivation for a Sweet Tooth, by Farrah

Broccoli carrots kale
.
Chopped
Garlic onion ginger tamari
Tofu
Cooked in a hot wok
Delicious food fast from my two burner hot plate
Aroma fills
My tiny apartment for many days
One room living makes for
A constant smell
Garlic permeating the whole place
Maybe I should have made some delicious Pi

***

This next one is one of a handful of personal professions of mathematical affinity that I received. (I'm happy that some of my students are willing to own up to a love for mathematics!) Carrie's not the only one to speak of a certain "balance" or beauty in math, as you'll see below.

For Me, by Carrie

Some see it as confusion
Something messy, some ugly
Something they are forced to do.
I am not these
I see patterns everywhere
A balance here that is seldom elsewhere
There are answers, solutions
And it's not hard
For me

But I talk to different people
They avoid it
Keep it at a distance,
Far far away
Higher levels
Fewer people
People think that you are crazy
It's still not that hard
For me

Everyone has things they are good at
Some have writing
Some have art and music
Some will go on to speak to the masses
These are the things that I find hard
These are what I avoid
Just give me a math problem
Because it's not hard
For me

***

Our next piece is another deeply personal one, a reflection on the author's feelings about not only mathematics but the way in which it is encountered. After it was written, I talked with the author about the fact that every line begins with an 'I': this pattern was accidental at first, but once noticed became an intentional goal that was harder and harder to achieve with each line. I find that pattern interesting, given the self-focus of the poem:

Imperfect, by Anonymous

I am ready to be challenged
Ironically this is the first thing which has been problematic
It is difficult for me to be patient and understanding with others
Irritation seems to be my main state of emotion
Intricate concepts are something I crave
Implicit differentiation as surprisingly stimulating
I fear my arrogance will be my downfall
I am ready to be challenged with others who feel the same way

***

I've selected a few of my favorite haikus by this next author, who turned in no fewer than ten of them! The slightly humorous, yet still insightful, tone I found in these poems was delightful. The author has asked to remain anonymous.

Confusion

A tall ladder falls
At twenty feet per second
Why would it do this?

The Number "Pi"

Three point one four one
five nine two six five three five
eight nine seven nine...

Math in Daily Life

Patterns on my bunks,
They resemble the graphs of
cosine and sine curves.

The Inventors

Did Leibniz invent
calculus or did he steal
the work of Newton?

My thoughts about math

Sometimes I like math
more than art; veggies and fruits
are boring to draw.

***

Here's another admission of fondness for mathematics; I have to admit that I feel much as the anonymous author does.

Math, by Anonymous

Math
To put into poetic verse
Seems quite contradictory
Free form verse
Clashes the firm ways of
Math
To put into words
Seems rather absurd
Verbs and nouns
Make no substitution
For the numbers and functions of
Math
Does not need words to explain
It stands on its own
Its equations speak their own verses of poetry
Numbers create their own images
In forms of graphs and shapes
Ideas are not conveyed through rhymes,
Or rhythms,
But through logic
And reason
Poetry is not needed to show the beauty of math
Its charm is found beneath its theories, proofs and functions.

***

Mark's pi-based piece, below, involves not only mathematics but biology as well, giving a brief natural history, and perhaps a glimpse of the future. His work was heavily informed by his view of the universality of mathematics:

The Universe to 44 decimal places, by Mark

Hydrogen and oxygen,
Bonding,
Create habitat for life.
Abruptly:
Orderless masses become perfect spheres
These fragments yield amino acids within fermenting, fruitful seas,
Beneath a
Young light and even younger sky.
Molecules converge as RNA grows,
Followed by DNA.
Double helixes entwined with data,
Data that unleashes its complexity upon every duplication.
Cells are born, and form intricate relationships of survival,
Synergies of molecules endure harsh primordial seas.
Mitochondria and bacteria converge, becoming what both couldn't alone.
Fins dominate waves,
Eyes guiding,
Gills supporting all,
Including gaping jaws, merciless to those who're obsolete.
But the sun touches
Not only the sea, but the land
As well.
The sun reaches leaves, creating new opportunities,
As a toughened foot blemishes
The smooth sand.
Tethered to water,
A bind that only the amniotic egg cuts.
A fiery rock flies,
Bringing chaos.
An instant unravels an eternity of adaptation.
Hairy hands grasp wooden clubs, utilizing new-found digits.
Musket drawn, soldiers hunt shadows.

