It's been a busy break.

Winter Break started, effectively, just under three weeks ago. In that time I've been almost completely submerged in writing theory and pedagogy, hard at work on my book. I've written about 17,000 words in the past two and a half weeks, so that I'm now roughly 60% done with a first draft of the damned thing.

I say "damned thing," but of course you know I'm loving every minute of it. Some paragraphs are slow going and take me an hour or more to kill, and others just fly from my fingers. It's all fun, though, and I'm excited about how the book's developing. I'm working away at several chapters simultaneously (I've started every chapter but the second and the seventh), and they're starting to grow and to grow together. By the end of next week I hope to have completed first drafts of Chapters 3, 4, 5, and 6. (That'll leave 1, 2, and 7 to go.)

While I'm at it, I'd like to give a massive shout-out to my writing colleague Libby of East Carolina University, who gave me fantastic feedback on my introduction, and excellent ideas for later chapters. She's the first of my colleagues in composition and rhetoric to read any substantive portion of the book, and her reading, and her response to it, has encouraged me greatly. Peace!

Unsurprisingly, in working so closely on the book I've begun to think about ways I can model to my students good writing and good writing process to an even greater extent than I already do. I'd like to offer them a window on my own writing process in whatever way I can. I'd like them to see rough drafts, wrong turns, dead-end ideas, reviewers' feedback, and revision, revision, revision.

Obviously they're not going to give a day-old donut about every single version I write of one or another dry math paper. But it can't hurt to show them some of my first-draft research notes that are little more than mathematical freewriting (see below), or the copious comments and suggested emendations my editor offers me on draft chapters of my textbook. If it accomplishes nothing else, at least showing these things to my students will show them that I too am human, that I too make (often very stupid) mistakes, and that I too am continually growing as a writer and a thinker.

Without further ado...here's a sample. Below are the first five out of nine pages of notes I scribbled out the other day as I was working on a paper dealing with the combinatorics of complex polynomials. In these notes, if you look closely (I don't recommend it) you'll see everything I indicated above: wrong turns, dead ends, and idiotic mistakes. (My Facebook friends may even be able to notice on one of these pages the (-1)(-1) = -1 mistake that was the focus of my Facebook status for several hours earlier this week.)

After all, I'm only human.

Coming soon: my thoughts on putting together my first-ever first-year seminar, on Ethnomathematics! This course begins in a little over a week, and I plan on writing my syllabus this weekend. Stay tuned...

## Thursday, December 30, 2010

### In the interest of full disclosure

Posted by DocTurtle at 10:26 AM 3 reflections

Labels: ethnomathematics, MATH 179, More Than Numbers, writing

## Wednesday, December 08, 2010

### On behalf of our esteemed colleague...

Now, I've had students get into the Newton v. Leibniz project in Calc I, but a trio of this term's participants took it to the next level.

Hermione, Nina, and Quinn, played the parts of Newton's colleagues John Collins, Isaac Barrow, and Henry Oldenburg, respectively, in my morning section. During the trial they filled the bill impeccably, offering accurate and convincing portrayals as they went, one by one, to the stand in defense of their colleague. Outside of class, too, they stayed in character as they wrote first-person apologia on Newton's behalf. Not only were these letters among the most creative and clearly-written responses to this exercise I've ever received from students...they were also rendered in startling form.

Executed in three distinct antique hand-like fonts (including "Emily Dickinson," modeled after the poet's actual handwriting) upon nine sheets of artificially-aged paper, the letters come to life in striking fashion.

Gimmicky? Perhaps, but cleverly so! The air of authenticity this medium lends is simply marvelous! See for yourself; here's Collins's letter:

The letters themselves delighted me (especially as I found them waiting for me in my office at the end of a particularly long workday), but more delightful still is knowing that the students had gotten this into the project.

Here's Oldenburg's offering, in bold diagonal strokes and terminating in a huge Hancockian signature:

Barrow's script is smaller and more timid, becoming the sort of man who would cede his chair as Lucasian Professor at Cambridge to his promising young pupil:

Who said there's no room for artistic creativity in the calculus classroom? Well done! (Incidentally, I did indeed get permission from my students to post their work.)

Posted by DocTurtle at 8:57 PM 0 reflections

Labels: Calculus I, MATH 191

## Tuesday, December 07, 2010

### Making waves

I'm still unclear as to how it happened (something having to do with Reddit), but I'll soon be appearing in an article in The Pipedream, the student-run school newspaper at SUNY-Binghamton. One of their reporters, Zelda, got a hold of my recent post on in-class exams and found it interesting enough to start a conversation around it. She and I had a very pleasant discussion this morning about the pros and cons of various assessment techniques, including in-class and take-home exams.

She raised good questions, and I hope I gave good answers. Many of my responses were much in line with the ideas I've put forth in the above-linked-to post and in the comments to that post, but there was one point I'd like to elaborate a little more thoroughly.

At one time Zelda raised a concern an unnamed faculty member at her own university had discussed with her when asked about elimination of take-home exams. In essence, the concern is as follows: "getting rid of in-class exams might work well in upper-level courses where students are familiar with the great responsibility placed on them by being granted a take-home exam, but first-year students simply don't have the maturity or the sense of responsibility to realize the gravity of an act of academic dishonesty. How can these students be trusted to take on this responsibility?"

My response was largely in line with elements of the above post and of my teaching philosophy: if we show our students that we trust them and respect them as adults and as co-learners, they will have a harder time betraying that trust. "Won't there always be temptation to cheat, on the part of the students given a take-home exam?" Zelda asked in follow-up. "Indeed," I agreed, "but if they've been made to feel respected, understood, and appreciated, they'll be less likely to follow through on temptation."

To be sure, this establishment of mutual trust, respect, and understanding is not foolproof: there will always be those who cannot resist the temptation to cheat and who therefore abuse the trust being placed in them, but I'm not convinced that the relatively few cases of cheating one might encounter make it worth abandoning take-home exams.

Moreover (and here's the point I didn't make clear in my interview), how are students ever to learn about mutual trust, respect, and understanding if not given an opportunity to demonstrate that trust, respect, and understanding? Put another way, how else will students develop the maturity we ascribe to upper-level students but by demonstrating that they can acquit themselves in a mature fashion when given tasks like take-home exams in courses like Calc I? It seems to me something of a Catch-22 to say that students aren't mature enough to handle take-home exams, while at the same time they can't develop the sort of maturity we're looking for without being given such opportunities.

That said, and as I told Zelda this morning, I don't believe that take-home exams are appropriate in just any class: it takes pedagogical skill, time, and a considerable amount of practice and patience to design and develop the sorts of learning environments in which young students feel trusted, respected, and understood. It takes just as much skill, time, practice, and patience to develop the robust assignments and assessments that abrogate the need for in-class exams. The course I'm plotting is not for everyone, as I indicated to Zelda when she asked if I'd support a campus-wide "no in-class exams" policy. "No," I said. "Not only do faculty as a rule not like to be told what to do (a fact which would make implementation of the policy horrifically difficult), but the policy isn't appropriate for every instructor."

Anyhow, that's the scoop. I'll post a link to the article in The Pipedream when it appears. I'll also soon be posting some delightful documents three of my students produced as part of their Newton v. Leibniz project (I've just gotten permission from all three to post). Stay tuned!

Posted by DocTurtle at 5:43 PM 2 reflections

## Saturday, December 04, 2010

### Observations: a follow-up

[Note: I first wrote much of this post as a comment in response to a comment one of my colleagues from grad school left on my last post; I thought it was strong enough to stand as a follow-up post in its own right. Meredith there left a link to an NPR story on teen neurochemistry some might find interesting.]

