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.

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.

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,
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.

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.

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 xn-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.