The moon:
A pearl-white foot disturbs dust, an eternal footprint.
Generations later, another foot climbs a sanguine summit,
Its owner re-imagining humanity.
Europa:
Ice-cold, alien, yet sharing same miraculous elements:
Hydrogen and oxygen, yet yielding entirely different organisms.
Sol
Keeps all of this life alive.
Precious, yet inconsequential, a speck in a milky-white galaxy,
Itself very insignificant,
One galaxy amongst countless, but all under one universe.

***

My final poem for this post (I've got a bunch more for which I'm waiting for permission to post) was one of the strongest of those submitted verses that worked off of the Fibonacci sequence. I love the way John makes use of the sequence's exponential growth to allow each line to elaborate further on the previous ones, until the final line is reached, where it seems a thematic turnaround is made. (I apologize for the decreasing font size in the final line; I've done this to try to accommodate the line on a single line of the screen...I think it's gonna fail anyway!)

Math, by John

Math
Math?
Math is...
Math is everything;
The nature of everything is math.
The dynamics of today still follow original patterns.
And all the patterns in life and nature are based upon a formula.
But I wonder: do the laws of nature dictate these math formulas, or does nature follow universal laws created by math?

***

That's all for now, folks! I'm glad that I've been able to fulfill one of my promises. Later this week I'll post some more poems, and I'll get around to dissecting that Newton v. Leibniz thingamajobber.

For now, please have a pleasant eve.

Very busy

That was the "learn Chinese" phrase printed on the fortune cookie fortune I got a few nights back at the Asiana Grand Buffet.

Fitting!

As was the flipside, the fortune itself: "It's tempting to make promises, but can you fulfill them all?"

It's as though the cookie knew to whom it was going.

A few days back the man in charge of planning for next year's Conference on Diversity sent an e-mail to the members of the "Ideals" Subcommittee (on which I am), indicating that the subcommittee is still without a chair; he was fishing for volunteers. My finger hovered over the mouse button, itching to hit "reply," but I thought to myself, "my wife will kill me. This is no exaggeration: she will kill me."

My finger stayed.

She should be proud.

Soon after posting yesterday, I finished grading the Calc exams. They did quite well. The average between the two classes was around 78%, and very few failed. There was one 100% (yay!), one 99%, and one 98%, and a smattering of slightly lower As. Most were Bs and Cs, with a few stray Ds and only a very small handful of Fs. I'm pleased.

I had a chance to finish writing responses to the second section's poems as well. As I did with the first section's, I've saved a number of these, hoping to receive leave to post them.

Yup, the semester's coming to a close. Tomorrow's the due date for the two final exams still out (both for independent study courses) and the final paper (for the third independent study I'm directing), and tomorrow from 11:30 to 2:00 we have the last of the 280 presentations. I'm excited to know that there will be pie, provided by one of the team members deriving a continued radical expression for pi. By tomorrow evening, I should be done.

Then what?

I've had a good semester. I'd give myself a B+/A- for Calc I: high marks for ingenuity, originality, classroom skills, and general derring-do, tempered with a slightly lower grade for just...well...you know, I feel like I never really clicked with a number of the students in my class, especially several in the second section. I think I'm probably overanalyzing and expecting too much of myself, but I feel as though there was a bridgeable disconnect somewhere...we'll see what kind of grade the students give me.

I'm giving myself an A for 280 this semester, I think I outdid last semester's performance. I rather like the decision to use homework committees, I think that came off rather well. I believe my conscious, directed efforts to improve mathematical writing worked wonderfully, and I'm particularly proud of the first day's assignment, as well as the Professional Proof Analysis. We had several truly exciting days of class, on which we simply said Screw it! and dove headlong into relatively unrelated mathematical ideas that struck our fancy.