My colleague Meredith left the following comment on my last post:

A bit more about those non-traditional students. I would love to see a push on research on the neuroscience of the teenage brain and the "wisdom" of pushing ever-more high school students to take Calculus, at the expense of time spent on functions, trig, etc.

As I hinted in my previous post, it's really no surprise that the nontraditional students are outperforming the younger ones: it happens every semester. The relatively high percentage of such older students at UNCA is one of the things that makes our school such a fun place to teach. Because of the relative affordability of UNCA we get far more than our share of returning students (something one couldn't count on at schools like Davidson, Drake, or Bucknell), and I can't count the number of more mature students I've had whose presence has measurably improved the dynamic of whatever class they're in.

"Measurable"? This semester offers an extreme case: my morning section of Calc I has about three times as many "returning" students as does my afternoon section. Yesterday I noticed that every single one of the fifteen or so questions (every one of them reasonable and deep) raised by students in my morning section during our discussion of integrals was asked by a nontraditional student. Every one. They're simply less fearful of appearing "stupid."

It's this fearlessness that's helped deepen the level of the discussion that goes on in that course and that has ultimately, I really believe, helped lead to the startling difference in grades between the two sections: my morning section's overall course average is almost a full letter grade higher than my afternoon section's. There are other factors which contribute to this differential, to be sure, but it's difficult to discount the involvement of so many older, more mature students.

I absolutely agree with Meredith's placement of priority, and I'll bet she'd agree that we need to stop pushing kids straight into college after high school unless they're truly intrinsically motivated to pursue college coursework on their own. Some kids (even very, very smart ones) simply are not ready for college, and go there only because they've been told by their teachers, their parents, and their guidance counselors that it's the "next step."

Often, though, the best next step is to burn off some energy taking a full-time job for a few years, traveling (perhaps in service of a humanitarian organization like AmeriCorps or Peace Corps), or even joining the military for a tour or two. Though I'm not a fan of many of the things the military is called on to do for our country, I believe one of its most beneficial functions is providing order and structure to kids who sorely need order and structure in their lives.

Posted by DocTurtle at 8:03 AM 0 reflections

Labels: Calculus I, MATH 191, theory

## Friday, December 03, 2010

### Observations

A few random observations about my courses this semester:

1. Ever since our department implemented a writing component in the Senior Seminar, the overall quality of the students' oral presentations has gone up as well. Of course, this makes perfect sense for anyone who understands the concept of writing-to-learn: the students are using writing as a means of engaging meaningfully the topics on which they'll later present...moreover, they're making that engagement earlier in the semester than they normally would, asked as they are to complete a rough draft of their written report by the halfway point of the term, at least a week or two before they must present.

2. As in every semester that's ever been and every semester that ever will be, the "nontraditional" students in my Calc I class are outperforming the young 'uns. The more mature folks (juniors, seniors, and returning students) are the first and most frequent askers of questions and the strongest showers on exams, and they make up the majority of those still completing the homework regularly, despite the fact that it's been optional for a week or more by now.

Though there are a couple dozen Cs and Ds scattered throughout my two sections of Calc I, none of the 15 (out of 65) older students has a grade below a B going into the final exam. While they might not have quite as fresh of math skills (or even intrinsic mathematical aptitude) as do the first-year students, the older folks have vastly superior time management techniques, study habits, self-direction, commitment, and sense of purpose. Those skills and attributes (among the most important ones learned or acquired in college) help them to be, by and large, far more successful than their younger peers in most of their courses.

3. This semester's end-of-term presentations in Linear (the first three of nine given today) are far and away better than those the students put together for the course the last time I taught it...and they students have had less time to prepare them than they'd had the last time, too. Though this is a particularly strong class, I humbly give myself some credit for doing a better job overall in leading this course than I did last time. The exercises and activities I've put together are more authentic and robust, and the ways in which the various components of the course were fitted together simply made more logical sense.

Incidentally, one of the presentations will very likely lead to a rich research project. Ino and I have already discussed (at great length) our plans to parlay her team's presentation on linear analysis of nutritional data into a full-scale publishable project. The sky's the limit, and I'm really looking forward to directing what will likely be my first-ever truly applied math research project.

4. As early as I can next term I need to make a point of helping students get past their own pride when it comes to asking questions in class. Students (especially younger ones) often have a morbid fear of "looking dumb" in front of their peers, and of course asking questions makes one appear ignorant. (As opposed to remaining silent, which leads not to appearing ignorant but simply to being ignorant.) I've got to more actively help students overcome that fear.

To that end, I made some remarks in my second (the quiet) section of Calc I today that seemed to have a positive effect on students' querulousness: before proceeding from a specific example of a definite integral computation to a general one, I said something along the lines of "any questions before we move on? This idea is a crucial one, and it's very important that you have a good grasp on it before we proceed. [Silence. Pause.] How many people are there in here? [Count out loud.] Thirty or so? In a class this size, I fully expect more than half of you, probably 15 to 20 of you, don't understand something about what we just did. I expect that. Typically at this point half of the class doesn't fully know what's going on. I expect that. I'm sure you've got questions. I'm just not sure why you're not asking them. But I really can't do anything about it if you don't ask, so we'll just move on."

Move on I did, and within a minute or so two or three people who almost never ask questions posed a few. I was ecstatic! It's the first time this semester that some of these people have come out of their shells.

I've got to remember that trick.

5. I will never ceased to be amazed by how wonderful are the students I work with on a daily basis. Our students are incredible. They're intelligent, devoted, hard-working, honest, and down-to-earth. They're smart, sassy, funny, and fun. They're some of my favorite people on the planet, and I cherish every moment in working with them. I am the luckiest man on Earth for getting to do what I do, and getting paid for it.

Posted by DocTurtle at 11:35 PM 3 reflections

Labels: Calculus I, Linear Algebra I, MATH 191, MATH 365, MATH 480, Senior Seminar, undergraduate research, writing

## Tuesday, November 30, 2010

### Shift, paradigm, shift!

This morning was painful.

I had the uncomfortable task of watching 30 of the students in my morning Calc I section puzzle their way through a difficult (but fair, I feel) exam, their last "mid-term" exam of the semester and a traditionally hard one (on applications of the derivative). This awful duty was the last straw.

The penultimate one came yesterday: I was called upon to help adjudicate a case of "academic dishonesty" (to use the lovely euphemism) in a colleague's class. This colleague wanted to know if the students in question had indeed "cheated" (to stop beating around the bush). After cursory (and then more in-depth) inspection, I agreed that yes, indeed they had.

However, to me the incident said more about the culture of the academy than it did about the "dishonesty" in which the students were involved. More specifically, the students were clearly guilty of "cheating," but to me the more crucial issue concerned why it is they felt the need to "cheat" in the first place.

To "cheat" requires that there be a "game," and that it matter that people follow the rules of that game, and further that it matter that in order to succeed at the game one must "do better" than anyone else playing the game. This was definitely the case in this particular course: the students had been given a (very) high-stakes exam, which to them was more than an assessment instrument; it was moreover one of a very small few means of receiving feedback on the degree to which they were mastering the concepts of the course the exam was given in. To them, "cheating" on the exam was a natural response, given the way in which they've been acculturated to consider the exam a must-win game in which success is measured by high marks.

The point I'm getting at is the following: the more we as educators eliminate, or to the greatest extent possible downplay, the competitive aspect of education (high-stakes testing, rigid and number-driven grading schemata, individualistic learning paradigms, etc.), the less likely we are to find our students "gaming" the system by engaging in "cheating." In some regards, "cheating" will cease to exist, as it simply will have been defined away.

