Thursday, October 29, 2009

Editing party

No fewer than six of my 280 students met with me for two hours in the Math Lab this afternoon to go over the "Watchwords" appendix to our textbook (or "TeXbook," as the students are starting to call it...the watchwords consist of fifteen simple words [like "and," "if," "since," and the like] with precise mathematical meanings and which are often misused, especially by beginning students) and Chapter 3, on set theory.

I had to squelch squeals of glee at some of the conversations that were going on, dealing with the logical flow of the sections in Chapter 3 ("No, Cartesian products have to come first, because the author of the cardinality section is making use of the product in their example." "But we could just modify that example, or get rid of it." "All that has to be true is that ordered pairs come before products, since they're defined by ordered pairs."). It was for the first time today that it fully struck me, head on, how much this exercise is forcing the students to become completely aware of in the interconnections between the various topics we've been studying.

"Are you going to try to get this published?" asked one of the students from last semester's 280 class.

"I don't know. I'm going to try to get some folks to talk about it at the conference at Elon in March, and I'm going to promote the hell out of it. It's a big project, and a big deal, and I think they should all be proud of their work."

"Yeah, I'm sure people at other schools would like to see it."

"Even though it's not going to be perfect, and it's going to be rough and have mistakes, and it's going to look like it was written by fifteen different people (mostly because it is written by fifteen different people), it's going to be authentic. And ultimately that's where its strength lies."

Keep it up, my young friends and colleagues, and I'll keep bragging on you!

Tomorrow: donuts!

Tuesday, October 27, 2009

School daze, part II

My visit to my former student Maria's Discrete Math class at SILSA was a refreshing experience.

Most striking were the similarities between her class and my own: some were engaged, others were not. Some were clearly interested, others were not. Some were quick to pick up the ideas we were talking about, others were not. There was a mix of interest, apathy, passion, and torpor.

I arrived about fifteen minutes before I'd planned to, running through the rain to the entrance just below the front door of Asheville High School's gigantic main building. In the door, down the hall, past the first vestibule, and just down another corridor, I found SILSA's office and was shown to Maria's classroom, just around the corner.

Most of her students were there (15 of the 17 she'd told me to expect), milling about, working away on the laptops they'd pulled from the giant wheeled cabinet at the front of the room. "Come up with at least one question," Maria requested as she walked around the room. "Come up with at least one question and write it down. Do you have a question?"


"Can you write it down please?"

"Do I have to write it?"

"Yes. Can you write it down, please?"

Some needed several requests. Meanwhile I sat and took it in. The room seemed smaller than most classrooms I teach in, and it felt more "lived-in," its walls more heavily decorated and its atmosphere homier. The comfy-looking couch at the classroom's rear was offset by the state-of-the-art Smartboard at the front of the room.

"All right, everyone, this is Dr. Patrick Bahls, who was very helpful to me when I was studying math in college. He's done a lot of research in graph theory, and written papers on it. He's very knowledgeable about it, so I hope he'll be able to answer your questions." After some obligatory applause, I assured the students that they were very likely to be able to stump me with their questions.

Just to buy a little time while I got a sense of the class's overall receptivity, I hemmed and hawed for a few minutes about graph theory and its applicability. As I warmed up, so did they, I think, and after a little while longer I segued into the topic I'd planned to speak on, channel assignment. I introduced the general ideas, using the real-life motivating example of radio station frequencies, and then passed out a worksheet which challenged the students to complete a valid channel assignment on a simple graph.

"Try to make the numbers you use as small as possible so that your choices of frequencies are as efficient as you can make them." Several students took this challenge on earnestly and bore into the task. Maria and I walked around the room watching as the students worked, much as I would in my own classes. In fact, most of the time I was there I felt very much as though I were walking the floor of my own class, stalking my own students. It was nice to know there wasn't much difference between our classes.

Ursina and her friend, sitting at the front of the class, were the first to complete what I suspected was an optimal solution, and Ursina wasted no time in acting on my invitation to share her solution on the Smartboard. We moved on to the infinite integer lattice, a graph whose span is 6 but for which the best channel choice the students could find at first had span 8. I asked another student to share her solution on the board, but she was less eager than Ursina had been, and took some cajoling.

After we'd done discussing channel assignments, we had time for me to field several general graph theory questions from the students. A few required some normalization of terminology before we could understand one another, but I think I was able to give reasonable answers to several questions on binary trees, hamiltonian cycles, and planarity. I think the students most appreciated the idea of realizing, without crossings, a complete graph on 5 vertices on the surface of a doughnut.

However I was, as I'd predicted, stumped by a question on Steiner trees.