All in all, the class has been great.

I've enjoyed my independent studies this semester, too. Nicodemus is fantastic at providing himself with direction, so I've hardly had to think about where to take him as he found his own way through our Abstract Algebra II course. And Bethesda's independent study on The History of Math Technology has come along nicely, too, though in retrospect two of the topics on which she's produced papers have dealt more with philosophy than technology, ultimately: her first discussed the advantages of a duodecimal (base-12) system over a decimal one, and her last (of which a final draft is due tomorrow) deals with the validity of computer-based proofs. She's really come along in her writing, I've enjoyed working with her. Felicity's managed to eke out a fine independent study, too, coordinating homework and exams over e-mail, scanning and sending me her completed problems in Linear Algebra II. I'm amazed at the ease with which I've been able to juggle all of them, their hard work has made my job an easy one.

No research students next semester (unless Sieglinde and Trixie make any major breakthroughs in graph labeling over the winter break), but I've got that one reading course in lattice theory, and I'll be working with three of my favorite blasts from the past on their 480 presentations (Fiona's let me know she's already had nightmares about her presentation).

I've got to start putting together my Moore-method notes for graph theory, and figure out a rubric and grading schema. My wonderful friend Griselda and her colleague Natalia have been kind enough to share their respective course materials for discovery-method "proofs" courses they've run at their institution. I'm sure those will serve as handy guides.

I'm going to bring this post to an end.

Coming this week (I promise promise promise): Newton v. Leibniz, in the students' words, and a bit of mathematical verse.

A cathartic post

It was one of those days yesterday.

One of which?

One of those.

As I write this, the dawn is creeping up in the east, grasping at the leafless trees with Homerically rosy fingers. Pink clouds are scudding on a periwinkle sea. It looks like it'll be a beautiful day, and the prettiness tempers the ugliness of the purely visceral righteous indignation I felt last night.

Yesterday at 5:00 p.m. was due my Calcsters' take-home final, their final obligation to me this semester. Their duties done, it was the last I'll see of most of them until next year.

I had some wonderful parting moments as I said my farewells for the semester. Many of these people will be back for more in Calc II, and I look forward to it. I'm very open about the fact that Calc II is my favorite class to teach. (Although now that I've got 280 down pretty pat, I'm going to miss it next semester!)

The partings were rendered bittersweet by a revelation made last night as I began grading the Calc I final exams.

Six of my students cheated off of one another.

Exams from three pairs of people (two pairs of folks from the first section, and a third pair straddling the sections) show incontrovertible signs of collaboration.

This hurts me.

It hurts me because it insults my intelligence (did they think I wouldn't notice?), and because it takes the respect I've shown them all semester (in treating them like responsible adults all semester long, in trusting them with a take-home exam in the first place when many of my colleagues would likely say "you can't trust freshmen with a take-home exam"), the respect I work ceaselessly all semester to build, and it throws it back in my face.

Did you think I wouldn't notice?

News flash, folks: I'm a pretty smart guy, and I've been playing this game for a decade now. It ain't hard to recognize when someone's worked with someone else: there'll be identical use of uncharacteristic phrases, identical idiosyncratic errors, identical stray notational boo-boos. What the cheatee writes, the cheater will often copy. If the cheater wants to try to cover her or his tracks, she or he will often modify a line or two here and there, but if said cheater has no idea what the cheatee means by a particular line or computation, the result can be a wholly incongruous modification. (I saw a couple of doozies last night, changes that rendered the corresponding sentences ungrammatical.)

Did you think I wouldn't care?

In class, I'm a nice guy.

I'm a nice guy in class mostly because I'm just a nice guy in general. I could never be a hard-ass. It's just not who I am.

But I'm also a nice guy in class because I truly believe that in showing you kindness and understanding, and in showing you respect as adults (young adults yet, to be sure, but adults nonetheless), you will return the favor.

And for the most part, you do.

Of the thousand-odd students I've had over the past ten years or so, only a very, very small handful have bobbled the trust I've placed in their hands. By and large, I receive little but respect, gratitude, and positivity from my students. Even those who don't like me so much generally agree that I treat them fairly, that I treat them like sane, rational, adult human beings capable of managing their affairs with me with all due maturity.