As I hinted above, most "cheating" (I truly believe) is undertaken as an act of desperation, a means of coping with failure as measured by receipt of lower-than-average academic grades. "Cheating" is a means of striving to succeed within a system which provides extrinsic rewards for optimal performance rather than intrinsic rewards for authentic mastery and authorship. I cannot but believe that the vast majority of students would welcome an academic system in which the goal is not to earn high marks but rather to learn, and that students accustomed to this system would see no need to game the system by "cheating." I'm not so naive as to suppose that every student will respond well: there will always be those so acculturated by thirteen-plus years of a largely competitive educational system that it's in their blood to fight tooth and nail for every last percentage point that might tip them from a B+ to an A-...but I believe that even those who are very comfortable with this traditional system will abandon it if given the chance to do so.

All of this gets me back to this morning, and the thoughts I had as I watched my Calc I students (even the strongest of them) wiggle and squirm in completing their in-class exam, stressing out not over whether or not they'd really learned the concepts we've been working on together for the past few weeks, but rather over the grades that will unmercifully adorn their papers when they get them back tomorrow morning.

My thoughts can be boiled down into two short words: no more.

No more in-class exams. Ever. I'm through with them. The one I gave this morning (and will give again in a couple of hours) will be my last.

For quite some time now I've not given in-class exams in my upper-level courses, feeling that little meaningful could be asked on such exams, aside from requiring students to parrot already-proven theorems or give short answers to requests for definitions. In these courses, in-class exams offer none of the opportunity offered by take-home exams to ask authentically engaging and probative questions, and therefore I've found them pointless time-sinks.

For quite some time now I've resisted the banishment of in-class exams from my first-year courses, thinking, I suppose, as I've heard some of my colleagues to think: "there are certain computational techniques the students will have to learn to perform, and to perform quickly." True, but even the most straightforward computational skills will be as well developed (and much more readily understood) if performed in the service of completing the more meaningful, authentically engaging exercises included on take-home exams. This is as true in Precalc or Calc I as it is in Abstract Algebra I or Topology. Even in lower-level courses take-home exams offer a much more meaningful sort of assessment, and even if those exams are still meant as individual exercises and not collaborative ones (of the sort with which I've been experimenting in Linear Algebra this semester) they go much further than do in-class exams in encouraging a culture of collaborative engagement, simply by downplaying high-stakes individualistic assessment.

Keep in mind that I'm not boo-hissing exams entirely: exams offer students a means of reflecting on the ideas they've learned for the past ___ weeks, and if properly responded to they're fantastic tools for giving students feedback. Well-designed and well-delivered exams give students a healthy way of furthering their learning and assimilating their knowledge as they think critically about it. Exams are here to stay.

In-class exams, however, for me will soon be a thing of the past.

I've already responded to a couple of concerns I anticipate colleagues (and even some students...very bright ones, in fact) might have about this decision of mine, but let me respond to a couple more hypotheticals.

Colleague/student: "If you de-emphasize high-stakes individualistic exercises like in-class exams, some students will game the system and prop themselves up on others' work without really learning anything themselves."

Me: "No matter how you set the system up, students are always going to find a way to game it. Gaming the system I propose means cheating themselves out of learning. Gaming the system as it currently stands involves engaging in a behavior that's viewed as sociopathic but is really little more than a symptom of deeper systemic problems. I find the former course far less pernicious. Sure, there'll always be students who frankly don't give a shit, but we're not going to serve them well in any system, and with a system more conducive to authentic learning, they might just pick up a thing or two along the way."

Colleague/student: "You can tilt at as many windmills as you'd like to in your own little class, but you've got to assign grades at the semester's end, anyway. Won't the system you propose result in massive grade inflation?"

Me: "My response to this concern is twofold. First, in courses in which I've begun doing away almost entirely with individualistic activities (begun in Foundations, Topology, and Abstract Algebra I and II in previous semesters, and taken to the extreme in my Linear course this semester), I've seen little noticeable change in the final grades for the courses. This was true even in Topology, in which students had unlimited opportunity to revise and resubmit all work, with no constraint on collaboration. I simply don't see evidence for grade inflation. Second, even if there were grade inflation, so what? It would be incredibly difficult to determine whether students' grades were made higher because they were bracing themselves on each other ("gaming" the system in the sense expected by my first colleague above), or simply because they'd actually managed to more richly and more fully understand the ideas addressed by the course. That is, maybe the grades are higher for a reason: the students are actually, for the first time in their lives, getting it."

I truly feel this way.

Moreover, I truly feel that I've become proficient enough as an educator that my courses offer the sort of rich collaborative learning environment wherein in-class exams no longer serve a meaningful purpose. They're simply anathema to my teaching philosophy, and they're counterproductive to my goal of establishing a safe, stressless, and supportive setting in which we can all learn from one another in robust and authentic ways. From here on out, they're gone.

Before I leave, let me pass my apologies on to the students currently enrolled in my Calc I course: I'm sorry that you had to suffer through the last iteration of this practice. I know how hard the topics you're being tested on are, and I'll keep that in mind as I respond to the exams tonight. I'll respond to them not with an eye toward giving you a grade, but with an eye toward letting you know how well you're learning. I hope you'll receive them back from me with the same thoughts in mind.

One last note: I should point the interested reader in the direction of Alfie Kohn's No contest: the case against competition, about which I've blogged before, and which over the years has probably proven to be the single book which has exerted the greatest influence on me as a teacher. It should be required reading for all educators, at every level.

Posted by DocTurtle at 9:32 AM 15 reflections

Labels: anxiety, Calculus I, cheating, Kohn, Linear Algebra I, MATH 191, MATH 365

## Sunday, November 21, 2010

### Sometimes dreams do come true

I'm on cloud nine plus seven.

I think it's fair to say that putting on the conference that concluded yesterday evening around 5:00 p.m. was one of my greatest dreams since coming to UNC Asheville over five years ago. Not only did we put the conference on, but almost everyone involved seems to have had a fantastic time and gotten a lot out of it. Moreover, I've already gotten some good feedback on how to make the event a better one, and a bigger one, next year.

I typically use pseudonyms on this site, but as published poets (for the most part) the principals involved in this conference are used to seeing their names in print, and I'm sure they won't mind me thanking them by name.

My heartiest thanks, then, go to Kristen Prevallet, for her marvelous midday performance and morning workshop, and to Michael Leong, for an eye-opening presentation and a delightful chapbook of occasional poems.

My thanks go to my colleagues Curt Cloninger, Sloan Despeaux, and Merritt Moseley, for offering their own idiosyncratic takes on the intersections between mathematics and art: Merritt ably introduced attendees to Oulipo, Sloan showed them how poetry can be used to communicate mathematical ideas, and Curt demonstrated a dizzying palette of generative visual art.

My thanks go to the quirky, catchy, and wonderful poet Lee Ann Brown, whose work was the centerpiece of the conference. Her presence brought interested parties from several states, and her presentation brought Oulipo and its works to a broad audience on Friday night.

My thanks go to my partner-in-crime in organizing all of this, my colleague Richard Chess, whose indefatigable spirit and helpful organizational skills helped make the event possible in the first place. Without his connections to the literary world, a dream come true would still be a dream.

At last, my thanks go to, collectively, the remarkable students at UNC Asheville, for taking the time to contribute their own work, to help with arrangements in advance, to spread the word, to generate interest, and simply to show up and participate on the day of the event itself. I am well aware that not every school has such dedicated students (here they made up roughly half of those attending), and this is a treasure I would miss were I elsewhere.

You're all wonderful people! I'm lucky to know and get to work with you all.

Posted by DocTurtle at 8:21 AM 0 reflections

Labels: conferences, poetry, writing

## Friday, November 19, 2010

### Great expectations

It's been a bit since I wrote anything truly original on this here site; I fear things have been rather hectic in Patrickland.

I've had a few straight weeks of mind-boggling busyness, but every one of them has been good, almost to a day. Seriously: things have been chugging along furiously but swimmingly, and I've been enjoying all of my classes immensely.

