The little gifts

It's the little gifts I get that make this job worthwhile.

Yesterday one of the hardest-working of my second section's Calc I students came in to ask for help with a few of the related rates problems we've been working on for the past few days.

She didn't need much help, really: she understood most of it quite well already.

In fact, she's needed little help for the past few weeks. She's redoubled her efforts in our class, and despite not having had calculus before (whether or not that's a liability is a topic for another post) she's clearly picking up on the new ideas far more readily than most of her peers, many of whom are much more experienced with these topics.

I was particularly pleased by what I saw on the rough draft of her homework: "Know:" and "Need:" appeared ubiquitously on her paper. In response to my constant exhortation "to identify what you know and what you need," a number of my students are making their responses to this exhortation explicit in their writing, just as I've done on the board before them. The sooner they appreciate how crucial those two simple bits of information (the needed and the known) are in solving a mathematical (or, for that matter, any) problem, the better.

This evening's review session brought me another little gift: little more than a week ago maybe one or two of a class of thirty students would have remembered to include the "dy/dx" at the end of the implicit differentiation of y², nearly every person present at the review called out for its presence in unison, as though their intonation might mark the coming of a mathy god.

I remarked: "did y'all notice that? A week ago almost no one understood what the Chain Rule meant us to do right here. By now it's old hat to most of you."

It's the little gifts.

I've learned to be more patient in waiting for these little gifts, but to be more mindful of them, to expect them and appreciate them, to know that they're bound to come.

It's only every now and then we're likely to win awards for our teaching, no matter how outstanding our teaching is.

It's only every now and then our students are liable to approach us once a course is done and say "I truly appreciate all that you've done for me" or "you've touched my life, in a good, good way."

But if we're doing right and we're doing well, nearly every day will bring us little gifts: one student finally grasps the difference between "equals" and "implies"; another (unaided) drafts a beautiful document in LaTeX; a third, in the middle of an in-class group activity, helps a fourth through an application of logarithmic differentiation. Elsewhere, a colleague borrows a thing or two from your teaching toolbox or asks to use a version of the rubric you'd written for assessing the quality of students' writing, while another asks you to come and have a talk with their faculty: "maybe you can show them a few of the things you're doing in your classes, and they'll understand that there are alternatives to the way they've been doing things for years now."

What little gifts will tomorrow bring?

Fall Break fantasy

As I make my way through the stack of grading I've set up for myself over Fall Break (it's not so bad, spread out as it is over four days), I can't help but fantasize about a course that will likely never be.

Let's call it MATH 301: Introduction to the Philosophy of (Mathematical) Feedback.

Audience? Math majors.

Prerequisites? Calc I, Calc II, Calc III, and MATH 280 (Introduction to the Foundations of Mathematics).

(Stated) purpose? To introduce math students to the important role played by written feedback on various assessment instruments given by the instructor to her or his students.

(Unstated) purpose? To offer a pool of qualified instructors to assist in the grading of homework, quizzes, and exams.

Let's face it: a public school the size of UNC Asheville simply does not have the resources to provide support for student graders for all of its calculus classes, and it's not likely that any of our majors are going to simply volunteer themselves as unpaid class lackeys, however rewarding the tedious experience might ultimately prove to them.

The solution? Offer a one- or two-credit class in which the students enrolled would first be trained in the offering of appropriate, effective, and meaningful feedback on a number of different assessment tools (homework, quizzes, and exams), before being unleashed on actual course work provided by the department's various instructors.

I envision three or four weeks of training, on topics including appropriate writing style; interpreting students' writing; common concerns, mistakes, and missteps; the basics of evaluation; and advanced grading philosophies (partial credit, curves, revision opportunities, feedback on qualitative work). Some of the training would take place through lecture, but much of it would be hands-on, in a "workshop" environment. The students' homework for the first few weeks would include sample grading of artificial homework and quizzes, as well as short papers on the various philosophies underlying grading and feedback.

In the weeks following the first few, students would be considered qualified to serve as graders for any of the following courses: Calc I, Calc II, Calc III, Precalculus, STAT 185, and Nature of Mathematics. In order to "earn their hours" for the course, students would be expected to meet for one (or two?) hours per week in order to grade together, sharing their travelers' tales of problems encountered as they offer feedback to their less-experienced peers. Of course, such "grading parties" could serve incredibly effectively as social events, and students in the course would be encouraged to meet in this manner on their own time, more often than is required.

Would it be a required course? Probably not, but as it would be most meaningful for those intending to pursue teaching or graduate school careers, maybe we could require it of the students in the Pure and Licensure concentrations.

If it weren't for the relatively light writing load, the class could even qualify as a writing-intensive course, so rich a picture of mathematical writing does it offer the enrolled students.

I really think it could be a rewarding experience not just for the students but also for the instructor who provides the training, and clearly for any faculty who take advantage of the resulting pool of student graders.

They may say I'm a dreamer, but what the hell, why not dream?

Observations

Having taught MATH 280 four times now (counting this current semester's installment) in the past four years, I've begun to notice trends in my teaching of it.

I noticed a few weeks ago that so far I've had two "on" sections of the course (Fall 2007 and the current section, Fall 2009), and two "off" sections (Spring 2007 and Spring 2009).

To say that a section is "off" isn't to say that it's full of bad students, or that the course itself isn't a pleasure to teach (it's still, along with Calc II, my favorite course to teach); it's merely to say that it doesn't run quite as smoothly as it would were it "on": certain handouts give the students more difficulty than they give students in "on" sections, committee reports don't have quite the snap that they would in "on" sections, and the general atmosphere isn't quite as jazzed.

I've also noticed trends that might one day help me predict whether a section will be "on" or "off." In descending order of influence on the "onness/offness" of the course, I've noticed that

1. both "on" sections are/were smaller (sometimes significantly so) than were the "off" ones (15 and 20 versus 24 and 27...I went from 27 down to 15 from last semester to this one). Obviously students in smaller classes will receive significantly more one-on-one attention than students in larger classes, and their in-class experience will be more meaningful and student-centered.

2. Both "on" sections are/were taught in semesters immediately following my having taught the course in the previous term. This "recency" allows me to be more aware of the difficulties students face with certain concepts than I would be were there a year or year an a half intervening between my teaching the course once and then once again. For instance, having just taught 280 last Spring, I remember how damned difficult students find the very idea of induction. This memory helps me more patiently coach them through their inevitable struggles with this concept than I would had I a year to forget just how hard they found induction. Whether it's fair or not, I find myself being more understanding of students' conceptual miscues and misfires this semester than I was last semester.

3. Both "on" sections mark/marked the occasion of a brand new major curricular component: in Fall 2007 I unveiled homework committees for the first time, and this semester I'm asking students to write a "textbook" for the first time. These overhauls may carry with them a sharpening of focus: since I'm making significant changes in the way the course is laid out, I pay more attention to the nuts and bolts of the course's functioning, and this greater attention leads to a more carefully crafted experience for the students.

Speculation, speculation: all speculation.

For now, I'm off to lead the first Super Saturday of the Fall 2009 term, my seventh term at the class's helm. Today's topic: fractal fun!

Something in the air (oh, and...verdicts!)

Second things first, here are the verdicts:

Section 1: Leibniz, by a vote of 3 to 1.

Section 3: Newton, by a vote of 4 to 1.

The second section could have gone either way, if you ask me; both sides were similarly well-prepared. In the first section, though Newton's team did very well, Nora, as Leibniz's lead attorney, was so on top of things that I think she stole the show. (She told me that she was on the debate team in high school. It showed.) Well done, Nora, if you're reading this!

The Calc I students are now setting out on their personal reflections on the project. As usual, I'm all anxious and atwitter as I wait to see how they've been affected by it.

Meanwhile...

...there must be something in the air they pump into Rhoades/Robinson. During the past week I've had no fewer than four current students and advisees come to me and profess some sort of passionlessness, dissatisfaction, or ennui with the courses they're currently taking. (For two of the students, it was the same class.)

One of my 280 students is wondering if the math major is the right decision for her. I'm not convinced that it is, and it won't break my heart if it isn't, especially since she's not declared yet and has promised to pursue at least a minor.

One of my advisees is just generally glum about her current courses. I'm a little worried about her.

Two others (a current 280 student and one of my brightest advisees) have diverse concerns about an education class they're both enrolled in right now...and it's not a course I've heard students complain about before. What's up with that?

With the latter advisee I had a long and at times laborious conversation in the wee foggy hours of Thursday morning. For various reasons she's not sure that she wants to stay in the teaching licensure concentration, and I'm not convinced she should, unless she's fair and squarely dedicated to it. For just as many reasons I suspect she'd be better off, from the point of view of personal satisfaction and fulfillment, if she went to graduate school in education (doing something like middle-grades learning or curricular development or educational policy) and became a superintendent or other high-level administrator: I think she wants to be in education, but she could affect so much more meaningful change if she were involved at a higher level.

Or, as I told her, she could go to grad school in math or math ed. She's got the chops to do just about anything she wants to.

We'll see how things go.

I just want her to be happy. I just want her to find something she's passionate about, something to which she'll dedicate enough time to do it well, something at which she can shine.

That's all I want for all of them.