So, believe it or not, it really does hurt me when a few folks pull this kind of crap.

I ask again, did you think I wouldn't care?

"Do they realize how much you care?" Maggie asked last night as I was ranting and raving about the six students who'd clearly gotten a little undue help from one another. "Do they know how much work you put into their classes? How much you care about how they do?"

Yeah, they do. I know they do.

Not a semester goes by without many students remarking (to me directly and anonymously on their evals) on the effort I put into class, the consideration I show them, the amount of work that goes into making sure they'll have a quality experience in my courses.

They do know that. I still believe that.

Why, then? Why cheat?

These aren't bad people I'm talking about. I like them all. They're smart, they're funny, they're kind. They're good people.

But, at least momentarily, they're also desperate, and they're inconsiderate.

Desperate times call for desperate measures.

For the most part, these folks are first-year college students. As such, though they may be sharp as tacks and quick as whips, they're also pretty piss-poor at managing their schedules. A year from now most of them will look back at their first year and realize just how much time they were wasting simply because they hadn't yet learned how to organize themselves and get their shit together to get it all done. They feel harried and hurried. They're continually, but artificially, pressed for time. They feel they don't have enough time to finish everything they need to, mostly because they've squandered their time through mismanagement. So when time's run out and they've yet to get it done, they'll resort to desperate tactics.

A couple of disclaimers are in order here.

First, I'm not claiming that you're not working hard. In fact, I'm sure you're working your tails off, I've seen evidence of your hard work all semester in my class. I'm rather claiming that you're working inefficiently; you've not yet learned how to write effectively and efficiently, you've not yet learned how to streamline your use of your supporting resources (textbooks, internet, outside assistance), you've not yet learned how to prioritize.

These things will get easier for you. By the time you graduate, you'll get better at doing these things, mostly because you'll have to: almost without exception, every one of you will look back on your first year of college as the easiest year of your college career. Hands down. To make it from here, you're going to have to get better at managing your time.

Nidra, a student from the first class I ever taught at UNCA, stopped by my office yesterday morning. She'd just finished off her thermodynamics final, and was on her way to go study for another exam. She was in the neighborhood and thought she'd say hello. We talked for a bit, and I made some cute remark along the lines of, "ah, freshmen!" and she agreed. She's sufficiently far away from her own freshman experience to look back on herself at that time and laugh.

Note to current freshmen: you'll laugh, too. You'll have your chance. The next few years will be hard as nails, but you'll love them, and you'll sincerely wonder how you ever thought your first year of college was difficult.

My other disclaimer? There are a few of you (and I'd like to think you know who you are) who truly do have heinous, untameable schedules. Some of you work forty hours a week, and/or have small children, and/or have ridiculously long commutes to campus nearly every day of the week. Some of you have had major personal crises that have unexpectedly affected your performance in your classes this semester. Some of you can gather no more than three or four measly hours of sleep each night, night after night after night. I know who you are, and I thank you for the hard work you've given me this semester; it touches me that you value our class enough to make it such a high priority.

But you know what? Almost without exception, these are not the people who cheat. Why this is, I don't know. Perhaps respect is born of their work ethic, perhaps they simply don't enough time to orchestrate the dishonest maneuvers cheating would require. For one reason or another, these folks, who at the semester's busy end are likely the most desperate people in the class, aren't typically the ones who cheat.

As I said above, the cheaters are also being inconsiderate.

I'm using the term in its somewhat literal sense: they're simply not considering the effects their actions will have on others. I cannot believe that they would act as they have if they'd thought about how strongly it would effect me.

Or so I think.

And so it goes.

So where do we go from here?

To those of you who cheated on the final exam through undisguiseable collaboration: you will be receiving a zero on the exam. If you feel this penalty is a bit harsh, please refer to Section 2.1 of the Student Handbook, dealing with "academic honesty": I would be well within my rights if I were to fail you for the course and report you to the Student Affairs Office.

And so it goes.

So.