During the Q 'n' A, with only five minutes to go before class was let out, the students got a bit restless, and several times Maria had to call for order and I had to raise my voice a bit to be heard over the steady rain of teenage titters. I finished up, and Maria's lecture to the students to take seriously their upcoming college placement exams and to treat the substitute teacher ("Who is it?" "..." "Ahhh, shit.") brought a stark reminder that this was high school and not, indeed, college. As soon as the bell rang, most of the students (all but the one who had further work to do) were out the door.

But Maria's work was not yet done; she'd be on the clock for the next hour or more, reading over students' exams, helping with exam revisions and retesting, going over lesson plans.

"I've got three preps, six sections, and since a lot of it isn't in textbooks, I'm making up a lot of the material myself," she told me afterward. She looked happy, but very, very tired. I can sympathize.

She was one of the two teaching licensure candidates our department graduated last year about whose careers I was most excited. She's smart, she's funny, she's wise, and she's good with kids. From all that I could tell of her class today, she's done a good job in earning her students' respect, and I'm sure she's a fantastic teacher.

I hope she doesn't burn out.

And she's at a good school.

It's a true dilemma, with sharply pointed, piercing horns: if they're to succeed, our nation's public schools will need nothing but the most passionate, intelligent, and dedicated teachers our colleges can provide them, yet the Herculean tasks these teachers will be asked to perform (for, frankly, shit pay and sadly little respect) are daunting to all but the most determined souls. As I've discovered recently, this makes it difficult to give career advice to students who may be thinking about teaching but who are not totally certain about the idea.

Decisions, decisions.

School daze

This afternoon I'll be paying a visit to an old UNCA student's Discrete Math course at Asheville High School's School of Inquiry and Life Sciences at Asheville (SILSA).

I haven't spoken to a high school class since grad school, when I gave a brief introduction to pure math to students at Nashville's Hume Fogg Magnet School.

I'm sure it'll be an eye-opener in many ways, especially given the cynicism with the American educational system I've developed since that last visit almost ten years ago. I hope the students' eagerness and excitement will counter that cynicism and beat back doubt, and that whatever inequities and iniquities I find will be offset by possibilities and opportunities.

I'll be sure to report back.

Monday, October 26, 2009

Back to the basics

I'm feeling a bit less stressed-out than I was this afternoon when I put together that last post. A good run always helps me out.

While handing back exams in both my morning and afternoon Calc I sections today I brought up the idea of using portfolios as a means of assessing student learning in mathematics courses. This idea was couched cozily inside of a conversation about the shock of receiving a "bad" grade on an exam (as some of the students no doubt experienced today). "I hate having to grade y'all," I told them. "I'm more and more opposed to grading in general, and to the simplistic distillation that goes into assigning a single letter grade to such a Gestalt as the sum-total of a student's learning activities throughout an entire semester."

There were many nods of agreement when I described how I'd like to be able to supply them with all of the same feedback I give them already...without the numerical rankings, the stigma-making marks that say "she's more highly-ranked than he is."

"I'm not going to do it this semester, since it wouldn't be fair to any of us, you all or me, to change the system midway through. But I'm seriously thinking about it for future semesters."

More nods of agreement. I'm convinced that students are not against this.

But if I were to move to portfolios, the first question would be, What goes into those portfolios? Clearly students would be asked to submit materials of various sorts that purport to demonstrate mastery of course learning goals. Ultimately, then, the question becomes twofold: What are the learning goals of the course? and What course activities (projects, exams, written assignments, homework assignments, etc.) would be sufficiently rich to demonstrate clear mastery of the learning goals selected?

As I reminded my 280 students today (quite forcefully, I hope), when you've got no idea what to do, you go back to the basics. In 280 in particular and in mathematics in general that usually means you'll want to take a long, hard look at the definitions. In course design, it means you'll want to take a long, hard look at the reason you want the students taking your class in the first place.

My current learning goals for Calc I (as stated in this semester's syllabus) are as follows:

1. Be able to explain to a peer the concepts of limit, continuity, and derivative.

2. Demonstrate how basic problems in physics, engineering, chemistry, and other fields can be couched in math terms using mathematical models.

3. Be able to follow confidently the course of a simple proof.

4. Be able to perform and properly interpret derivatives.

5. Demonstrate (through informed question-asking) a healthy skepticism regarding mathematical and scientific arguments.

6. Demonstrate how to approach a (not necessarily mathematical) problem effectively by breaking it down into smaller problems, arguing by analogy, and applying other basic problem-solving techniques.

I think it's clear that mastery of some of these would be very difficult to assess using "traditional" assessment instruments. While (1) and (4) could be got at with a well-designed traditional test, assessing (2), (3) and (6) would require a more robust (and likely highly nonstandard) project of some sort, and (5) would require something extremely atypical...maybe a dialogue of some sort, or some other "creative analytic practice" (to use Laurel Richardson's term).