Okay, I've got to be off. My father's visiting from out of town (I see him about once every other year) and the only time I've got to grade is right now.

Liveblogging Newton v. Leibniz, Round 2

Below is the "transcript" of the Newton v. Leibniz trial, as enacted by my second section of Calc I studets. See here and, more recently, here, for accounts of other iterations of the same assignment.)

1:51: The court is called to order. Newton's attorneys make their opening statement: "There's a lot of controversy surrounding these two titans of the mathematical world. What we're here to do today is prove beyond reasonable doubt that Newton has absolute priority in the discovery of calculus, and that Leibniz did plaigiarize from Newton." And: "When somebody sees someone else's work and tries to make it their own, this is wrong and constitutes plaigiarizes."

Leibniz's opening argument: "When I came into contact with Newton's colleagues, I was first learning mathematics and didn't understand much of Newton's work. Leibniz subsequently developed my own work."

1:54: Newton begins his case; Johann Bernoulli is called to the stand.

"Is it true that you were the only one who thought well of Leibniz's work?"

"No!"

"Then who else?"

"My brother Jakob, Ehrenfried Tschrnhaus, and hundreds of others in Europe and China."

"Did you deny writing a letter to Leibniz?"

"I don't recall."

There are no further questions for Johann Bernoulli.

1:56: A historical expert is called to the stand.

"What do you know about the Great London Fire of the 17th century?"

"It took out most of London, including printing presses."

"Did this affect the price of paper?"

"It did. It made paper more expensive."

"And because of the plague, weren't rags more often burned rather than being pulped for paper?"

"That may be true."

"Good. I am attempting to establish that Newton did not publish for reasons other than simply not knowing calculus."

There are no questions from Leibniz's team, and the historical expert is dismissed.

1:59: John Collins is called to the stand.

"What is your relationship with Sir Isaac Newton?"

"We met in 1676, through Isaac Barrow."

"Yes, in 1669."

"What were the contents of that letter?"

"It dealt with Newton's research. Not on tangents and curves, but earlier work."

"Did you copy this letter at all?"

"I don't believe so."

"Did you come into contact with Leibniz?"

"Yes."

"Did he see the letter?"

"He saw me in 1676 and he did see the letter at that time."

"Did he copy that letter?"

"I don't know, but I know Leibniz did see the letter."

Leibniz's team cross-examines: "We never received the entirety of the supposed letter. Do you know what we're talking about?"

"I know that when he came to see me, we talked about Newton's work."

"No further questions."

Collins is dismissed.

2:03: Henry Olderburg is called to the stand.

"Did you deal with Newton's writings and relay letters between Newton and Leibniz?"

"I did, in my capacity with the Royal Society."

"Were some of these letters rather cryptic?"

"I don't know. They were highly mathematical, including facts about the binomial theorem and other aspects of calculus."

[There is confusion from Leibniz's side of the courtroom: "how could one expect a 25-year-old not yet fully immersed in the world of mathematics to understand the convoluted mathematical writings of Isaac Newton?" "Do you have any questions for this witness?" "Not at this time."]

The witness is dismissed.

2:06: Newton's team rests. The court is in brief recess.

2:21: Court returns from recess, and Leibniz's team calls their first witness. Henry Oldenburg is called to the stand again.

"We have proof that there was a letter sent from Newton to you on October 24, 1676, that remarked that Leibniz had developed a number of methods, one of which was new to Newton (on power series). Is this true?"

"I'm not too sure. I might have seen such a letter, but I'm not sure."

Oldenburg is dismissed from the stand.

2:24: Johann Bernoulli is called back to the stand.

"Were you in contact with Isaac Newton?"

"Indirectly, yes."

"Did you ever receive a letter from him, talking about his calculus, and ways of coming upon calculus?"

"I did; I saw a paper he published. In Book 2, Proposition 10 of that paper, he made a mistake that I pointed out to my nephew, Nicolaus, who then corrected the error."

"This is after Leibniz had published his work on calculus, right?"

"Yes."

Cross-examination begins.

"What is the name of this book?"

"I believe it was his Principia."

"And this was not about calculus, was it?"

"I don't know."

"And who is this nephew? Do you have a copy of the book? Do you know what was in it?"

"It wasn't on calculus, it was more physics."

"What does his making mistakes about physics imply about his knowledge of calculus?"

"I merely want to point out that he's not infallible."

"No one's claiming that he's infallible. Not even the members of the Royal Society."

"But the Royal Society is claiming that he is the sole discoverer of calculus."

"Yes it is."

"But was Newton not on the board of the Society?"

"Yes, he was."

"Is this not a conflict of interest?"

"..."

Johann Bernoulli is dismissed.

2:31: Gottfried Leibniz is called to the stand.

"Do you know anything about Newton's development of notation, and his theories? And can you say more about your own notation?"

"Newton's notation came primarily from physics; he worked with vectors, velocities, speeds, and so forth. Whereas I, on the other hand, tried to use basic graph definitions. Newton did a lot of the same things, but his methods were harder. Furthermore, I tried to understand convergence of power series representations of functions, and he did not. Notationally, I used differential notation, whereas Newton used dots."

Cross-examination begins.

"Newton and Leibniz corresponded, is this not true?"

"Yes, but most of the time I learned of something from him only after I'd done it myself."

"So you did talk about mathematics, such as tangents and whatnot?"

"Yes, but Newton was concerned more with physical quantities."

"But even though you had your own ideas, the letters could have possibly helped you to expand on your work, is this not true?"

"Yes, because I'd often work through problems using both his work and your own?"

"Did you not admit in a letter to Conti that you were aware of Newton's work?"

"I did. But I'd already developed it on my own."

"Did the Royal Society not officially charge you with plaigiarism in 1715?"

"They did, but there was bias in that action."

"Do you have proof that you had discovered calculus first?"

"No, but I published first."

Leibniz is dismissed, and Leibniz's attorneys rest.

2:40: Newton's team gives their closing argument: "Keep in mind that what we're dealing with here is a very serious issue. Although there are dirty tricks played on both sides, the bulk of the evidence supports Newton's claim."

Leibniz's closing argument: "We're not claiming that Leibniz created calculus solely on his own, but only that he did not plaigiarize the work of Newton."

Fin. We'll see what they say tomorrow!

Liveblogging Newton v. Leibniz, Round 1

In just a few minutes Section 1 of my Calc I class will begin their re-enactment of the controversy between Isaac Newton and Gottfried Wilhelm Leibniz, and for the first time ever I will bring it to you "live"! (Compare the last iteration of the trial, in which I merely provided a transcript after the fact.)

9:33: Leibniz's lead attorney begins with an opening statement: "Our defendant deserves credit for calculus's development. Newton's statements contain inconsistencies, and he was okay with Leibniz's credit until Leibniz started to gain credit for the discoveries. Moreover, Leibniz's students and colleagues made great strides in furthering mathematics, directly from the work of Liebniz. We challenge Newton's team to present discoveries coming from the work of Newton. We also question Newton's motives in charging Leibniz with plaigiarism."

9:36: Jakob Bernoulli is called to the stand. "We understand that you are familiar with my client, Mr. Leibniz."

"Mr. Leibniz and I have worked side-by-side on this process."

"Can you elaborate on the personal character of my client?"

"He's very friendly, very trustworthy. We've never had any difficulties."

"Can you tell me about any of the other mathematicians you were familiar with at the time?"

"My brother, for one."

"Was it common for you to meet up and share papers, that this was normal and not particular to Leibniz?"

"Yes, it was common."

There is no cross-examination.

9:39: Johann Bernoulli is called to the stand.

"Can you tell us about your relationship with our client?"

"I've worked very close with him and with my brother. We've been working on calculus problems together."

"Is he well-versed in calculus?"

"Yes, definitely."

"Anything else you'd like to add, toward his character?"

"There were problems we were able to work out that he was not able to work out on his own, and he's given us credit. Why would he not do the same with calculus?"

"What's this problem you presented to the Royal Society?"

"It was sent to different mathematicians, including Newton. The answers Newton submitted were different from those that my brother and I submitted. The methods we used came from Leibniz's work."

"So the methods you used, coming from Leibniz, were different from those coming from Newton's work?"

Cross-examination: "Our sources say that I [Newton] solved the problem asked that day, whereas he Bernoulli's solutions came later."

"That is true."

9:43: Ehrenfried Tschirnhaus is called to the stand.

"You met Newton, and later on that year you met Leibniz."

"That is true."

"Did you not share techniques you were familiar with to John Collins and others, and later, when you met Leibniz, he showed you some unpublished papers by Descartes, correct?"

"That may be, but we were both mostly concerned with ethics and other issues at that time."

"But it was common to share material at that time?"

No cross-examination at this time.

9:45: John Collins is called to the stand. "The letters Leibniz had given to you, did they fall into someone else's hands after your death?"

"I don't know."

"Who was the President of the Royal Society at the time the Epistolarum Commercium [sic]?"

"Oldenburg?"

"No, it was Newton. Do you think this may have led to bias?"

"Perhaps."

Cross-examination: "Is it true that you Isaac Barrow?"

"Yes."

"Can you point him out?"

"The man in the green."

"Did he not share Newton's papers with you?"