So yesterday wasn't all bad, in spite of those late-evening grading revelations (and no, I'm not yet done grading. I didn't want my choler to color the remainder of my job, so I've put it off until after I'll have written this cathartic post, at which time I'll set to the task once more with renewed alacrity!), and in spite of a somewhat disappointing mid-morning meeting on grant-related stuff about which it would be inappropriate to say more (actually, I lie: the meeting was a productive and pleasant one, I just wish we'd had it three months ago!), I had a good day yesterday.

I shared a group hug with Quincy, Olivia, and Davina, three of my tip-top 280 students, in the Math Lab. I have had a blast working with those three, truly with everyone in the class, this semester. I've got one more meeting with them, on Monday.

Early in the morning Sieglinde came by to drop off her Calc final. She was touched by the e-mail I'd sent her letting her know how happy it makes me that she's happy she's chosen a Math major. She'll be a great student. We met for a half hour or so on Thursday, at which time I gave her a crash course on graceful graph labelings, enough to get her started over break in thinking about the gracefulness of lobsters and other closely-related graphs.

I gave a similar problem to Trixie yesterday afternoon. On Thursday Maggie and I had lunch with five of the folks who'd helped with Super Saturday this semester, in order to show my gratitude for their efforts. Trixie was among them. Half-(but only half-)jokingly she told me she was going to miss not having any math to do over break, so I promised her I could give her something to think about. "Not calculus," I said. "Real math." She agreed to consider it, so I spent an hour or so yesterday morning typing up much of the information I'd given to Sieglinde in person the day before, indicating a related problem Trixie could work on, dealing with asterisks instead of lobsters.

As she left I put the notes in Trixie's hands. "I'm excited," she said, "I don't know why."

"Because, one, you like math," I told her, "whether you admit it or not. And two, you're good at it, whether you know it or not. And three, you, maybe not over break, maybe not by the end of this coming semester, but likely within the next year or two, if you get started now and work hard at this stuff, you, Trixie Muddleston, have a chance to discover new, original mathematical results."

"That's exciting," she said.

"I know! And that's why you're excited!"

I've got her, I hope. I've got her hooked.

After all, if I care about it, so will they.

If I'm excited, they'll be excited too.

To all beginning teachers out there: be conscious, confident, and competent, to be sure, but above all else, be passionate, and show your students that you truly care about what you do. There's no better way to do this than to get them involved in your own efforts, to give them a spot on the team. The sooner this is done, the better for them.

"But freshmen?" the traditionalists cry. "Some of them can barely add, and they butcher the Product Rule, can't expand a binomial to save their mothers' lives! What business have they got working on research?"

Here's something I've noticed: many of the students who did well at math in high school and who in their first year of college elect to pursue a Math major truly have no idea what math is really about. They're toeing up to the starting line without even knowing how to run the race.

How do they get themselves into this? Many of them believe math is little more than an extended elaboration of the almost entirely mechanical classes they went through in high school. (True quote from a first-year Math major, earlier this semester: "I'm going to be a math major. I love derivatives!") They're wholly unaware that doing math well takes more than attention to detail and carefulness in computations, that in addition it demands creativity, originality, analytical perspicacity. They're unaware that math is an art as much as it is a science, that at heart most active math researchers are no less artists than are poets, painters, and concert violinists. Though they may know that math has to do with physics and engineering, and perhaps even chemistry, they're unaware that math shares intimate bonds with biology, sociology, economics, and philosophy. They're unable to see past the numbers and plusses and minuses, to see the abstract patterns and structures that underlie it all.

Sometimes by the time they've finished off a year or two of a college mathematics curriculum (and have thereby learned a bit more about what math is about), they've discovered they're in over their heads, they don't particularly like what they're doing, they're not very good at it, but if they want to finish their degree in time, there's no time to change majors.

They've no means to see past the numbers, for the numbers are, for the most part, all they've ever known. How on Earth can you convey to a first-year college student the depth and breadth of mathematics without letting them involve themselves in its enterprise? And the sooner, the better: they'll thereby learn early on what math is about, and they'll grow stronger at it.