As you can tell from the two most recent posts (this one and this one), Newton v. Leibniz came off without a hitch. Both sections of my Calc I class prepared well and performed well. Though the debate wasn't as heated as I've seen it in the past, both sides were ready and the back-and-forth was steady and confident. I was particularly impressed with the woman playing John Collins in the second section, and the woman playing Henry Oldenburg in the first. The lead attorneys all did splendidly, with well-prepared arguments and clear questions. Objections were at a minimum, and everything was civil (for the most part...there was one somewhat heated exchange in the second section). I enjoyed it a lot. Judging from a preliminary reading ("skimming" would be more apt) of the students' reflection papers dealing with the project, they got a lot out of it, too.

Since then (that was a week ago this past Tuesday), my primary occupation has been getting ready for the Conference on Constrained Poetry. Everything's in order now, with all of our speakers in town (aside from my colleague who's driving over from the Sylva-Cullowhee area tomorrow morning), almost 80 conference-goers pre-registered, recently-run articles on the event in the campus literary mag and the local free weekly, and a well-known journal (Critiphoria) ready to print attendees' creations. Not bad for a first time event. I'm quite sanguine about our chances of repeating next year.

Mimicking our guest Michael Leong's construction of occasional constrained poems in honor of our conference (check out his chapbook, The Hoax of Contagion...also, my thanks to Michael for linking to a handy on-line n + 7 generator), I applied my finite-state automatatistic (what a wonderful word!) constraint to the list of last names of students in my Linear class, obtaining the following automaton:

The word list I obtained from this constraint was short enough to offer a challenge and long enough to give me a rich source of words to describe the joy I've had in working with the students in this course, one of my favorites since coming to this school five and a half years ago.

I'll leave you with the resulting poem, but I'll be sure to check in again soon to let you know how things go tomorrow (because I know you give a damn).

We are one

Here we were married.

We were wedded,

welded

one to one

on a vessel on a charted curve

etched in every memory

in marked meter.

We were battered hard

at intervals,

in imagined rain,

in grey tones.

We are free, we veterans,

sores patched, keels buried,

meet, reserved, nerves set.

We are free, we veterans.

We are one.

Posted by DocTurtle at 5:38 PM 1 reflections

Labels: Calculus I, conferences, Linear Algebra I, low-stakes writing, MATH 191, MATH 365, poetry

## Tuesday, November 09, 2010

###
*Newton* v. *Leibniz*, Section 2

Aaaaaaand...Section 2!

Judge 1. We will begin with opening arguments.

Newton's lead. Thank you for being here today. I'm here today to ask you to give credit where credit is due. Newton was first by more than a decade, and he deserves this credit. We have proof that our opponent, though a great man, did in fact see some of Newton's work and did not give credit. We are here to get to the bottom of this case. Please recall that my client spent many years working on this material, and he deserves credit for it.

Leibniz's lead. We believe that our client is not guilty. He came to his theory of calculus based upon his own philosophy. Some describe him as a "universal genius." He is skilled in law, philosophy, and mathematics. We will here testimony to this end. Newton has a history of responding unhealthily to criticism, and that's why we're here today: Mr. Newton has it out for our client.

Judge 1. Newton: witnesses?

Newton's second. We'd like to call Isaac Newton to the stand.

Newton's lead. When did you begin working on calculus?

Newton. About 1665, right before the university closed.

Newton's lead. How much work did you put into it?

Newton. I spent two years getting the basics down. I worked hard.

Newton's lead. Did you develop it in service of some other scientific purpose?

Newton. I enjoy science and math as well, and I pursued math to help my scientific studies.

Newton's lead. You had a goal, then?

Newton. Yes.

Newton's lead. Why did you not publish?

Newton. As was stated, I was indeed leery of publishing. Moreover, there was a large fire which hampered publishing.

Newton's lead. It was difficult to publish, then?

Newton. There were difficulties.

Newton's lead. Who knew of your work?

Newton. Mr. Barrow, my mentor, and Mr. Collins, as well as Henry Oldenburg.

Newton's lead. So others knew about it?

Newton. Others knew.

Newton's lead. So Leibniz may have heard about it?

Newton. That's possible. My ideas on calculus came out about ten years before Leibniz's work.

Newton's lead. What is your opinion of Mr. Leibniz?

Newton. I believe he's intelligent, but I hope that he honestly came up with his own ideas on his own.

Newton's lead. Could just anyone have come up with calculus?

Newton. Not just anybody. There was a lot of work, regarding geometry and algebra.

Newton's lead. Many have referred to you as "the greatest mind of all time." What else have you done?

Newton. I worked on optics, color, and light, as well as in space...regarding the work of Galileo and others. My laws of motion are also important.

Newton's lead. Have you tried to destroy others' character?

Newton. Not really. I'm pretty much a homebody.

Newton's lead. Does fame matter to you?

Newton. Not really. I do what I do for the sake of the knowledge.

Newton's lead. You find these matters stressful. So why take this to court?

Newton. If Leibniz had simply given me credit, we could have avoided all of this.

Newton's lead. Thank you.

Leibniz's lead. Mr. Newton, to you believe that Leibniz plagiarized you?

Newton. I actually do not know that. This is largely up to the witnesses. I just want to be known as the first to study it.

Leibniz's lead. When, exactly, did you do your work?

Newton. I began in 1665, around February, as the records show. In 1669 and 1671 I wrote two papers which were not then published.

Leibniz's lead. In the first edition of your Principia you mention Leibniz, and his name disappears later.

Newton. I don't deal with the publishing matters, so I don't know.

Leibniz's lead. What did you discover?

Newton. Basically, the rate of change.

Leibniz's lead. It seems like a lot of your calculus is similar to that of Fermat. His work was not exactly right, but he was the first to discover his ideas, in the 1640s.

Newton. And why was he not considered the first?

Leibniz's lead. I don't know, but why can he not claim to be the discoverer?

Newton. I can't say. I can't say that Leibniz didn't work off of my work, and I can't say that he plagiarized. I just want credit given where credit is due.

Leibniz's lead. That's all I have.

Newton's second. We'd like to call Leibniz to the stand. When did you start your work on calculus?

Leibniz. In 1675.

Newton's second. So, after Newton?

Leibniz. Yes.

Newton's second. What's in the letters you exchanged with Mr. Newton?

Leibniz. There was no mention of differentiation, and that's what Newton's claiming I took from him.

Newton's second. So there was no influence on you?

Leibniz. None.

Newton's second. We believe Newton's colleagues will dispute that. We believe that you did not cite your sources when referencing my client.

Leibniz. I took rooms full of notes. and I can't be sure that I didn't have notes that should have been cited.

Newton's lead. What's your opinion of my client, Mr. Newton?

Leibniz. I believe he suffered from mercury poisoning.

Newton's lead. Did you ever work with chemistry?

Leibniz. No.

Newton's lead. So you didn't take as many risks as Newton did? Hmmm...did you ever credit the work of Newton which you saw?

Leibniz. I did see some information in his work.

Newton's lead. Why did you not give him credit?

Leibniz. I could ask the same of Newton: he took information from Barrow and gave no credit.

Newton's lead. What did you get from Newton's work?

Leibniz. I was only interested in infinite series, and I didn't find any information on this matter, so didn't feel it was worth citing.

Newton's second. Did you, indeed, plagiarize?

Newton's lead. Sometimes we do not cite people because we want people to believe that we did more original work than we had. Do you have something to hide?

Leibniz. I may have been influenced by Newton's work, but I did not steal his ideas. There is a difference between being influenced by someone and stealing their ideas.

Newton's lead. What is your nationality, Mr. Leibniz?

Leibniz. I am German.

Newton's lead. And my client is English.

Leibniz's attorneys. Objection.