"Yes."

"Is it possible that Leibniz may have seen these papers?"

"Yes."

"No further questions."

9:50: A historical expert is called to the stand.

"Are you qualified to rule on the personality of one Nicolas Fatio de Duiller?"

"de Duiller knew Newton, and they had exchanged papers. Is there more that you'd like to know?"

"Did they have a personal relationship or a professional one?"

"It cross some borders."

"Can you read this letter from Newton to de Duiller, please?"

[An excerpt of a very personal letter is read.]

"Hmm. That's interesting. It seems this de Duiller may have been romantically involved with Newton?"

["Objection!" "Sustained."]

"Was de Duiller the first to charge Leibniz with plaigiarism?"

"Yes."

"Might his motive have been a personal one?"

"That is possible."

Cross-examination begins: "Where have you gotten your history from?"

"Books."

"Therefore we could assume that you are taking the work of others and 'regurgitating' it, that you are not doing any of the work yourself."

"Well, I've read it all."

"Can we assume, then, that this information is accurate? Were you there?"

["Objection! Was anyone in this room alive when this all took place?"]

"No further questions."

9:57: Leibniz is called to the stand.

"Were you interested in math originally?"

"Not originally."

"It was only after you met Henry Oldenburg that you became interested in math."

"Yes."

"Was it common for you all to meet and have conversations?"

"Yes."

"Was there any secretive exchange? Was there anything going on behind the scenes?"

"No, it was all very open."

"Ideas were discussed in the open?"

"Yes."

"And you saw Newton's letters, right?"

"I saw the letters, but I couldn't understand his notation, so I could get anything from it."

"So you couldn't learn anything from it?"

"No, not really."

Cross-examination begins.

"It's nice to meet you, after hearing so much about you. Have you published any works of your own, before this controversy?"

"Yes, the Acta Eruditorium was published in 1684."

"Did this work contain work on calculus?"

"It had my notation for derivatives in it."

"Was it before or after 1666?"

"It was after."

"And this is after you saw Newton's notes and after you talked to his colleagues?"

"Yes."

"Therefore you published this book after you had spoken with Newton's colleagues and after you had traveled to Britain, and after you had seen Newton's work?"

"That is true."

"No further questions."

10:03: The court recesses for five minutes.

10:10: Newton's attorney makes her opening argument: "Though Leibniz developed notation for calculus, he did not in fact perform any of the work. Although he changed the notation and terminology around, he did not in fact discover any of it. We will show that the facts of the case bear this out."

10:11: John Collins is called to the stand.

"Mr. Collins: you knew Leibniz and Newton."

"I was good friends with both."

"Did you ever feel as though you had wronged Newton?"

"Yes. He didn't know about it until the day he died. But I had taken his work and distributed it, since he was so reluctant to publish it. I showed his work to Leibniz."

"No further questions."

Cross-examination begins.

"What were the contents of the letters you shared with Leibniz?"

"Newton's work."

"Calculus related, or did it concern more infinite series?"

"Is there anything more than your word to support your claim?"

"You have my word, as well as Barrow's and Oldenburg's."

"Other mathematicians saw Newton's work too, though, right?"

"That is the case: I and Barrow saw them, but no one else is claiming that they invented calculus."

"But you are a mathematician, and could have discovered calculus having read those letters, right?"

"Yes, but I wouldn't do that to my friend."

"No further questions."

10:14: Isaac Barrow is called to the stand.

"Would you say that you are the sole person who was allowed to distribute Newton's papers to the outside world?"

"Well, I only shared his work because of his reluctance to publish himself. I shared it with Collins."

"Is it safe to assume that such intellectual news would travel quickly and would seen as a 'bright light' at the time?"

"This is true of Leibniz, as well, who was smart enough to understand Newton's work himself."

Cross-examination begins.

"You worked with tangents, right?"

"And geometric functions."

"Were you familiar with the work of Pierre de Fermat?"

"I don't remember."

"My sources show that you saw another's work and developed further upon it?"

"I don't remember."

"Did Newton himself not elaborate on others' work?"

"Well, Newton invented calculus from other works that were geometric and algebraic and put them together."

"Is it not possible that Leibniz could put together another's ideas and do the same?"

"It is possible."

"Did you lie to Newton?"

"I published behind Newton's back."

"No further questions."

Barrow is questioned on redirect: "I'm gathering that you built on the ideas of other mathematicians?"

"Yes, you could say that."

"When you say 'build on their ideas,' is it not the case that these people were long dead after you used their work?"

"That is true, and I developed many of my own ideas in my travels."

"When did you first meet Newton?"

"When I was Lucasian Professor of Mathematics at Cambridge University."

"So you saw his talents early on?"

"Yes. And I told him to publish early on, but he didn't."

"Later on, you did learn about Leibniz's work?"

"Yes."

"When was that?"

"At least ten years...seventeen years...after Newton's work appeared."

Barrow is finally excused.

10:23: Gottfried Leibniz is called to the stand once more.

"You did see the work of Newton, right?"

"Yes."

"But you did not understand his notation?"

"I didn't get his fluxions, no."

"Did others understand his work?"

"No, others didn't understand it either."

"How can someone become excited about something that person does not understand?"

"It can happen."

"I find that very hard to believe. [There is a brief conference with Newton's colleagues.] Why were you excited about something you couldn't understand?"

"Because I knew that we were working on similar ideas."

"So you were excited about information you couldn't understand and couldn't see a use for?"

"I knew we were both working on calculus at that time."

"Was it called 'calculus' then?"

"No."

[There is a bit of confusion and a couple of objections.]

"No further questions."

There is no cross-examination.

10:26: Isaac Newton is called to the stand.

"Mr. Newton, could you please inform the ladies and gentlemen of the court and jury of your achievements?"

"Even as a young child, I was very intelligent. I made many devices and discoveries."

"So from early on you displayed a keen intellect?"

"Yes, of course."

"Can you tell me about something you published in your adult life?"

"I didn't actually publish on calculus until Opticks in 1704, because I got in a controversy early on with Robert Hooke. I did write letters to colleagues containing my work, and I referred to these letters when publishing Opticks. Leibniz got wind of my ideas and ran with them."

"That must have been hurtful."

"Yes, it was."

"You were knighted by the Queen, right?"

"Yes."

"So it's safe to say that you made meaningful contributions to science?"

"Yes, it is."

Cross-examination begins.

"Can you please read the date of this letter you wrote?"

"October 24, 1676."

"This is before Leibniz's work was published, right?"

"This is true."

[An excerpt is read.]

"By your wording, it would appear you were aware of other people's methods for solving the same kind of problems?"

"But it was all based on my work."

"Do you have proof?"

"Collins and Barrow shared my work with others."

"Leibniz claims there was no calculus in those letters that he was able to understand."

"That's not conceivable: how could Leibniz not understand this work, if he's so smart?"

"But you were aware that other people were doing work in calculus, right?"

"Yes."

Redirect: "There were other mathematicians out there working on these problems that you'd already solved, right?"

"That is correct: Johann Bernoulli's problem, for instance. I solved it in a day, and only years later did Leibniz publish his solution, after he'd seen my work."

From Leibniz's attorney: "How did you submit your answer? Was it not anonymously?"

"I don't recall."

"Was your name on the solution?"

"No, because I was afraid of criticism, after my experience with Robert Hooke."

No further questions.

10:36: Closing arguments begin.

From Leibniz's attorney: "We have shown that a good deal of math came from both Leibniz's colleagues and students, and there was no proof that Newton's calculus played a major role, but rather it was Leibniz's work that formed the basis for these discoveries. We also showed that Newton had motives, personal and professional, for claiming Leibniz was a plaigiarist. We don't feel that Newton's attorneys have proven Leibniz was, beyond reasonable doubt, indeed a plaigiarist."

From Newton's side: "Thank you for your time. I believe that my client has a true and valid case, and that his letters were circulated and recognized long before Leibniz published his work, and that there's no way Leibniz would not have understood the import of these papers. He clearly took this work, changed the notation, and claimed it as his own. Newton, as you've seen, was a brilliant a man, and clearly capable of inventing the calculus."

10:40: Court is adjourned.

The editors are in

I'm in the middle of the first "editorial meeting" with two of the students from the current 280 class as we go over the seven sections their classmates have provided for inclusion in the first "chapter" of our course "textbook." (I've never before so horribly abused quotation marks.)

It's going wonderfully: we've had great conversations about exposition and style, and about the appropriateness of various arguments and explanations. They're making wonderful and cogent points about various aspects of their peers' writing, and they're suggesting meaningful additions to the current draft, which will be taken up again this afternoon by two more of their peers.

I have to get back to it now, but I thought I'd check in. I'll report more later.

Things students say that generally don't impress me...

...particularly when they turn out not to be true (which happens at least 50% of the time):

1. "I already know how to [perform a straightforward, utterly formulaic and therefore not difficult, acontextual, and unintuitive computation] from taking this class in high school."

2. "I'm really good at writing papers, so I don't have to start them until the night before they're due."

3. "My high school teacher forced us to memorize all of these formulas."

(I hope not) to be continued...

Pretrial discovery

You may recall that in addition to my dedicated service to the Writing Intensive subcommittee and my interest in writing and writing pedagogy, I also double as a math professor.