I'd like to put together a one-credit course titled something along the lines of "Mathematics: a Survey," whose content would comprise an overview of the mathematics discipline: what are the major branches of mathematics, how did they evolve, how do they relate to one another, how do they touch on other fields of study? It would introduce the idea of mathematics as an extended exercise in abstraction and pattern recognition, of deduction and induction. It would also showcase some of the beauty of mathematics, and so might make an effective article of propaganda. It would have no prereqs, and would be recommended to anyone majoring in or thinking about majoring in Math. "Take this course," one could tell one's students, "and it'll help you decide if math is really for you."

Anyway, I've been writing this for two and a half hours now. I'd best be getting back to grading. I'll check in later, perhaps, and follow up on some of those loose ends I keep meaning to tie up (Newton v. Leibniz, math poetry, and so forth).

For now, take care. To all of my students, even those desperate few: thank you for your work, thank you for your efforts. I look forward to working with you further.

Mathematical muses

Last night I had a blast reading through my Calcsters' contribution to the world of letters. Many of their math-themed poems are positively delightful.

They run the gamut from simple, insightful haiku to epic poems in which the hero plays a pivotal role in the battle between the fractions and the decimals (one wonders what sort of heroic epithets would have to be invented to make this poem fit a classical hexametric form?). Some made math the thematic centerpiece, while others simply used it as an informant for the (as often physical as numerical) structure of their poems. Some were funny, witty, light, others were dark and brooding, still others intriguing and mysterious.

Many of them truly introspective and personal, and I thank my students for allowing me the chance to peer through this window onto their mathematical ideas.

I would like to post some of the poems, either in part or in their entirety, but I've first got to get permission from the authors. (Yes, yes, I know: I've still got to post some of the comments on Newton v. Leibniz...it's been a busy semester, all right??!)

I mentioned to Maggie the other night that I'd thought of compiling the students' poetry to produce a small bound volume they'd be able to keep as a memento. I think she liked the idea.

What say you, students? Sound like a good idea? Feel free to comment and let me know.

In other news: several students have mentioned how useful they've found 280 in improving their writing skills...the second round of 280 presentations came off rather well...I've started toying with the idea of using mathematical origami as a medium for guerrilla mathematics...and I've got two students from my Calc I classes who sound interested in looking at a little bit of graph theory over the break, just to make sure they get a head start on math research. The way I see it, the sooner they learn that math is much much much much much more than crunching numbers, the more they'll love it, and the better they'll be at it in the long run.

I am off.

I must run, quite literally.

Mathematical masturbation

Professor Alexander Ol'Shanski'i worked wonders with his magical constants.

He'd begin a proof by laying out several numbers that were related to one another through precisely premeditated proportions: "we let epsilon [always pronounced "ep-SIGH-lon"] be given, and we choose lambda less than 53 epsilon, and we let c be chosen so that c squared is greater than 7 lambda minus 3 epsilon over 2..." These numbers were always given from memory, as though he were performing a familiar liturgical paean in some gnostic ritual.

As the proof progressed, the pieces of the proof, including the magical constants, fell into place like tumblers in a lock. If you waited patiently and bothered to put the pieces together you'd see why it was each particular choice was made, you'd see why every constant had been chosen to have exactly the value that it had, no more and no less. The tapestry was woven so finely as to deflect the sharpest logical blade. The resulting proof was a thing of unrivalled beauty.

Professor Ol'Shanski'i was the architect of several other, relatively constant-free, of my favorite proofs from my graduate school days. I recall a day on which he led our Representation Theory class through the verification of some theorem or another whose content now escapes me. At the time the proof left me with a warm feeling of empowerment, of being in on a deep secret. That's how such a proof made me feel: with its careful construction, its meticulous and methodical progression, its ultimate culmination in an elegant and often surprising result from the most arcane abysses of mathematical thought, it could hardly fail to leave an avid acolyte like me spellbound, basking in the warm glow of its ethereal nimbus.

Thus, it seems like sacrilege to me to say that I worry about the message the non-mathematically inclined might get from all of this mathematical masturbation.