Judge 2. Sustained. Relevance?

Newton's second. Getting back to the letters, why is it that you struck up correspondence with Newton?

Leibniz. I was interested in his ideas.

Newton's second. What kind of information did you get from these letters?

Leibniz. There was information on finding maxima and minima, but there was nothing which I could cite.

Newton's second. That's all we have for this witness.

Leibniz's lead. On the matter of the mercury poisoning, I should note that we can document this allegation, going to Newton's state of mind. I have another question for Mr. Leibniz, though. In one of the letters from Newton to Leibniz, he mentions his own "fluxional calculus," and Leibniz replied with a description of his own calculus. Why would he have replied at all, if he knew he was merely stealing Newton's ideas? The papers Leibniz wrote dealt with series, correct?

Leibniz. Correct.

Leibniz's lead. Why is this in dispute, then? ...

Newton's second. We would like to call John Collins to the stand. Mr. Collins, can you tell us about the relationship you had with Newton?

Collins. I was a publisher based in London, and carried on correspondence with many scientists both in Britain and on the Continent.

Newton's second. So you knew of Newton's work?

Collins. I knew of De analysi, for instance, and tried to get him to publish.

Newton's second. But he didn't?

Collins. He was a great thinker, but he was worried about criticism, after his experience with Hooke and Optics.

Newton's second. So Leibniz saw Newton's work?

Collins. When he visited, I showed him copies of De analysi.

Newton's lead. When did Leibniz visit England?

Collins. He visited Oldenburg, and he visited me in 1672 or 1673?

Newton's lead. Was he interested in Newton's work?

Collins. Yes, he was keenly interested. He asked questions about Newton. I knew he was carrying on a correspondence.

Newton's lead. What was his opinion of Newton?

Collins. Respectful. But Mr. Newton has his idiosyncrasies.

Leibniz's lead. You said that Leibniz got to look at Newton's De analysi, right?

Collins. Yes.

Leibniz's lead. But this paper dealt primarily with series, and not differentiation, right?

Collins. Nevertheless, Barrow thought the paper brilliant enough to give up his Chair to Newton.

Leibniz's lead. But Barrow was very interested in series, so of course he was impressed with Newton's work. Do you know anything about Robert Hooke?

Collins. We corresponded a bit.

Leibniz's lead. Did Newton's work have anything to do with Hooke's work?

Collins. Likely yes, given the interdisciplinary work we all did.

Leibniz's lead. It seems that Newton's work was very similar to that of Hooke.

Collins. Are you suggesting that Leibniz plagiarized off of Hooke? Or that both did?

Leibniz's lead. Nothing further from me.

Leibniz's second. One more question: were you aware of all of the convergent ideas at the time? You do realize that Hooke accused Newton of plagiarism?

Collins. You're referring to the optics paper?

Leibniz's second. Yes. There were a lot of ideas coming to a head then.

Collins. I agree.

Judge 2. We will now take a five-minute break.

Leibniz's second. We would like to call an historical expert. Talking about the idea of convergent ideas, can you indicate any other examples similar to Leibniz's and Newton's?

Expert. There were trends in advancement of these ideas. For instance, there was the Kerala School in India, where these ideas were put forward.

Leibniz's second. They had accurate calendars and were trying to compute the instantaneous motion of the moon, right?

Expert. Correct.

Leibniz's second. So these people, 300 years before Newton, invented some of the same ideas that would later become calculus? Is there any way this information could have gotten into the hands of Mr. Newton?

Expert. It was in a book, written in Sanskrit. It's still there, if you can read it.

Leibniz's second. And missionaries took this back to Europe? And they were schooled in mathematics. It was a possibility, then, that Newton could have deduced planetary motion from these texts, or at least been influenced by this?

Expert. This is possible.

Newton's lead. Is there any proof that Newton saw this work?

Expert. Not that I know of.

Newton's lead. So this is pure speculation?

Expert. It is possible.

Newton's lead. Is it possible that they plagiarized off of me?

Expert. I suppose.

Newton's second. Why are we arguing that Leibniz plagiarized off of Newton, if we could just as easily argue that they both plagiarized off of the Indians?

Expert. It was a very simple form of calculus, it could have done little more than influence people.

Newton's lead. 300 years is very different from 10. Isn't it more than coincidence, don't you think, that calculus was a simultaneous discovery?

Expert. It's odd, yes.

Newton's lead. Does Newton read Sanskrit?

Expert. Not that I know.

Newton's second. No further questions.

Leibniz's second. We call Johann Bernoulli. You are a prominent mathematician and long-time friend of Leibniz, yes?

Johann Bernoulli. I am his student.

Leibniz's second. What was his influence on you?

Johann Bernoulli. My father wanted to become a doctor, but after reading Leibniz's work, I became a mathematician.

Leibniz's second. This man changed your life?

Johann Bernoulli. Yes.

Leibniz's second. You expanded on Leibniz's work?

Johann Bernoulli. And Newton's. And I worked to spread their knowledge.

Leibniz's second. Was Leibniz smart enough to invent calculus himself?

Johann Bernoulli. No doubt. He could easily have come up with it himself.

Leibniz's second. No further questions.

Newton's second. We're not trying to argue that Leibniz did not make advances; we're trying to argue that Newton did it first. What can you say to this?

Johann Bernoulli. Newton's gravitational theories can be improved by using Leibniz's theories.

Newton's lead. The terms you're using sound similar to those used by Newton. Isn't it possible that Leibniz got these ideas from Newton?

Johann Bernoulli. Leibniz was able to use his methods to solve problems Newton posed.

Newton's lead. So Newton's work came first?

Johann Bernoulli. Newton did it wrong. I improved upon it using Leibniz's methods.

Newton's lead. You say that you love Mr. Leibniz.

Johann Bernoulli. Well, look at him!

Newton's lead. Don't you think you're a bit biased?

Johann Bernoulli. Yes: I'm his champion; I thought we went over this.

Newton's lead. Nothing further.

Leibniz's second. We call Newton back to the stand. Do you have any secrets you've kept as an adult?

Newton. No.

Leibniz's second. I have copies of a collection of your notes, and in them it was read that you studied alchemy and kabbalistic studies, and you hid this in order to avoid charges of heresy.

Newton. I like science, and I was curious. This isn't a secret.

Leibniz's second. So you weren't rather reclusive?

Newton. I admit to that.

Leibniz's second. But you ingested mercury?

Newton. Yes. What does this matter?

Leibniz's second. According to the Royal Society of London, you suffered two mental breakdowns?

Newton. Yeah. It was a rough time; I was somewhat stressed over the allegations Hooke made.

Leibniz's second. According to the Royal Society, your temper was extraordinary.

Newton. I had a temper, but it didn't affect my mathematical work.

Leibniz's second. But I believe that it affects your state of mind, and may lead to the reasons you raise the accusations you raise against my client.

Newton. Does this affect my mathematical work?

Leibniz's second. Doesn't this shed light on your view of his plagiarism, though? If he's found guilty of this charge, might it not destroy his reputation?

Newton. I agree he did something very good, but I only want credit where credit is due.

Leibniz's second. The work Leibniz published was only a small piece, though, and because this was simply a tiny result 10 years after his own discovery, might it be possible that he didn't think it necessary to cite his sources, especially since he might not even have been aware of the existence of those sources?

[General cross-talk and disorderly nonsense.]

Leibniz's lead. Between your work and Mr. Leibniz's, there are crucial differences, especially concerning integration. Your methods are very different, is that not the case?

Newton. This is correct.

Leibniz's lead. You used very different methods, so how can you claim that he plagiarized your work when his method was fundamentally different from yours?

Newton. It may be that he didn't plagiarize. I only claim that I did it first.

Leibniz's lead. Nothing further.