Today my Calc I classes spend their respective class periods meeting with one another in groups as they took part in the first-ever "pretrial discovery" I've organized for the Newton v. Leibniz project. The brainchild of one of the students who took this course last semester, this activity was meant to give the various parties a chance to meet with one another (Newton's team with Leibniz's, Leibniz's team with Leibniz's colleagues, both litigating parties with the historical/mathematical experts, and so forth) and coordinate arguments and defenses.

For the most part I think the hour or so was well-used. Certain parties dove into the project with gusto. Nora, Leibniz's lead attorney in the first section, was champing at the bit as she met with Newton's team. I'm eager to see how valiantly she defends her client next week. Meanwhile Nicolas (playing one of Newton's colleagues), though a bit more subdued than Nora, clearly had victory on his mind as he talked through various arguments Newton might use in order to win the case. With each jab I threw his was as devil's advocate, he feinted feistily and jabbed back. I think he'll make a good witness.

While I'm actually on the subject of math, I should say a little bit about a project in 280 that's threatening to come off the rails. Though for the most part the course is running along smoothly and the students are doing marvelously, the only new component to the course, the student-authored textbook, has stalled on the semester's roadside. It's partly my fault, as I've been lax in instituting deadlines and laxer still in spurring the students to work. This is in part because they've already got leviathan tasks facing them with Exam 1 due tomorrow, a new homework set to be handed out in class tomorrow and due next week, and various high-level handouts to digest and deliver in class.

I hope to spend a little bit of time tonight in helping the students get their shit together:

1. There are several sections of the first "chapter" already written in languishing on the "textbook forum" on Moodle. I'll collate them into a single document and ask one of the students to take a stab at editing over the weekend.

2. Though hesitant to do this at first, I'll bow to the suggestions of one of the commenters on this blog and one of the current students and put together a "checklist" of issues that should be addressed in the first chapter. I'll pen similar checklists for the second and third chapters and distribute those as needed. (We're in the middle of the third chapter, on sets, right now.)

3. I'll put off asking students to work on the second chapter until a bit later in the semester; it might make a good review topic at the semester's end, when we're likely to have at least a little free time.

4. I'll firm up a clear and coherent schedule for the writing of the third chapter, to commence at the end of next week.

So that's 280 these days, folks. It's really a joy of a class this semester. Last term's class was so large that it was overwhelming and unwieldy. Though I loved many of the students in the class, teaching the course was a tiring enterprise. This semester's class has rejuvenated me.

Before I go I should mention that I invited my new colleagues from the College of Charleston to collaborate with me on my assessment of REU students' technical writing. I'm excited to see what comes of this project, and delighted to get to work with my new friends.

Liveblogging CWPA 2009: Part 3

1:46: Heavy with lunch, we enter our afternoon session, a recap of this morning's small-group breakout sessions on various topics of concern to WPAs and other practitioners of writing (sustaining programs, organizing writing centers, organizing WAC/WI programs, and designing and directing sophomore writing courses).

1:48: Chris Warnick and Jessie Moore update everyone on sustainability and writing centers.

One issue is setting aside time for reflecting on objectives and successes.

Very poorly developed in many WPAs is the ability to say "no," and this attribute can effect program sustainability: if the precedent is set that a few people will do it all, then once those people are gone, the program may falter.

There is substantial dovetailing of both expertise and responsibility in working with other departments and programs on common writing-related goals.

1:52: The WI group (the one in which I played a part this morning) reports back. Much of our discussion focused on faculty development, achieving faculty buy-in, and managing WAC/WI programs with minimal mission, resources, and oversight. We also recognized that the bridge between first-year composition courses and more advanced disciplinary writing courses is built on the backs of writing centers and sophomore-level writing courses.

2:07: Will Banks (East Carolina University) directs his sophomore-writing group members to share their ideas.

Tony Atkins (UNC Wilmington) speaks on the idea of bringing in folks from other disciplines to teach more discipline-specific writing courses as sophomore-level offerings: not only does it help to stretch a tight budget, but it also helps to involve faculty from areas not typically involved with teaching lower-level general education courses. Moreover, it helps faculty to avoid the academic stagnation that can occur when a person teaches the same course year after year after year.

2:12: Jessie Moore returns to ask people to center themselves on some patches of common ground: (1) the issue of sustainability, (2) the encouragement of principled decision-making rather than purely logistical or fiduciary decision-making, (3) the need for physical space for meeting, planning, and reflecting, (4) the need for cross-program conversations, (5) the issue of expertise: who brings what to the conversation, and how can everyone feel and be needed?, and (6) the need for time for faculty to pursue other academic interests without feeling their lives are dedicated solely to oversight of unwieldy programs.

2:16: Our own Dee James discusses UNC Asheville's Lorena Russell's idea of instituting an "expertise barter" system by means of which area colleges and universities can effect short-term exchanges of faculty in order to more widely spread faculty expertise around the region. Jessie and Mary Alm (UNC Asheville) follow up by indicating other ways in which expertise can be exchanged.

2:24: Jessie Moore asks what CWPA can do to encourage this sort of cross-fertilization.

Observation du jour

The energy a student puts into the revision process is directly proportional to the sense of ownership and authorship the student feels she has regarding the content and structure of the piece of writing (or mathematics, for that matter) being revised.

Liveblogging CWPA 2009: Part 2

9:06: The folks from the College of Charleston are describing their adjustment of the first-year writing requirements. Chris Warnick begins...

The new program replaces a two-semester FYC sequence with a single semester course. Problems with this set-up? The second semester often saw too much focus on literature and de-emphasis of composition. Students often perceived it as redundant.

The new set up features a one-semester, four-hour course, streamlined for pedagogical and financial reasons. It is hoped that this one-semester course will see more intentional instruction of composition, and it's certainly helped financially: the college's dependence on adjuncts has been greatly reduced by the drastic reduction in the number of required instructors.

They've had to face and conquer various myths about writing instruction, including (1) the notion that writing skills "transfer" from one discipline to others (reality: this is nonsense) and (2) the notion that two semesters of writing instruction are better than one (reality: no number of first-year courses will adequately prepare students for disciplinary writing).

9:17: Amy Mecklenburg-Faenger continues, playing "Negative Nancy," addressing challenges faced with the changes they've made...

The appearance on the surface is that the department is very much on the same page with regards to writing instruction. However, there are always different perceptions of the nature and effectiveness of curricular change, and in reality there has been a good deal of resistance to the changes, and the WAC component concomitant their overhaul has yet to take hold.

There's a leadership vacuum, with no one there to oversee the program directly, resulting in inconsistent quality of instruction. In particular, one could not assume that students in the second-semester course had been taught particular skills in the first semester-course.

There's very little training and faculty development, and since the responsibility for maintaining the courses is shared throughout and governed by the entire English Department, when pivotal decisions are made regarding composition instruction, the votes of the "non-experts" get swamped by those of the "experts."

There's no centralized place (physically speaking), leading to a defocused existence in space and a lack of coherence program-wide.

There's the notion that writing about literature is superior in some way to basic composition, and the notion that students will "learn to write by reading."

There are antiquated grading systems still in place that punish students for almost arbitrary, acontextual compositional errors.

The nature and quality of thesis-driven research-based writing is highly inconsistent from section to section.

There's fear on the part of faculty that they're being micromanaged, being told what to teach. In reality, a well-designed, shared curriculum ensures that faculty don't have to start at zero, and can benefit from a well-developed and standardized curriculum. Students, too, can be assured that they will receive similar instruction, no matter who their instructor is.

9:35: Meg Scott-Copses continues...

They've amassed a goodly pile of materials in the process of redesigning the curriculum.

A list of student goals for the single-semester course helped reify the intentions of the course's faculty. A list of recommended readers help as well. Rather than mandating a particular reader, they've attempted to standardize an assignment sequence: (1) summary and response, (2) analysis, and (3) synthesis.

They've been partnering with the library in coordinating instruction of research and information literacy skills, and have developed a list of expectations regarding this instruction. These expectations focus on instruction on critical evaluation of sources.

Now, what to do with the fourth hour of the course? Conference with students? If so, in class, in small groups, or individually? This fourth hour gives instructors extra time to fit in instruction they would ordinarily have had to do "on their own time."

But how is it all working? Hard to say. To say something about it,

9:53: Jennifer Burgess continues, on assessment...She's built an assessment program from the ground up. It's a hefty program!

Common ground

I noticed a long time ago that I tend to have better conversations about pedagogy and academic theory in general with people outside of my own discipline than I have with people in it.

It was only tonight as I was talking with a couple of newly-met colleagues in rhetoric and composition that I finally figured out why this likely is: when I'm talking shop with folks from wildly different disciplines, I'm forced to seek out common ground with them, and that ground is usually centered upon academic fundamentals like classroom practices, curricular development, pedagogical theory, and the like.

With the aforementioned colleagues (one of whom teaches at the College of Charleston, and the other at Montreat College) I shared a wide-ranging discussion on college pedagogy, especially as regards writing, and we had what I felt were some excellent insights on college students and their education.