Of course, the above learning goals are merely my own...I'd love to see what students could come up with for learning goals of their own. Maybe I should ask them? Yes, I think I shall.

Clearly there's a lot of thinking left to do, on many persons' parts.

For now, I'm off to eat dinner. I hope that if you read this, you'll reflect on it for a moment or more and offer me a few thoughts of your own in the comments section.


I don't have time to write anything meaningful.

I'm not sure I have anything meaningful to say, anyway.

We're about two thirds of the way through the semester, but it feels like it should be just about over, already.

We're all tired.

We're frustrated.

We're close to tears (some of us are already there).

We're weakened by increasing demands on decreasing time. You don't have to do a single derivative to know that that's not a very good recipe for success.

Sometimes we all feel like the only way we'll make it is to start sounding like Clover the horse in Orwell's Animal Farm, who says to himself over and over and over and over "I will work harder," only to find himself on the way to the knackers' before the book is over.

If I'm at all short, if I'm at all curt, if I snap or snip or sound at all bitchy, please know that it's not with you.

We're all in this together, and, as I told my Calc I students in both sections today, all I want for you is for you to learn as effectively as you can. I sincerely hope that this is a goal that we share. Whatever you need from me in order to help you meet that goal, I'm willing to give.

Okay, that's all the time I've got right now for a pep talk...I'm off to the next time-demanding task. Just remember: you're not alone.

Thursday, October 22, 2009

Another thought while reading Kohn

What might a portfolio-based Calc I class look like?

Will the (Learning) Circle be unbroken

I'm in Clemson for a couple of days, very much enjoying the 24th Annual Miniconference on Combinatorics and Algorithms. I'll be speaking tomorrow about a topic I've thoroughly enjoyed working on for several months now, with the help of one of this past year's REU students. Despite the research-heavy atmosphere I'm breathing down here in Tigerville, I've been reflecting deeply on teaching philosophy and practice in the last several hours.

You see, I've decided that I'd like to offer to lead a Learning Circle for Spring 2010, and I'd like to base it one of two books, both by the same author. I'm trying to decide between Alfie Kohn's No contest: the case against competition (1986) or the same author's collection of 1990s essays, What to look for in a classroom. The former I read a couple of years back when it came up in the reading I was doing to supplement the reading a Learning Circle I was taking part in (I don't remember which Circle it was now); the latter I'm reading now. (I took in about 50 pages as I whiled away a couple of hours this morning by the giant fountain at the center of Clemson's lovely campus.)

Both books are compelling and offer frank examinations of questionable classroom practices that leave our students in the lurch. I think both would make for wonderful multilogues between faculty and staff.

Which to choose?

While working my way through one of Kohn's more challenging essays in What to look for, I began to envision a way in which Calc I could be taught authentically and in a purely problem-based and student-centered manner. It might go something like this:

Day One. "Holy shit! Here's an economics problem we've got to solve! XYZ Megacorp, LLC needs to find all of the maxima on this profit curve!" [Shows students real data, with very messy messy that guesstimating by eyeballing a graph ain't gonna cut it.] "What's all that mean?" "Well, let me brief you on XYZ's portfolio..."

Day Two. "So, how are we going to find these maxima, y'all? Any suggestions?" "I dunno...those are the places where the graph goes up and then comes down..." "Yeah, if we could find the places where the slope goes up and then down..." "Well, how do you find slope?" "Uhhh..."

Day Three. "Okay, so we need to find slopes, right?" "Uh...yeah." "How?" "Uh..." [A timid voice, as yet unheard, comes from the back.] "It's rise over run...right?" "Cool...can you compute those?" "I guess. Let's try to get some formulas..."

Day Four. "So now we've started computing rises over runs. But so far they're giving us rises over runs on intervals, not at specific points. How's that going to help us find the exact point where the slopes go from increasing to decreasing?" "Uh..." "We've got some formulas we can work with..." [Another timid voice, this one from a different student.] "What if we take two points that are really close together?" "All right, what does that do to the formulas?" "Is that the right way to go?" "I don't know. What do you think?" "Uhhhhhh..."

Day Five. "Okay, so you've got these formulas...I give you two points...two points that are really close together, and you can give me the slope. Let's try it out for a bunch of different functions. Which ones would you like to try?" "How about just y = x?" "That's too easy, man...let's do y = x2." "And square root of x!" "Dude." [There is much scribbling and crushing of paper.]

And so forth.

This whole enterprise would be messy, sporadic, anxiety-inducing, and would require the patience of a boatload of Jobs to do it effectively. Moreover, I can't see it working with a class of more than about 12 students.

I wanna try it!