Newton's lead. They make a big deal out of your chemistry. Do you regret the work you did with chemistry?

Newton. No.

Newton's lead. Do you feel this work has harmed your judgment?

Newton. No, it does not.

Newton's lead. Do you believe this matter, all of this, is worth taking to court?

Newton. No. I'm only here because I believe there's an important case to be made here.

Newton's lead. Thank you, Mr. Newton.

Newton's second. You weren't intentionally sniffing mercury, right? Just to make this clear. You were studying this legitimately?

Newton. That is correct. But my mathematical work came early in my life.

Judge 1. It's time for closing arguments.

Leibniz's second. You heard a lot of evidence today, and the argument is essentially a matter of probability: is it likelier that my client stole his ideas from rough mention of "fluxions" and "fluents" or that he came up with the ideas on his own? There was a good deal of convergence in ideas during that era, and my client was certainly intelligent enough to come up with calculus on his own. Telescopes, logarithms...why not calculus? Our point is not to show that Newton stole his ideas from the Kerala School; it's to show that Newton may have come upon the same ideas at a different time in a different place. The maps of our ideas may look very similar, even though we've come upon those ideas separately from one another. Finally, consider that the notes Leibniz saw dealt only with a small chunk of calculus, nothing like the grand theory with which he later came up.

Newton's lead. We do believe that Mr. Leibniz did indeed plagiarize, with all due respect to his genius. We were unable to call sufficient witnesses to show what we've needed to show. Central to our argument is that Leibniz saw Newton's work, and didn't credit him as needed. Leibniz may have come up with some of his ideas independently, but he still needs to give credit for the ideas which he got from others, including Newton.

[The court recesses while the jury deliberates.]

Judge 1. The jury has reached a decision?

Jury foreperson. We find the defendant not guilty of the charge.

Posted by DocTurtle at 1:41 PM 1 reflections

Labels: Calculus I, MATH 191

###
*Newton* v. *Leibniz*, Section 1

Below is a (rough) transcript of my first section's rendition of the Newton v. Liebniz trial!

[Court is called to order at 8:00 a.m.]

Leibniz's lead. As our opening argument, we should indicate we are stating that Leibniz did not plagiarize Newton's work. There were letters written between Newton and Leibniz, but the colleagues were as much to blame for the debate as the characters themselves.

Newton's lead. Let us begin by saying we are accusing Leibniz of plagiarizing Newton's work on two separate occasions. We are here to make sure Newton gets primary credit for his discovery.

Judge 1. Prosecution may call the first witness.

Newton's lead. Henry Oldenburg, please come to the stand. Mr. Oldenburg, have you ever shown Leibniz any of Newton's work.

Oldenburg. Indeed. He visited in 1667 and then I showed him Newton's work.

Newton's lead. Which work?

Oldenburg. I believe it was Epistola prior and Epistola posterior.

Newton's lead. These were geometry-based, the building blocks for calculus, right? Could Leibniz have seen the beginnings of calculus here?

Oldenburg. I believe so.

Newton's lead. That is all. Thank you.

Leibniz's lead. Why do you say that it's "safe to say" Leibniz saw the work?

Oldenburg. He did look over it: it's got information concerning binomial series, curves, etc.

Leibniz's second. Did you read it yourself before giving it to Leibniz?

Oldenburg. Yes.

Leibniz's second. Didn't they have different methods?

Oldenburg. Yes, they were different in notation, in the end.

Leibniz's second. Didn't Newton also encode his work?

Oldenburg. Yes.

Newton's attorneys. Objection! That was a different letter.

Judge 1. Sustained.

Leibniz's second. That is all.

Newton's lead. Call John Collins, please. [Collins takes the stand.] Isn't it true that you also showed Leibniz some of Newton's work?

Collins. Yes. In 1667, the Royal Society was in recess at the time, and I wanted Leibniz to see how great British scientists were. He looked at my work and at De analysi, on which he took notes.

Newton's lead. He took these notes back to Germany.

Collins. Yes.

Newton's lead. He could have developed this into calculus, correct?

Collins. I can't say directly, but it's possible. I never told Newton that I'd shown his work around, either.

Newton's lead. Sketchy.

Collins. I guess. I felt bad about what I'd done.

Newton's lead. The beginnings of calculus were there, in De analysi, correct?

Collins. Yes. Leibniz could have gotten information that inspired him in this work.

Newton's lead. Indeed, Leibniz could easily have discovered calculus from this work? No other questions.

Leibniz's lead. So you're saying the you showed Leibniz Newton's work. How do you know what Leibniz knew before this?

Collins. Since the early 1670s Leibniz had been in touch with me. Leibniz had been writing to me and to Oldenburg asking about mathematical ideas. We didn't give him any real information; we only gave him methods. He sent us information, as well, but none of us sent complete information.

Leibniz's second. Do you know why Newton didn't publish his work right away?

Collins. I pushed him to publish, but he wouldn't. He was shy, and he had been burned: he'd published a work on optics and had been embarrassed, so he was reluctant to publish until his critics died. Moreover, after the Great Fire nobody published for a long time. Mostly, though it was because of public criticism. There's clear evidence, though, that he has priority. I don't know if you could call it plagiarism.

Leibniz's second. Was Newton angered by Leibniz's publishing first?

Collins. Personally, I was dead at that point. But I know that the colleagues were the one who had the most beef.

[There is grumbling from the Leibniz people.]

Newton's second. We call Leibniz to the stand. We hear that you were not popular with your employees.

Leibniz. That is all hearsay.

Newton's second. Didn't you have business schemes that ended in failure?

Leibniz. No.

Leibniz's lead. Objection!

Judge 1. Sustained.

Leibniz's lead. I have Leibniz's response to the allegations. [He reads from a formal statement which indicates that it would be too much trouble for him to respond formally to every point.] To me, this says that even though he's being attacked, his integrity is such that it drove him to continue his work rather than respond to specious claims.

Newton's second. Didn't Leibniz have a hard time corroborating his work, and he had a hard time indicating his sources. How do you respond to that?

Leibniz's second. We have no response to that. Is this line of questioning relevant.

Judge 1. Sustained! The sort of allegations being made by Newton's side are immaterial to to the case at hand.

[There is grumbling from the Newton bench.]

Judge 1. Any more witnesses for Newton?

Newton's lead. We call Newton to the stand. Mr. Newton, are you the sole originator of calculus?

Newton. With no doubt.

Newton's lead. When did you start work?

Newton. 10 years before Leibniz...about 1665 or 1666.

Newton's lead. This had to do with "fluxional calculus," correct? It was very unwieldy and hard to understand at that time. But could not Leibniz simply clean it up, make it more efficient, and claim it as his?

Newton. Yes.

Leibniz's second. Is it not true that Leibniz received your letter after he created his own method of calculus?

Newton. He received the letter in 1666, and he hadn't published anything at that point.

Leibniz's second. But he had developed his method.

Newton. There's no proof.

Leibniz's lead. [Reads statement on Leibniz's development of calculus before publishing, indicating the elegance of Leibniz's notation.]

Newton. I would say he changed my notation, but that he stole my ideas.

Leibniz's second. What proof do you have that he took your notation and changed it?

Newton. No response.

Leibniz's second. Nothing further.

Judge 1. Any further witnesses for Newton?

Newton's lead. We call Barrow. Mr. Barrow, you knew when Newton came up with calculus, correct?

Barrow. Correct. I was very close to Newton, and I suggest that he become Lucasian Professor at Cambridge after I left that position. He showed me a lot of his work.

Newton's lead. What was he doing then?

Barrow. In a letter in 1666, he announced his binomial theorem, just a decade before Leibniz published his work on calculus. I know also that Newton had developed De analysi before Leibnis published.