Among them: that first-year college students are delightful people with whom to interact because they're liminal beings in so many ways. They're positively brilliant one minute and downright stupid the next. They're jaded, arrogant, and self-assured in some ways, but naïve, immature, and credulous in others. They're truly passionate about learning, but they don't want to take the time to take it on, and while they've got enough energy and enthusiasm to change the world, their piss-poor time-management skills can barely get them from breakfast to lunch on any given day.

They're awesome people.

And as my new friend Nicola said tonight, whenever you're interacting with them, you've got the power to truly change their lives and offer them an eye-opening, life-changing experience. As lovely as they are, our upper-level majors, as a rule, have drunk the flavor of Kool-Aid we've offered them over and over, and they don't need their hands held any longer. They'll come to class willingly, and they'll often interact passionately once they've done so...but there's something about the wild-eyed eagerness with which freshmen feast on the ideas and concepts of a course which has first inflamed their passions.

"It's always exciting to be the first to spark that flame," Nicola said.

Indeed.

On that note, I'm going to hit the hay. It's a long day tomorrow (on the agenda: a presentation by my new friends from the College of Charleston, and break-out groups on various areas of concern to writing program administrators...and socializing...plenty of socializing...), and an even longer one on Wednesday.

Liveblogging CWPA 2009: Part 1

7:34: I'm sitting in the basement of the North Lodge at the Wildacres Retreat in Little Switzerland, NC, at my second Carolina Writing Program Administrators workshop. As energized as I was by last year's conference, I'm more than eager to see how this year's program shapes up. You may recall that in the wake of last year's workshop I had planned a twelve-part series dedicated to writing-related issues...the realities of the semester set in and only five (?) parts materialized.

If I write as I go, maybe more magic will happen!

So here we go...

7:37: Jessie Moore (Elon University) is introducing "The WPA Game" to help direct us toward our objectives. Who are our allies? What are our resources? "The game is collaborative; everybody is on a single team, working together to achieve an objective." Sounds like fun!

7:46: The party has been momentarily distracted by the presence of a gigantic spider.

8:00: After an introduction to the Retreat, we return to "The WPA Game"...self-introductions now commence.

8:02: Objectives, going around the room: "keep everyone employed," "firm up WAC plans," "developing a writing fellows program," "successfully implement a new WI program," "get tenure for the current writing program co-director," "revise second semester of composition (by introducing interdisciplinary features, for instance)," "retain the second semester of a writing sequence in the face of budgetary cuts and administrative opposition," "keep a writing program afloat," "pass comprehensive exams," "get other programs/schools at the institution to buy into WAC," "pass the torch to the next WPA," "assessment, assessment, assessment," "bringing pedagogy into the 20th century," "implement the new one-semester composition curriculum," create the groundwork for a future writing program," "managing a CAC program without resources or direction," "make IWIn work institutionally," "create a sustainable life, personally, professionally, and in all other dimensions."

8:20: Random observation: it's a young bunch this year compared to last year! The torch is being passed.

8:24: We're about to play the game! More later...

"You can call me 'Patrick...'

But if you're uncomfortable with this, 'Dr. Bahls' is fine."

These are the words with which I begin most classes on the first day of any given semester. They're uttered shortly after I rock on in wearing vividly patterned shorts, Chaco sandals, and a plain uncollared T-shirt.

To my more dressed-up and buttoned-down friends in management or accountancy departments, or to my colleagues from cognate math departments in more conservative universities, I look a caricature of liberal academic hippiedom, a stereotype of the left-leaning professor. My dress is casual, my manner with my students more casual still. I'm unconcerned with formal titles, encouraging my students to think of me more as an equal than as an expert. I'm laid-back, easy-going, and down-to-earth.

And I can get away with this because...

...is it because I'm a straight white male?

I talked about this with new friends I met during a visit to Clemson University yesterday (shout-out, Nanette!): I've known for a long time now that a number of my equally-if-not-more-well-qualified female colleagues have a hard time dropping the word "doctor" from the phrase by which they ask their students to address them: the word is an extra layer of armor plate that protects them from charges of academic inadequacy. ("Of course she's qualified to teach this course, she's a doctor!") I've only recently begun to think about the privileges bestowed upon me by other markers of my sociological makeup.

In the "plus" column: male, straight, white, non-disabled, holder of an advanced (terminal) degree.

In the "minus" column: atheist, untenured, left-handed.

If it were a matter of toting up points to determine my net level of privilege and power, I'd probably come out ahead: my pluses are more numerous than my minuses, and generally exert more force. I'll never have to worry about going through my career being defined as a "white male mathematician," while some of my finest colleagues have had to put up with appellations like "one of the best female research mathematicians in the game." (Allan G. Johnson, in Privilege, power, and difference, p. 33: "People are tagged with other labels that point to the lowest-status group they belong to, as in "woman doctor" or "black writer," but never "white lawyer" or "male senator.")

What assumptions, fair or unfair, intended or unintended, privileging or oppressing, am I taking with me into the classroom?

For instance: having a septuagenarian student in one section of my calculus classes reminds me that I can't make generational assumptions, and even the laid-back level of discourse with which I often interact with students of more typical college age in an attempt to convey tricky technical ideas in a down-to-earth fashion might confer unearned advantages to them to the disadvantage of someone less familiar with today's slang and pop cultural references. This same highly fluid "code-switching" that I do in my classroom, which puts many (most?) students at ease, may also put up unintended obstacles for students whose first language is not English.

What about race? At a school like UNC Asheville (which is woefully undiverse from a racial point of view), it's easy to fall into normative assumptions about the ways in which people of various racial, national, and ethnic backgrounds perceive and receive new ideas and new information, even ideas and information as abstract and "objective" as those found in mathematics. Am I unwittingly privileging the vast majority of my calculus students, who are white, to the disadvantage of the meager few (no more than five or six out of sixty-five this semester) who are of color? Obviously I'd like to think that I'm not, but privileging assumptions are often subtle. What would such assumptions look like in the mathematics classroom?

More to come, certainly.

Making space

It's incredible how much the physical configuration of a classroom (or other gathering setting) can influence the dynamics of the gathering that therein takes place.

As I settled into the seat I'd chosen for myself at the outset of a committee meeting I would be chairing this morning, I noticed that I'd unconsciously chosen to position myself across the table from the others who had already arrived, going so far as to reposition a small classroom table to that it angled up against those already occupied by my colleagues, forming a small triangle with theirs.

I told myself at the time that by rearranging the seats into more or less a circular formation I was just trying to create a physical environment that would facilitate greater inclusion, but soon after a few others arrived and seated themselves at the tables across from me, I realized that unwittingly placed myself in a position of physical "dominance" relative to them.

This sort of physical domination goes on in our classrooms every day, even in those classes which are most intentionally designed to avoid such domination. Permanent classroom configurations often reinforce this domination: even if the classroom's desks are not bolted into the floor, quite often the classroom's communal writing surfaces (whiteboards, blackboards, et cetera) are placed in such a fashion to privilege one side of the classroom over the others, and it is at this side that anyone (instructor or student) hoping to take lead of the class for any length of time must stand, distinguishing that person for a time as "the authority."

How might this be avoided, short of eliminating all manner of written communication during class? (While this may be possible in some courses, it's well-nigh impossible in mathematics for any sustained length of time.) Install communal writing surfaces on every wall of a classroom? Provide resources for fully electronic communication in the classroom setting?

To be continued, I'm sure...

But wait!, there's more...

Already this semester I've had two 280 students ask me about the use of the word "but" in mathematical proofs. This is remarkable if only because none of the roughly 80 students I've had in this class in the past few years have ever asked about that word choice.

"I don't get it," one of the class's sharpshooters said this morning. " 'But' seems to have negative connotations. Why would you use that word?" She'd been puzzled by my writing something along the lines of "The lefthand side gives us blah blah blah...but this value is nothing other than..."

I had a hard time explaining it, and I hemmed and hawed. "It's like you're leading the reader to a surprise ending...or..."

"It sounds like an infomercial," one of the other students broke in, "You know, like 'but wait! There's more!' "

"Exactly!" I said. "That's exactly it! You're injecting a little bit of theater into the proof, indicating that you shouldn't be satisfied with what's already come, you're about to raise the stakes a bit and up the ante."

Only after a handful of jokes about Billy Mays doing mathematics were we able to continue.

I'm impressed with this group's acuity and attention to subtle details. The semester just keeps getting better and better.

On maturity

I'll likely be putting up a rather lengthy post later today concerning thoughts I had during the first meeting of my latest faculty Learning Circle (on Allan G. Johnson's Privilege, power, and difference, 2nd ed., 2005), which took place yesterday.

For the time being, I just wanted to share a simple observation I made last night as I worked my way through this week's Calc I homework: almost unfailingly, seniors in any major (no matter how closely related to math, no matter how great their raw math talent) will do well (B or better) in a Calc I class. Nowhere is this more true than on the homework, on which seniors typically outscore first-year students by a substantial margin.

Why is this?

I believe it's because to do well in a relatively low-level course like Calc I, one need do little more than pay attention in class, take good notes, and complete homework cleanly and correctly and punctually. These things done, quizzes and exams will generally take care of themselves. Seniors excel at these things because they're mature, they're well organized, they're adept at managing their time. (In fact, I would say that the number one difference between first-year students and seniors is their ability to budget their time effectively.)