More later, to be sure. And I'll update you on the Learning Circle situation once I've figured out which Kohn I'd like to roll with.

Tuesday, October 20, 2009

The little gifts

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

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

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

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

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

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

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

It's the little gifts.

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

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

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

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

What little gifts will tomorrow bring?

Sunday, October 11, 2009

Fall Break fantasy

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

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

Audience? Math majors.

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

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

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

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

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

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

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

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

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

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

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

Saturday, October 10, 2009


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

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

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

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

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

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

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

Speculation, speculation: all speculation.

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

Saturday, October 03, 2009

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

Second things first, here are the verdicts:

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

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

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

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


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

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

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

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

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

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

We'll see how things go.

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

That's all I want for all of them.

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

Thursday, October 01, 2009

Liveblogging Newton v. Leibniz, Round 2

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

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

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

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

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


"Then who else?"

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

"Did you deny writing a letter to Leibniz?"

"I don't recall."

There are no further questions for Johann Bernoulli.

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

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

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

"Did this affect the price of paper?"

"It did. It made paper more expensive."

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

"That may be true."

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

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

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

"What is your relationship with Sir Isaac Newton?"

"We met in 1676, through Isaac Barrow."

"Yes, in 1669."

"What were the contents of that letter?"

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

"Did you copy this letter at all?"

"I don't believe so."

"Did you come into contact with Leibniz?"


"Did he see the letter?"

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

"Did he copy that letter?"

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

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

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

"No further questions."

Collins is dismissed.

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

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

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

"Were some of these letters rather cryptic?"

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

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

The witness is dismissed.

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

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

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

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

Oldenburg is dismissed from the stand.

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

"Were you in contact with Isaac Newton?"

"Indirectly, yes."

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

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

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


Cross-examination begins.

"What is the name of this book?"

"I believe it was his Principia."

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

"I don't know."

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

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

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

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

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

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

"Yes it is."

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

"Yes, he was."

"Is this not a conflict of interest?"


Johann Bernoulli is dismissed.

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

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

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

Cross-examination begins.

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

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

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

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

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

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

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

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

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

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

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

"No, but I published first."

Leibniz is dismissed, and Leibniz's attorneys rest.

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

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

Fin. We'll see what they say tomorrow!

Liveblogging Newton v. Leibniz, Round 1

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

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

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

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

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

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

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

"My brother, for one."

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

"Yes, it was common."

There is no cross-examination.

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

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

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

"Is he well-versed in calculus?"

"Yes, definitely."

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

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

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

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

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

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

"That is true."

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

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

"That is true."

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

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

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

No cross-examination at this time.

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

"I don't know."

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


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


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


"Can you point him out?"

"The man in the green."

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


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


"No further questions."

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

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

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

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

"It cross some borders."

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

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

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

["Objection!" "Sustained."]

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


"Might his motive have been a personal one?"

"That is possible."

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


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

"Well, I've read it all."

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

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

"No further questions."

9:57: Leibniz is called to the stand.

"Were you interested in math originally?"

"Not originally."

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


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


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

"No, it was all very open."

"Ideas were discussed in the open?"


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

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

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

"No, not really."

Cross-examination begins.

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

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

"Did this work contain work on calculus?"

"It had my notation for derivatives in it."

"Was it before or after 1666?"

"It was after."

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


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

"That is true."

"No further questions."

10:03: The court recesses for five minutes.

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

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

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

"I was good friends with both."

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

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

"No further questions."

Cross-examination begins.

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

"Newton's work."

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

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

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

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

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

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

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

"No further questions."

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

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

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

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

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

Cross-examination begins.

"You worked with tangents, right?"

"And geometric functions."

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

"I don't remember."

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

"I don't remember."

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

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

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

"It is possible."

"Did you lie to Newton?"

"I published behind Newton's back."

"No further questions."

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

"Yes, you could say that."

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

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

"When did you first meet Newton?"

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

"So you saw his talents early on?"

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

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


"When was that?"

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

Barrow is finally excused.

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

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


"But you did not understand his notation?"

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

"Did others understand his work?"

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

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

"It can happen."

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

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

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

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

"Was it called 'calculus' then?"


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

"No further questions."

There is no cross-examination.

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

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

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

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

"Yes, of course."

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

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

"That must have been hurtful."

"Yes, it was."

"You were knighted by the Queen, right?"


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

"Yes, it is."

Cross-examination begins.

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

"October 24, 1676."

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

"This is true."

[An excerpt is read.]

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

"But it was all based on my work."

"Do you have proof?"

"Collins and Barrow shared my work with others."

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

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

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


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

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

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

"I don't recall."

"Was your name on the solution?"

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

No further questions.

10:36: Closing arguments begin.

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

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

10:40: Court is adjourned.