Newton's lead. So it's fair to say, based on your testimony and the others', that Newton clearly developed his work ten years before Leibniz, and because of the fire and because of personal reasons (Newton's a shy man), Newton was leery of critics.

Leibniz's second. Don't the letters between Newton and Leibniz talk about the different methods the two men came up with.

Newton's lead. Objection: those letters were privy only to Newton and Leibniz.

Judge 1. Sustained, unless you can show the substance of these letters.

Leibniz's second. [Reads from a 19-page letter from Newton to Leibniz indicating his method of fluxions, written in code.] Did Leibniz know how Newton came about his calculations?

Barrow. Both of these men are incredibly intelligent, and either could have deciphered the code and understood the work. Both were moving along the same path, in the same direction.

Leibniz's second. They moved along the same path, different methods?

Barrow. But the question is who developed it first; second discovery counts for nothing.

Judge 1. If there are no further witnesses, let's take a brief recess.

[The court is in recess for five or ten minutes.]

Judge 1. The court is called back in session. Leibniz's side may begin their defense.

Leibniz's lead. We would like to call Ehrenfried Tschirnhaus. How would you describe Newton's character.

Tschirnhaus. I didn't know know Newton well.

Leibniz's lead. How would you describe Leibniz's character?

Tschirnhaus. I knew him well. I met him in 1675 in Paris, and we developed a rapport, both professional and personal.

Leibniz's lead. Could you go into greater detail?

Tschirnhaus. Leibniz had great integrity, and these attacks are groundless. He allowed me to study unpublished works by great philosophers, and it helped me along professionally. Also, after we left Paris and went our separate ways, we kept in touch. If I were working out a problem, I'd write to him for advice, and he was always willing to help me out.

Leibniz's lead. So the Newton team is trying to distort the view of Leibniz?

Tschirnhaus. Indeed.

Newton's lead. Are you not also biased in this regard, being a close friend of Leibniz?

Tschirnhaus. True.

Newton's lead. Does the defense have any neutral parties to testify on their behalf?

Leibniz's second. We call Jacob Bernoulli to the stand. Could you please tell us the difference between Leibniz's work and Newton's work?

Jacob Bernoulli. I'd be delighted. Let it be known up front that whatever work these two men had was not entirely original. It all contained bits and pieces of work men prior to them had been working on, like Wallis. The work on infinite series had been done long before. It was these two men, though, who really went to work in solving problems with calculus methods. The differences were in the ways they handled things like infinite series, and in the ways their work was capable of doing different things. Newton's work, only shared in code, didn't give a clear-cut method for solving problems, but Leibniz was the first to give it a solid algebraic method. Newton's work in the '60s didn't have this foundation, but Leibniz's work on infinite series did. The clearest indication that Leibniz's work was original came in 1675: Leibniz could actually find closed sums of infinite series, whereas Newton could only find approximations. Leibniz took two years to develop this method. In fact, in 1677, after Leibniz shared his work with Newton in a letter and Newton looked it over, that's where the problems began.

Leibniz's second. So the correspondence began after they had both developed their methods?

Jacob Bernoulli. Leibniz had developed his method, but Newton had yet to formalize his method. Principia did not contain anything new, even though Newton knew of Leibniz's methods.

Leibniz's second. Could we not say that Leibniz did not plagiarize Newton's work?

Jacob Bernoulli. Yes.

Leibniz's second. No further questions.

Newton's lead. Didn't Newton's work precede Leibniz's?

Jacob Bernoulli. Yes, but he had yet to develop a systematic method, which is necessary for dealing with things like integration. He had the foundation of his work down, and he could solve a few problems, but he hadn't developed his work to the extent that Leibniz had. Moreover, Leibniz, like Newton, was a very intelligent man, and was able to discover these ideas on his own.

Newton's lead. When was it that Leibniz had developed his method?

Jacob Bernoulli. It was in the 1660s. My brother Johann and I were among the first to work with Leibniz and make use of his methods.

Newton's lead. But when did Leibniz first publish his work?

Jacob Bernoulli. I believe it was in 1675. It was not uncommon then to have such long gaps in communication.

Newton's lead. I just find it strange that Leibniz had gone to London to see Newton's work just before he begins his later-published work.

Jacob Bernoulli. But you can't say he wasn't working on developing calculus during that time. The letter from Collins in 1673 was more of an update on British mathematics at the time, it's not like it was singularly about Newton. To say that Newton was the centerpiece of that work would be to misconstrue it. So, yes, there was information in that letter Leibniz may have used, but Newton's work didn't lead to a particular method. Leibniz could not have discerned Newton's method from his work.

Leibniz's lead. We call Johann Bernoulli. Mr. Bernoulli, can you tell us about the problems you sent to various mathematicians?

Johann Bernoulli. First of all, hail Leibniz! I should be up front about things: in 1696, I submitted a problem which could only be solved by those who really knew calculus. The problem was out there for six months without being solved. A year later, finally, there was a response from Newton ["that bloody British!'].

Leibniz's lead. Do you have any evidence in support of Leibniz?

Johann Bernoulli. In 1684, my friend and colleague found a method for solving certain differential equations. [He writes on the board: "dx n = n x

^{n-}

^{1}."] This formula was given in Acta eruditorum before Newton had determined it, two years later.

Leibniz's lead. Do you have any more evidence in support of Leibniz?

Johann Bernoulli. Indeed. I can't understand why Newton would impugn the character of a man who attempted to unify philosophy and science.

Newton's lead. How did his attempt to unify Protestantism and Catholicism go?

Johann Bernoulli. He was unsuccessful. But he had many interests, he tried to diversify his interests. He had good intentions, but most of his work was mathematical and philosophical.

Newton's lead. Here's a quote regarding Acta eruditorum and Principia. [Read quote.]

Johann Bernoulli. Can I respond in the form of a question? Which equation is now used by scientists and engineers?

Newton's lead. The one you've written.

Johann Bernoulli. So!

Newton's lead. It's a cleaner form, no doubt. But Newton did develop fluxional calculus, much earlier. There's enough time in there for Leibniz to take this work and polish it up.

Leibniz's second. Objection: doesn't plagiarism require publication?

Judge 1. Sustained.

Leibniz's second. It's a legal impossibility: Newton's work never appeared in print.

Johann Bernoulli. The moral of the story is: "don't be shy."

Judge 1. Any more witnesses? No? Closing statements?

Newton's lead. Newton's impact has been profound. Newton has been knighted; Leibniz has not. Newton's Principia is a foundation of science; Leibniz has no such counterpart. Newton's formulas concerning gravity and planetary and tidal motion, and his laws of motion are all used today. Newton is a genius: he singlehandedly developed physics, whereas Leibniz failed at many practical endeavors. How could Leibniz have developed calculus while Newton, scientific genius, did no precede him? This seems strange to me. Leibniz plagiarized.

Leibniz's lead. Leibniz was also a genius, as was Newton. There is also considerable evidence for the originality of Leibniz's work. Moreover, as we've seen, Leibniz's methods are those currently in use today, and his notation is superior and current. This was developed independently from Newton's work. It's also important to point out that the Royal Society itself cleared Leibniz of the charge.

Judge 1. As the statements have been given, we are now adjourned. The jury may deliberate.

[UPDATE: After five minutes of deliberating, the jury returns its verdict.]

Jury foreperson. We find that Leibniz is not guilty of plagiarism: the charge is insubstantial.

Posted by DocTurtle at 7:56 AM 0 reflections

Labels: Calculus I, MATH 191

## Sunday, October 24, 2010

### What's it worth to me?

19 hours, apparently.

19 hours, 7200 words in response to students' drafts, several follow-up e-mails looking for students' sources (ah, unintentional plagiarism!), countless handwritten notes (sorry for the scrawl, y'all), and, it must be said, a good number of sighs and head-shakes...but a few smiles as well.