I've got two seniors in my Calc I classes right now (one an Economics major, the other a Health and Wellness major), and despite their relative inexperience in calculus, they're both riding roughshod over most of the classes' first-years. Why? Their homework is gorgeous; it's clear that they both take the time to do it right, and I know at least one of them does it in multiple drafts. In doing the homework well, they're solidifying their understanding of every last concept we discuss in class, and thus they're able to do pretty well on the quizzes, too.

They're no exception: in classes past I've noticed that most seniors exhibit similar success.

A tip for the young 'uns: start the homework right away. Seriously. I don't say that over and over and over for the fun of it. Take time to do it neatly, legibly, correctly. The homework really is where you do much of your learning, as you puzzle through problems and sort the ideas out from one another. (This is especially the case in classes less like our Calc I classes, wherein you wouldn't see as much hands-on activity as we do.)

For now, farewell. Next up: "The privilege to call myself Patrick."

Throwin' the book at 'em

So we've taken the first few tentative steps toward writing a "textbook" for our 280 course, as described in an earlier post. To get the ball rolling, I've asked students to share, on Moodle, the on-line course management software to which we've all got access, a topic on from the first "section" of the course on which they'd like to write a paragraph or two, and for which they feel they could construct an example or an exercise.

So far only 1/4 of the class has responded to my call for topics, so I'm going to have to ride them a little harder.

Once they've all made their suggestions, they'll each choose a topic and submit their work (done in LaTeX) on Moodle, and one or two students will volunteer as editors to sort it out and piece it together into a single narrative to which the others will be able to respond.

I'm a little concerned about how long this process is going to take: I have a feeling we'll be done with the course's second "section" (proof methods; we're halfway through it now, about to begin induction on Wednesday after the Labor Day weekend ends) before we have a chance to finalize the first chapter of the textbook, and so the concepts therein contained may not be as fresh in their minds by the time they're asked to write about them.

On the hand, constructing the textbook material could be a good review.

We'll see how it goes.

For the time being, I'm just going to wait for a few more entries to come in via Moodle.

Signs of student learning, or, twenty-five minutes of time well-used

I've got a reputation for moving a bit more slowly than my colleagues through the same course material, and this semester's proving to be no exception. While one of my fellow Calc I instructor's classes is already pounding away at trig derivatives (judging from the terminals bits of his lecture I was able to catch before following his act with my own in the classroom we hold in common), students in my class are just now starting to get a feel for limits and an understanding of how they might be useful.

Why the slow pace? While student-centered learning makes for meaningful learning opportunities, it's certainly lacking in celerity.

Sure, I could have lectured our way through the multistep radioactive decay problem in five minutes, but I'm pretty darned sure my students got more out of it by working through the difficulties themselves, one small but crucial point at a time...even if it took twenty-five minutes to work through the same example that could have been done in 20% of the time in a "traditional" fashion. Evidence for the exercise's success was immediate and evident, and right away I took note and pointed it out to the students in my first section.

"You know what I just noticed, three minutes ago?" I asked them, as we drew near to the exercise's end. "I saw thirty-two pencils scribbling away, without pause, on thirty-two sheets of paper, not a single one in sight at a standstill. Every one of you was busily working away at a solution, because as far as I could tell, every one of you understood what you needed to do to calculate the half-life we're now looking for.

"Compare this with the situation at the outset of the exercise, at which time most of you were staring blankly at your papers, wondering timorously what first step to take. Something happened between then and now: you gained confidence, you gained understanding. Something happened."

I was very happy with the exercise. There'll be more like it tomorrow.

More braggin'

Below is the most marvelous dialogue ever written by a student in response to one of my 280 homework questions. This particular question challenged the students to construct a dialogue in which the interlocutors got at the ideas underpinning quantifiers: what exactly is the difference between universal and existential quantifiers? The ensuing dialogue is thorough, clear, and exceedingly well-composed. It hits on just about every single point, subtle or not, you can expect a first-year prover to pick up on. Put simply, it's gorgeous.

Enjoy!

***

“I just don’t get it, any of it. It’s like, if universal means all, then shouldn’t the opposite one mean nothing? It doesn’t make sense that the opposite of ‘everything is this way’ is ‘something else exists.’ It feels like the opposite should be ‘nothing is this way.’”

“I can see where you’re coming from, and it seems like a lot of the confusion comes from thinking in terms of opposites rather than negation. In math, you don’t really need the opposite of a statement to disprove it, you just need the negation. Like, if I were to make a statement with a universal quantifier and say ‘all frat boys are jerks’—"

“That’s not true! David’s pretty nice.”

“Exactly! You just constructed a negation to my statement without even thinking about it. To prove that I was wrong in saying that all frat boys are jerks, you didn’t say ‘no frat boys are jerks,’ you just said that there exists a frat boy who is not a jerk.”

“Well, David’s not the only one. There are plenty of decent guys in fraternities. So it’s not an existential quantifier, because it’s more than one.”

“It’s still an existential quantifier if you’re talking about some instead of one. We use existential quantifiers when we want to talk about at least one, sometimes more than one, but not necessarily everything in the set we’re dealing with. I could say ‘there exists an integer that is not two’ and that would be an existential statement.”

“But it’s not true. There are a lot of integers that aren’t two, not just an integer that is not two. Actually, most integers are not two. Almost all of them are not two.”

“That’s true, but the point is that at least one integer, but not necessarily all integers, is not two. As long as the statement I’m making isn’t claiming anything about the entire set, it’s existential instead of universal.”

“So would the negation of that statement be ‘there is an integer that is equal to two’?”

“Not quite. To negate an existential quantifier, another existential quantifier won’t do. There’s plenty of room in the integers for things that are and aren’t two, so you can’t disprove my original assertion by saying that there is an integer that is two. You would need to say that in the set of all integers, there is no two.”

“That’s false. But I guess it’s supposed to be false, because what you said about an integer being not two was true, and to try to disprove it would be false.”

“Yes, the negation of a statement is always the opposite truth value of the original statement.”

“But why do I have to make a statement about everything in a set to disprove a statement about one thing in the set? Can’t I just say that the one thing doesn’t have the property that the original statement claimed it did? Before, you said I didn’t have to prove that every frat boy was a decent guy, just that there was one who was. Why is that wrong now?”

“To say something about the one thing in the set not having those properties, you kind of need to say something about everything in the set. Let’s try another non-math example. If I were to tell you that The Hop sells beef-flavored ice cream, how would you respond?”

“Eew. No it doesn’t.”

“What you said is the negation of my statement, and it uses a universal quantifier.”

“Didn’t sound like one.”

“No, it didn’t, but it will with some re-phrasing. To tell me that The Hop doesn’t sell beef-flavored ice cream, you are saying that none of the ice cream that they do sell is beef-flavored. So what you’re saying is that in the set of all the flavors sold at The Hop, every flavor shares the quality ‘not beef.’”

“Leaving no room for your false statement about beef-flavored ice cream existing!”

“That’s right! In this case, trying to use an existential quantifier would be like saying ‘they sell a lot of flavors that aren’t beef,’ and that still leaves room for the possibility that they also sell beef. But they don’t, because all of the flavors are not beef.”

“Thank God for that.”

“Amen. So does this whole quantifier thing make more sense now?”

“Yeah, some. I think I’m ready to tackle the homework now.”

Week Two is in the can

We're just about done with the second week of classes, and things are going great. All of my classes are engaged this semester (I had perfect attendance, 66 out of 66 between the two sections, for the first quiz in Calc I), and there's profound and evident eagerness to learn on the part of many, if not most, of my students.

I hope it doesn't wear off!

Further bulletins as events warrant, but here are the highlights:

• I've already had great conversations with 280 students about the effectiveness of dialogues in conveying deep mathematical concepts,
• the very first homework committee volunteers, in their presentation to class, mentioned (without prompting from me) "audience" as a major consideration in constructing a solid solution to a math problem,
• several students are already all-upons regarding Super Saturday,
• I've passed out almost a dozen of the newly-printed math major booklets,
• several of the Calc I students have already read up on the Newton v. Leibniz project on-line, even though I've not yet handed out the project description,
• dozens of the Calc I students are reporting that they're doing rough initial drafts of their homework before crafting a clean final version, and
  
• at least two 280 students have already begun TeXing the entirety of their homework.

Off to my first section of Calc I!

The Little Professor

It was still dark. Like most parents, mine were eager to spend the better part of their Saturday mornings in bed, except on the mornings when my dad had a yen to hit the trailhead early. At six years old, I was incapable of sleeping in past seven.

It was 1981. Space Ghost took turns with the standard stable of Warner Bros. characters on the giant and brilliantly-lit Curtis Mathis console television that dominated our tiny living room. The volume was turned down very, very low so that the sound wouldn't wake my mother, snoring away on the couch in the next room.

My attention wasn't on the TV screen, but rather on the toy I held clumsily in my hands, the Texas Instruments Little Professor, several ounces of hard yellow plastic skin surrounding high-tech electronic guts. Made to help children drill themselves on arithmetic, the toy spat out problems involving addition, subtraction, multiplication, and division. By then I'd learned the first three operations and begun to master them, but was still befuddled by the fourth.