Now, when I exhort my mostly mid-level math students in 280 to make their proofs clear, readable, and intuitive, I'm asking them to construct the antithesis of a "magical constant" proof. A good proof should let its reader in, not shut her out. It should beckon to the reader and invite understanding through a gentle mental climb. It should provide toeholds and grips, enough for the reader to reach its peak. Don't say "let epsilon be at least 7 delta...," say instead "notice that our goal is to find a value of epsilon so that if this condition is met...because of this, we must choose an epsilon that is at least 7 delta..."

Mathematicians on the whole do a whole lot of mutual ego-stroking and chest-thumping. "We're so damned smart that no one really understands how smart we are."

Mathematicians on the whole are lousy ambassadors for their field.

What are we doing to let non-math-type people in on the tricks of our trade? It's clear that such in-letting needs to be done: have you ever known another subject in which people are so proud to boast loathsome incompetence?

"What do you do?"

"I'm a mathematician."

"Oh, I hate math. I'm so bad at it."

I'd kill to hear someone say in earnest, "You're a sex therapist? Oh, I hate sex. I'm so bad at it."

Many view math as an inaccessible sanctuary, a place where only geniuses may tread, and there with only the lightest and carefullest steps. Instilling confidence in my students, convincing them that yes they too can do mathematics, encouraging them to embrace their inner mathematicians (yes, we all have one)...these are the hardest parts of my job. Anyone can add, subtract, multiply, divide, differentiate, integrate...given the will to do so and the dedication to give it an honest effort.

I really believe that.

Maybe I'm just naive. Maybe I'm just an idealist, seeing the world through rose-tinted glasses.

Can you blame me? I'm having a really good week. Every teacher will recognize those "breakthrough" moments, at which a crack appears and a shaft of sunlight blazes through. I've had a dozen or so in the past three days.

Today's most momentous breakthrough came when one of my students dropped by my office with a rough draft of her poem, titled simply "Frustration."

"This is really cool," I said after finishing the first read. "This is really cool," I said again. And then again. "I realize I just said the same thing three times," I said, "but...this is really cool!" Then we had a somewhat more lucid and fruitful conversation about the poem.

It spoke sincerely of her feelings on mathematics, how sometimes its "cumbersome intricacies" frustrate her to no end. "Cumbersome intricacies": I love that choice of words! I wonder how conscious she was of that choice. As I noted to her, the words "cumbersome intricacies," cumbersomely intricate themselves, phonetically capture the sense of building tension and frustration she's attempting to convey; in those words the form and the content of the poem combine and give her writing the energy it needs to reach the climax at the piece's middle.

The only suggestion I could offer her was to think about anchoring it more firmly to mathematics by providing a metaphorical object for her frustration: is there one thing she could point to, specifically, that frustrates her?

We both hit upon the same concept, simultaneously: "those functions!" we declared, in unison. We'd worked on those together.

"Don't be afraid to say something like 'why can't those fucking functions behave!' " I told her, knowing from a prior conversation that she'd not object to the salty language. (One of my colleagues was walking by my office at the time and made a comment about how violent math had become.) "Like I said in the prompt for this assignment, you should feel free to use any words you want, obscenity, profanity, whatever you want to say, as long as it means something to you." She loved the idea, and she set about revising her work right away.

I was positively tickled by her work. As I see it, she's gotten out of this project exactly what I'd hoped my students will get. I hope they'll find out something about themselves, as students, as students of math, as human beings. I hope they'll explore the way they think about math, and how it makes them feel: is it frustrating? Is it warming? Is it exciting, sexy, fun? I hope they'll learn how it is that they learn, I hope they'll learn that writing about math can be useful, it can help them better access and organize their own ideas, it can help them make sense of their own thoughts, and to fit them together with others' thoughts as they work together to construct new knowledge.

She's done that.

This is really cool.

The first two of several student presentations in 280 came today. Although Dewey could stand to work on eye contact a little bit, he showed great mastery of the content he'd chosen to work on (a couple of countability proofs involving unions of countable sets), and his organization and boardwork were solid. With only a minor slip here and there, I thought his presentation was splendid.