This weekend was the perfect storm of grading. I had a look-see at the roughly 25 first drafts of Newton v. Leibniz papers my Calc I students produced, the 9 "sections" from Linear Algebra for Dummies the Linear students wrote, as well as two problems sets from the former class and one from the latter, and assorted exam revisions and older homework sets. There were times during the past 48 hours at which I cursed myself for setting a precedent for such quick turnaround, at which I thought, "is it really work it to me?"

The answer, I think, now that I've finally got a few hours of free time before things start up again tomorrow, is...yes.

That's a simple answer which masks the earnest reflection I've done over the last couple of days on a number of weighty pedagogical issues: (1) meaningful instruction of authentic disciplinary writing (or even basic academic writing at the collegiate level), (2) the role played by homework completion (and feedback received on same) in the learning process, (3) the relative uselessness of computer-generated/computer-graded homework in providing meaningful conceptual instruction, (4) the role of numerical grades, especially as they pertain to the establishment of an extrinsic rewards system that encourages students to become number-crunchers at the expense of real learning.

Et cetera.

How've these issues come up?

Newton v. Leibniz offers students an imposing and unanticipated challenge: most of my students don't expect to do much writing for a math class, and I'm quite certain that (whether they admit it or not) they don't expect their math professor to be a stickler when it comes to writing. My suspicion is that if they've ever had to write much for a math class in the past (and they likely haven't), whatever writing instruction they've gotten from their math teacher was half-assed and half-baked.

So maybe I shouldn't be surprised when I receive inchoate two- or three-page drafts in which ideas are scattered and half-formed, or reference lists peppered with websites whose authorship is unidentifiable and with textbooks which are (literally!) over a century old. Four or five of the drafts were marvelous: well-researched and well-organized, clear, correct, and easy-to-follow. Five or six others have obvious potential, and rest on a foundation of appropriately-chosen references. These might make a few slips compositionally, and they may have a few logical lacnuae here and there, but they'll be solid with a few more sources and some good ol'-fashioned elbow grease. The rest (another five or six) have a ways to go.

I've been playing this game long enough that I know almost exactly what's happened as the students put those last few together. Eight days into the nine-day period they've been given to get their shit together, the members of the group putting one of these papers together met after class (seven hours to the due date) and said, "hey, when are we going to work on this?"

By this time it's too late to make use of the fifteen-odd recent and well-reviewed books on Newton and Leibniz and their fellow-travelers I put on hold at the library, so they opt for the next-best...er...well, still a halfway-decen...um...well, at least an okay...well, all right...a pretty piss-poor stopgap solution: Google.

Pulling up the first two websites they can find when they put "Newton versus Leibniz" into the Google search field, they read.

Of course, one of those websites (this one, written by someone I can only refer to as Mr. Angelfire) is utter crap:

1. It's impossible to tell who wrote it, although it's likely a term paper written by a high school or college (I'm betting on the former) student with limited skills for selecting references:

2. the only print references it cites are at least (not kidding...wish I were!) 40 years old, including one that's over a hundred, and

3. the only relatively recent source is a website...it's a good one, but it's a website nonetheless, and unless you're familiar with that site (I am; my students are not), you wouldn't know this from the way it's cited (incorrectly).

4. It's a perfect example of that boilerplate "five-paragraph essay" nonsense they teach kids how to write (for some godawful reason) in high school these days. It takes no stance, it offers nothing real. It has no voice. I want my students to take a stand, stake a claim, and fight for it tooth-and-nail. This essay is a lousy model for this sort of behavior.

The second-most-cited website is this one. It takes a little effort, but you can find the author of this site, one Robin Jordan, Professor Emeritus of Physics at Florida Atlantic University, whose website (featuring marvelous animated .gifs written, no doubt, circa 1998) makes him seem to be a pretty decent teacher, actually. Not only is Jordan's paper much more well-written than the other, but it's richer and relies on stronger sources. I'm okay with my students drawing on Jordan's paper, but I'd still rather they use him as a stepping stone to get back to the print sources on which he himself draws.

All of that aside, what's the next step in our hypothetical students' last-minute writing process?

Reference (singular) found, they devour it, in a matter of...well, minutes...because that's all the time that's left to them during the lunch period on this last day. A few choice quotes plucked from the "paper" they've just read, they begin writing.

At this point it's too late to develop a thesis of their own, so the students opt for that old standby, "we just can't tell who it is who invented calculus, doncha know?" Asserting that there's just not enough evidence to tell which man has the greater claim (or a claim at all), the students hammer out two or three pages in which they eventually get around to saying that Newton did it first, but maybe Leibniz did it on his own anyway, who's to say?

Well, you're to say.

One thing these last-minute Larrys and Lauries don't do is say anything, at least not anything meaningful. But I so much want you to do this, my young friends! More than anything, this is what I want from you: I want to hear your voice.

Make a claim...make a bold claim. If you feel like putting it in boldface letters 18 points high, then do that. But make a claim, and make it your own. Make it your own by finding the sources that help you say what you want to say, that help to prove that, by goodness, you're right. Find the sources which lend support to your claim, and lead me through an analysis of those sources, step by step. Prove to me that you're right, sentence by sentence, page by page.

I don't want to know what Dr. Robin Jordan thinks, I sure as hell don't want to know what Mr. Angelfire thinks...I want to know what you think, and why you think it. That, my young colleagues, is the essence of academic writing (and, indeed, academic thinking of any kind): saying something intelligent, and saying it in an intelligent way as you insert it meaningfully into conversation with all of the other thinkers who have come before you.

Is it worth 6 hours of poring over drafts and 7200 words (that's 27 pages of 12-point, double-spaced text, by the way) of responding to those drafts, if it helps you all become better academic writers?

Hell, yeah. I'd do it again in a heartbeat. And I'd be delighted to look over further drafts if any of my students care to hand me some between now and the due date next Friday.

The rest, the other 13 hours? Problem sets, problem sets...yeah. The Linear problem sets were fine, and those for Calculus I...were great, actually. I put 'em through the wringer, computationally. I've no one but myself to blame if I go blind from having to puzzle through the first six or more derivatives of e^{x} sin(x). The extra work is worth it to me, if only so I don't have to read through thirty or forty poorly-transcribed copies of the solutions manual because I made the mistake of assigning problems from the textbook (a pedagogical practice I'll never again adopt as long as solutions manuals are readily available).

Indeed, in the end the students did really well on the two problem sets (one on the Product and Quotient Rules, the other on trig derivatives), given their relative difficulty. The only gripe I might make about them concerns, as above, evident procrastination: if you don't get going until Friday, a few hours before they're due, you're not going to do that well.

All in all, though, these problem sets too are worth the time I put into grading them: I feel strongly that feedback (frequent and full) is essential at this stage in the students' engagement of higher mathematics, and I feel strongly that graded homework is the best way to provide that feedback. (Students know this, too: almost without exception my former students remark to me how helpful graded homework is once they've gone on to a class with one of my colleagues who doesn't require it...and when given a chance to assign their own grading weights to the various activities we take part in in my classes, they always give homework a substantial boost.)

This post, which began with a gripe, now draws to a close with acceptance and contentment. I've lost a weekend, in some regards (although I did take in several really good college football games yesterday), but I've come through to the other side a better teacher, having reflected a bit more carefully on, and asked myself to remember, the reasons I do the things I do.

Before I go, here's a postscript for my Linear students (in particular, for Ino and Iris, to whom I was complaining on Friday, about having to assign numerical grades to their written projects): I plan on asking you all to assign your own grades to your Linear Algebra for Dummies sections. FYI.

Posted by DocTurtle at 5:50 PM 1 reflections

Labels: Calculus I, Linear Algebra I, MATH 191, MATH 365, theory, writing, writing-intensive