On that morning I drilled myself with multiplication for a while (I'd only just learned that one), working problem after problem as the Little Professor added points and upped the difficulty of the problems it gave me. I missed a few here and there, but it wasn't long before I bored of that operation and decided to move on to something more challenging.

I'd not yet figured out division. Addition and subtraction were old hat, and multiplication I'd learned by conceiving of it as repeated addition, the only means I then had of computing products of multidigit numbers, being at that time unfamiliar with the formal method of "long" multiplication. But division? Fuhgeddaboutit.

That morning I had a hunch I wanted to follow up on: I suspected that just as subtraction served as an inverse operation to addition, "undoing" what addition did, division must serve as an inverse to multiplication. Thus, for example, if 6 x 8 were 48, a request to divide 48 by 6 would be answered by finding the number of sets of size 6 it would take to comprise all of 48 when unioned together. (I'm sure that I wasn't able to so clearly and succinctly summarize the process at the time, but my recollection of those general thoughts is quite clear, even now, nearly three decades later.)

Simple enough in principle, the details of this inverse operation were hard for me to carry out as soon as the superset had more than a few elements in it. Therefore when I switched the Little Professor over to division mode, the only questions I was able to answer unfailingly at first were those in which the divisor was 1. Nevertheless, my success with these simple problems lent credence to my theory regarding division's nature.

After a good deal of trial and error and after even more practice in the quick computation of products that it took me to "unwind" the multiplication once more to obtain the quotients I sought, I began to get better. Soon I was able to tackle the problems in which 2 appeared as the divisor, counting the number of 2s it took to make up the dividend I was given. Soon after that, 3s posed no problem, and I was on to 4s, 5s, and beyond.

My method was clumsy: it would be a long time before I learned long division, and until that time I'd have to resort to the protracted multiplication through repeated addition, done backwards, in order to solve even reasonably complicated division problems.

But I'd done it. I'd fucking done it: I'd uncovered the mystery of division.

The revelation was too exciting to keep from my mother, and I ran into the room next door to wake her and show off what I'd done. Obviously I can't recall her exact words, but I'm sure they were something along the lines of "that's nice, kid. Show me in a few hours, when normal people are up."

I'd hardly be human had I not felt a jolt of euphoria on making the discovery I'd made for myself, for I believe that much of what makes us human is our desire to seek order and understanding of the world around us. There's no high on Earth like the one that making such a discovery gives.

N'est-ce pas?

The new face of literacy

My thanks to the tireless director of the University Writing Center for calling my attention to this article on the directions in which technology is pushing literacy.

Question for discussion: what might math students stand to gain by employing Twitter-like brevity in describing mathematical phenomena?

Preparation, meet Perspicacity

Already I've noticed several signs that this semester's 280 students are a sharp lot, and with a little poking and prodding I'm sure they'll be capable of great things this semester.

For instance, they've already picked up on the subtleties of the proof and disproof of universal and existential statements, which subtleties might make fine fodder for a paragraph or two in the "textbook" chapter they'll be writing on mathematical statements. Namely, they recognized almost immediately, and on their own, that one can disprove a universal statement through counterexample, but not an existential one, paving the way to an easy understanding of negation, the topic for this coming Monday.

I like to think that their effortless mastery of these concepts comes as much from their preparation as from their perspicacity: it certainly helped to make sure all (or at least most) of them had read through the class's worksheet before class had been convened. I'm thinking it's going to pay off handsomely to designate discussion leaders as I've begun to do, also. Today the three discussion leaders provided the majority of the examples on the board, and did a fine job of explaining the choices they made.

This course's structure is in continual flux, but I think I'm more and more closely approximating an ideal design. My only worry at this point is that the "textbook" component of the course, though I'm quite sure it'll work pretty well this semester, would prove too complicated a project to put together in a larger class.

We'll see. I've admitted to the students that I'm not even sure how that aspect of the course will come together, and I'm as curious as they are (probably far more so) to see what comes of it.

Gumption

These kids rock.

It's only Day Three of Calc I and already two of the students in Section 3 have taken it upon themselves to organize and advertise (to the whole class) a study group.

Rock on.

I need to ask them to take pictures.

Day Three, going strong

Today was the second day of actual classes for me, and they went very well, substantially more smoothly than Monday's classes overall.

My 280 students did a remarkably good job in poring over the "Does good writing matter?" handout I've now used four times to encourage students to generate criteria for strong writing in nonmathematical subjects: they made a few observations no previous students have made, and they weren't the least bit shy in making them. For instance, they pointed out that the response I'd clearly intended to be the strongest one suffered from overuse of jargon and generalities, which can both be used to disguise ignorance; the "balance of power" buzzphrase was the one at which one group's ears perked. "People tend to write like that when they don't really understand what they're talking about," one student said. Nevertheless, most agreed that the second passage was still superior to the first, which despite its attention to detail frequently rambled far off topic. (Of course, the third, intentionally written to be sparse and grammatically disjointed, was everyone's pick for last place.)

Yes, the conversation regarding that handout was a lively one, and the ensuing one on the mathematical counterpart, "Does good mathematical writing matter?," was equally spirited. Already they're picking up on some important aspects of well-written proofs (complete and grammatically correct sentences, utmost clarity, etc.; as Belinda asked at one point, "so we should be writing as though we're writing to aliens who have no understanding of what we're talking about, trying to make it as clear as possible?"). When the time came for me to solicit volunteers for discussion leaders for the first "official" handout, several were more than willing.

So far the students seem sharp, outgoing, and eager to work.

Both sections of Calc I went well, too, as the students took turns presenting (in pairs) on the precalculus topics they'd been asked to review for today. Some of the presentations were understandably a little rough, but all were satisfactory, and some were positively outstanding. In the first section, those presentations dealing with the Vertical Line Test, asymptotes, and rational functions were particularly strong, involving appropriate and explicit examples. Ino and Nadia, for instance, were spot on in their discussion of asymptotes in the first section.

Today also brought me the chance to meet several more students during their "meet 'n' greet" interviews. A few were shy, as many were eager to set on the semester. It turns out that several (5 out of 15) of my 280 students are math licensure students, a disproportionately large number, as I mentioned a few posts back.

Tomorrow will be the third day of classes for Calc I, and a break from 280. Further bulletins as events warrant.

Oh, and: I've just put together a rough draft of my proposal for the 10th International Writing Across the Curriculum Conference in Bloomington, Indiana in May 2010. I hope to speak on the intentional disciplinary writing instruction I've been doing for the past two summers during the REU.

Day Two

What a difference a day makes! Today's languor lies in stark contrast to the tumult that was yesterday. I've had a chance to get caught up on several little nagging tasks in during the free hours in between my "get to know ya" meetings with MATH 280 students. (I've got another meeting in just a few minutes.)

So far they're all professing eager anticipation for what's to come in the course. There's a little trepidation about LaTeX, but that's only natural. One student expressed slight annoyance at having to do a good deal of work in groups, indicating that he's the sort of person who prefers to do everything himself if he knows he's the one who's apt to do it best. (I understand the feeling: I was like that in college, and every now and then I still slip back into that mode.) Thankfully, though, he recognized the importance of collaborative work and seems completely willing to give it a go in our class.

Another benefit of having such a small class is the brevity of the course's "mixing time": suppose, for simplicity, that every group project will involve 3 students. If groups are assigned completely at random each time a group is convened for a group project, in a class of 15 students after a person has served on 5 groups, she or he can expect to have worked with roughly 51% of the people in the class. In a class of 30 students, 5 rounds of service only puts the one student in touch with roughly 29% of her or his classmates, on average. (Yes, I'm enough of a nerd that I just wrote some Mathematica code to generate the desired expected values for arbitrary class size and group size.) Put simply, a student more quickly comes into contact with a greater proportion of her or his class in a class of 15 people.

Not shocking, but it's nice to know that the numbers back it up!

Okay, that's all for today. I'm homeward bound quite soon, but I'm excited to find out what tomorrow holds for me, class-wise.

Oh, and: this is my 300th post!

Does not compute

No time like the first day of class for the campus internet system to go farfufket. Connectivity to off-campus sites has been spotty since noon, and it's wreaked havoc with class activities, e-mail, and class prep as I'm trying to put out the fires set during the first section of Calc I this morning.

So what happened in that today that got my ever-lovin' panties in a big ol' knot?

My colleague who directs our Math Lab came in to give his Math Lab spiel, and to give the students pretty explicit instructions regarding the precalculus review "pretests" we've asked them to complete for placement purposes. These instructions, detailed and demonstrative as they were, took a good deal of time, and by the time I finished going over the syllabus (in what I felt were fairly broad strokes), there were precisely three minutes of class time left.

I didn't get a chance to solicit contact/background info from the students, and I didn't get a chance to ask them to sign up for a "meet 'n' greet" interview with me sometime in the first couple of weeks. This may not seem like that much of a big deal, but those of you who know my teaching style can vouch for me when I say that it's crucial to me to establish a good rapport and simpatico (for lack of a better term) with my students, right out of the gate. Both teaching and learning are optimally done in an environment of familiarity, support, comfort, and mutual respect, and such an environment is built by early and frequent efforts on the part of both instructor and students to meet one another on common ground.