Twyla and Calliope, my pair of graduating seniors, followed this with a flashy (methinks at times distractingly flashy?) PowerPoint presentation on fuzzy logic. I appreciated the central example they chose: how might two people of different ages construct different fuzzy membership functions for the same concept, namely, age itself.

"Twyla, who is 21," began one slide, "views anyone under 25 as definitely young." After that, your youth slinks quickly away.

"Calliope, who is 31," began the next slide, "believes you're definitely young if you're under 40." For her, youth slid downward, but not nearly so precipitously as in Twyla's measure.

By Twyla's measure I had partial membership in geezerhood, but I was still a spring chicken in Calliope's eyes, which are only a year younger than my own.

After their presentation, they told me how glad they were that I'd suggested the topic to them, that they'd both learned a lot and had had a lot of fun in preparing their presentation. I told them that their presentation had made me want to get back into fuzzy logic, with the aim of teaching a special topics course on the subject at some point soon.

I'm going to miss them. I've had the pleasure of having Calliope in three of my classes now, and I'm sure she'll do well in grad school.

It is now after 11:00 p.m.

I've had no more than five hours of sleep on any given night this week.

I am going to bed.

Fare thee well, fair reader, fare thee well.

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!

Quickie

I'll be off to my second section of Calc I in a little while here, but I just wanted to check in and let y'all know that I wasn't eaten by grizzly bears in Montana. I've made it back home safe 'n' sound, and ready to tackle the last (less than a) week of classes.

Wow.

What's up with teaching?

I spent a cumulative 20 hours or so over break in grading Calc I exams and 280 homeworks, and in reading my Calcsters' reflections on Newton v. Leibniz. I've extracted 19 excerpts, ranging in length from short sentences to entire paragraphs, from these papers, in order to quote them here. I'm in the process of gathering my students' permission to do so, but I hope to publish those excerpts soon, along with my own interpretation of the events that elicited the comments.

Many wrote on discovery versus invention: what does it mean to "discover" something? How can one claim credit, or is it even worthwhile asking who gets credit for what?

A number pointed out the humanizing effect that the project had on their understanding of mathematics, and of mathematicians: suddenly these titanic personages from history seem more lifelike, in all of their humility, failing, and pettiness. Through their faults are magnified their successes, and all of math becomes a more "human" enterprise.

Several reported periods of intense focus and excitement about the project. It came as no surprise to me that the most passionate reflections came from the second section, whose reenactment was decidedly more intense and authentic. Not a single person from the second section reported any disappointment in the trial, other than that it should have been allowed to take up two class periods instead of one. (I'll be sure to budget time accordingly next time around...several people explicitly said that this project should be retained in future incarnations of the course.)

In these reflections at least two of the Calc students expressed excitement about penning a math poem, and I've received a request to read over one students' rough draft.

Who knows what I'll learn from these poems?

What does math mean to them?

Is it wild, or warming, or simply terrifying?

And how can writing help them access their feelings about math, and then express them?

How is it that they make mathematical sense of the world, how do they fill the math-shaped holes that pop up around them?

This morning I picked up the post-surveys to be used in math 280 class to wrap up the writing assessment study we're currently working on. I can't tell you how eager I am to see what we can find out from the pre- and post-surveys and the differences between them. I typed up an additional question to which I'd like to know an answer: "Has this class affected your perceptions about writing in disciplines outside of mathematics? Please explain."

My hope is that my students, in reflecting consciously on writing in mathematics, will actually have learned something about writing in other disciplines.

The way I see it, it's kind of like how I had no earthly idea what in the hell the future indicative tense was in English until I learned what it meant in Spanish. Ditto the genitive case: German helped me out there. In such a way I hope that in asking students to focus on making their math writing correct, complete, clear, and well-composed, I've actually asked them to do the same with their writing in general.

Random observation for future elaboration: this semester, more than in all previous semesters of teaching combined, I've become more consciously aware of the appropriate pacing and spacing of assignments, from a developmental point of view.

Random note to self: I must write to Profesora Bornstein and let her know how the trial went...

...now, off to class: avanti!