Like I said, farfufket.

Moreover, we had no time for review, as it took all of the measly amount of time we had left simply to explain the homework for Wednesday. (Doing this was of paramount importance, as they simply have to be prepared for Wednesday's class if we're going to get anything out of it.)

The second section of Calc I ran much more smoothly: the Math Lab/pretest spiel was dramatically abridged due to the aforementioned network outage (silver lining, anyone?), leaving more time to answer student questions and then proceed more leisurely to the homework for Wednesday.

Nevertheless, though I'm much more satisfied with my experience in the second section today, it was still far from ideal.

If I had to rate the first day, I'd give myself a B for 280, a C for Calc I, Section 1, and an A- for Calc I, Section 3. Combine that with an A for damage control and coping with chaos, and I might just make a B so far.

It'll all be calmer on Wednesday, folks, believe you me!

P.S.: I've already had a few very pleasant out-of-class exchanges with new students, and I've no doubt they'll be joined by myriad more as the semester goes on. Keep at it, my friends, keep at it!

Night and day

My second section of Calc I went faaaaaaaaaaaar more smoothly than my first section. (Hooray for not having to deal with Educo at all!) Although it would have been nice to have had a bit more time (five minutes would have been really nice), everything went more or less according to plan, and already I feel much more comfortable with that section than I do with the first.

If any of my new students have followed up on my hints and have come to this blog for the first time, welcome! I hope you'll check in again every now and then to see what I'm thinking as we go forth into this new semester, and I hope you'll feel free to offer me feedback on the activities we design together in the classroom.

For now, get ready for Wednesday, try to get into some sort of groove, and have fun!

Two down...

...Well, that went...meh.

Not enough time!

Too much to say!

Is it just me, or does 50 minutes not go as far as it used to?

Argh.

Okay, one more to go. Let's learn from our mistakes and press onward.

One down, two to go...

MATH 280's in the can.

I think it went all right. The one drawback to the "complexity" of the course (HW committees, discussion leaders, now a student-authored "textbook") is that it now takes about 25 minutes to give a reasonable explanation of the course's components, and that's without going in to much detail.

14 people attended today, including two who aren't yet registered, so signs still point to a small class, roughly half the size of last semester's 280. I'm glad for that: it's conducive to a more intimate and supportive classroom dynamic. So far there are a couple of people who seem to be a bit more outgoing, inquisitive, and bold. If I can get them all to rally around a few natural leaders, it'll be a great team.

One point of note: most of the math majors in the class are in the Licensure concentration. Interesting.

Now I'm off to get ready for my first section of Calc I, and so must for the moment say adieu...

Day One, encore

Though I've been eagerly anticipating today's arrival for the past few weeks, the first wave of true excitement didn't hit the shore until I crested the hill on the way into campus this morning. The sight of students and colleagues alike converging on the campus from all directions reawakened my enthusiasm.

280's up first, at 9:00, and then Calc I sections at 11:25 and 1:45.

Let's do this thing, shall we?

Ping

As the first day of class in the Fall 2009 semester (too quickly) approaches, and as this blog passes its third birthday and nears its 300th post, I wanted to take a moment to find out who's out there.

I began writing this blog a little over three years ago more for myself than for anyone else. I intended it to serve as a means of organizing my thoughts on my teaching, and as a way of playing out various pedagogical scenarios as I experimented with student-centered teaching methods that were new to me at the time.

Quite early on many of my students began reading the blog and using it to learn what it is that goes through my mind as I design my courses. I realized then that the blog could serve as a new means of communicating with my students, as a way of making clear my intent and of making my techniques transparent. It could also serve as a means of disseminating my students' work, and of discovering their thoughts on the issues we faced together in the classroom.

But the blog's been still more than that to me. It's been a vent for my frustrations (when students cheat, or when classroom technology goes awry). It's been a forum for my own and my students' creativity (as it was here, here, and here). And it's been a place where I can simply speculate on random math-related (or even non-math related) thoughts.

For all it's done, writing this blog has been very relaxing and relieving exercise for me, and even if not another soul were to read it, I'd still feel it's worth the time it takes to write it.

But I know I've got some readers out there; I'd just like to know who you are.

So I'd like to ask my regular (or even non-regular) readers to ping me back in the comments section to this post, anonymously if you'd like. Just write me a few words to let me know that you're out there. Feel free to share a bit about yourself, if you'd like to, and tell me what road it is that's brought you here, or what it is that brings you here again. I'm very curious: knowing full well that there are hundreds of far funnier and far more entertaining blogs on the interwebs, I'd like to know what it is that holds your interest enough to make you keep coming back for more.

Where will I go from here? We'll see. I don't intend to change much about the way I put the blog together in the coming months, although I'm toying with the idea of beginning a new blog devoted to poetry, so here you may see a little less of that in the future (unless there's tremendous call to keep it here). I wouldn't have dreamed three years ago that I'd be where I am today in my teaching, so I hesitate to speculate further about what Change of Basis might look like in a year or two. Time will tell.

With that, fair reader, let me end this post. I hope to hear from you, if only in a few words, in the comments section.

Let the jitters commence!

Technically, I'm ready for Day One...and Day Two, for that matter...in both of my classes, aside from actually printing out the syllabi for Calc I. The first-day worksheets are ready for Foundations, the first-day activities are mapped out for Calc, and both websites are up and running and fully functional.

Nothing to do now but try to get a little bit further ahead in class prep, and maybe fret a bit for the next week and a half.

I suppose there'd be something wrong with me if I didn't worry a little bit.

Syllabus

This afternoon I put an hour or so into revamping my 280 syllabus for this oh-so-imminent semester.

As I hinted in my last post, I'm going to be asking the students to write their own textbook, in a sense: at the end of every "section" of the class (of which there are seven, listed in the syllabus excerpt below) I'll ask for several students to serve as the primary authors of a "chapter" of the "textbook" that will deal with the ideas we've just finished discussing as a class. That chapter will include examples, explanations, exercises, and proofs.

Here's the relevant snippet from the syllabus:

***

I feel that it will be a far more meaningful exercise for you to construct your own textbook for the course, instead of following along in a textbook that's already been written. Therefore, at the end of each "section" (of which there are 7 we'll discuss throughout the course, and they're listed below) I will ask a group of three to five of you to spend roughly a week composing your own definitions, explanatory text, examples, computations, and proofs to create a "chapter" of text. Once you've typeset your chapter using the LaTeX software, you will post it to the course's Moodle [the University's online course management and social networking platform] site, where others will have a chance to offer feedback: do the definitions seem right? Do the examples make sense? Are the proofs correct? Others may ask to permission to edit your chapter (or at least ask you to perform edits). In this sense the textbook you create is a wiki, communally created by you as a class.

Incidentally, although there will be a due date for each initial chapter posting, there will be no due date on revisions once a chapter is posted; therefore once a chapter is up, it's fair game for modification for the rest of the semester.

Here is a list of the "sections" we'll discuss during the semester; you will each be expected to work on two chapters:

1. Introduction to mathematical statements, quantifiers, and basic logic

2. Basic proof techniques (direct, contradiction, contraposition, induction)

3. Set theory (including set operations and cardinality)

4. Combinatorics (basic counting principles, permutations and combinations)

5. Relations (including equivalence relations and order relations)

6. Functions

7. Combinatorics redux

Some of these sections are more involved than others; I will assign more of you to collaborate on the more complicated ones, allowing you to more effectively divide the work. However, the writing of each chapter should be a well-coordinated task yielding a well-composed whole; therefore I strongly advise you to select a "chief editor" for each chapter once you're placed on a given team of authors. I'll give you some more tips as the time nears to write the first chapter.

***

I recognize that this addition to the class is going to impose on the students a substantial amount of work, and therefore I'll be looking for ways to streamline the ordinary homework assignments and create meaningful connections between those homeworks and the textbook chapters the students will be asked to write. (For instance, what can I do in writing my own homework problems that will help the students to write their own homework problems for the chapters?) Moreover, I'm thinking of scrapping the end-of-semester presentations in order to help students free up time for "textbook" revisions at the semester's end.

As another time-saver (and means of encouraging student self-authorship), for each worksheet I'll be looking for volunteers to serve as "discussion leaders"; these people will be expected to digest each worksheet as it's handed out and before the class addresses it, and they may be called upon to lead the class in completion of the worksheet's activities in class. This slight change in classroom operations should help in-class activities to run more smoothly.

We'll see how it goes. It should help a lot that this class is by far the smallest of the 280 sections I've taught at UNCA so far: only 14 people are registered right now, and I don't think that's likely to go up.

For the MATH 280 alumni among my readers, what would you think about the above changes? I'd really like to know your thoughts.

They could write the book on this stuff

Rising from the depths of my summertime silence, I surface only to say this: this coming semester I'm going to try to get my 280 students to write their own common set of course notes, suitable for use in a future version of the course.

I've yet to work out the details, but I've realized that it's only a short step or two from the writing and peer-review activities they've been doing for the past few iterations to this more ambitious project.

Than this one no task could better challenge them to understand key concepts, construct their own apt examples, and clearly communicate their discoveries.