Newton v. Leibniz, Section 1

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

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

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

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

Judge 1. Prosecution may call the first witness.

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

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

Newton's lead. Which work?

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

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

Oldenburg. I believe so.

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

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

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

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

Oldenburg. Yes.

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

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

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

Oldenburg. Yes.

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

Judge 1. Sustained.

Leibniz's second. That is all.

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

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

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

Collins. Yes.

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

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

Newton's lead. Sketchy.

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

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

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

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

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

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

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

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

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

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

[There is grumbling from the Leibniz people.]

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

Leibniz. That is all hearsay.

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

Leibniz. No.

Leibniz's lead. Objection!

Judge 1. Sustained.

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

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

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

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

[There is grumbling from the Newton bench.]

Judge 1. Any more witnesses for Newton?

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

Newton. With no doubt.

Newton's lead. When did you start work?

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

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

Newton. Yes.

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

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

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

Newton. There's no proof.

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

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

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

Newton. No response.

Leibniz's second. Nothing further.

Judge 1. Any further witnesses for Newton?

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

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

Newton's lead. What was he doing then?

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

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

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

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

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

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

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

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

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

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

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

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

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

Tschirnhaus. I didn't know know Newton well.

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

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

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

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

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

Tschirnhaus. Indeed.

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

Tschirnhaus. True.

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

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

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

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

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

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

Jacob Bernoulli. Yes.

Leibniz's second. No further questions.

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

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

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

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

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

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

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

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

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

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

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

Johann Bernoulli. In 1684, my friend and colleague found a method for solving certain differential equations. [He writes on the board: "dx n = n x^n-1."] This formula was given in Acta eruditorum before Newton had determined it, two years later.

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

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

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

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

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

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

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

Johann Bernoulli. So!

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

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

Judge 1. Sustained.

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

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

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

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

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

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

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

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

What's it worth to me?

19 hours, apparently.

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

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

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

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

Et cetera.

How've these issues come up?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Well, you're to say.

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

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

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

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

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

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

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

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

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

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

Hubbub

The room is full of noise, eight pairs of students hip-deep in peer-reviewing one another's drafts of their Senior Seminar written reports. I know some teachers are afraid of this sort of semi-structured clamor, but to me the din is marvelous: the class is boiling with activity, with life, with authentic knowledge-building. This is where ideas are born.

"I think that what concerned me about your project was this point, right here..."

"I'm not just saying this: I really want to read your paper when it's done. This topic is interesting, and you write about it really engagingly..."

"I can understand how you might read that sentence that way. I think what I was trying to say was..."

"I see what you're saying. I was trying to be really explicit, but..."

If you don't yet include peer review in your classes, start. Start now. I never cease to be amazed by the insightful comments my students offer to one another when primed prompted to do so.

Theorems don't have numbers

To tag a theorem is to label it an artifice, to suggest that it has no meaning beyond the cardinal place it occupies in one author or another's text. To number it is to catalogue it, to render it little more than a specimen or a reference point, against which some other theorem may be propped. To number it is to abridge it, to downgrade it, to curtail its conceptual power. To number it is to encourage its verbatim memorization, to make it impotent, to remove its powerful poisoned teeth.

Theorems in their natural state (in the great mathematical wild) roam unnumbered and numberless. They are ideas, notions, metaphors, all only marginally tamed, and tamed, if tamed at all, not by breaking them and beating them but instead by learning them well enough to leap upon their backs and let them take you to where it is their fellow theorems lie.

Theorems don't have numbers.

Don't ask!

There might not be such a thing as a "dumb question"...but there sure are questions better left unasked.

If by this point in your educational career the only thing you find the need to ask your professor* in class is "Is that going to be on the quiz tomorrow?," do us both a favor and don't bother asking questions.

I know many (most, I might hope!) of my students dream, and dream big, imagining the many wonderful things they'll be able to do with the knowledge they'll gain in their classes...even their classes which are sometimes more challenging than they might like them to be at the time.

I know that many of you are working hard to make sense of the tough, tough concepts we talk about day in and day out. (Tough they are: it took brilliant minds centuries to piece together the puzzles we're assembling and disassembling every day in class.)

I also know that some of you are here only because your parents (or your high school teachers, or your guidance counselor, or...) told you that it's the next step that you're expected to take in life: you're not here because you want to be; you're here because you're told to be.

Of those of you reading this who find yourselves in the last group, I might ask the following question (which might, after all, be better left unasked), and I might expect a serious, well-thought-out answer: Why are you here?

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* ...your professor who spends more time and effort than you can imagine in plotting a course replete with rich and authentic examples and opportunities for robust, hands-on engagement of central course concepts, for the benefit of students like you, who, I might add, very obviously (whether you know it or not, you're not very skilled at hiding your apathy, my young friend) couldn't give a rat's ass about what you're getting.**

________________
** ...bitter? Nahhh...

Audience is everything

As good writers know well, audience is everything.

Well, maybe not everything, but it goes a long way.

I've only just now realized a crucial (in fact, almost defining) characteristic of nearly every one of my teaching practices: in teaching my classes I try to take as my audience every single student in the classroom, from the strongest to the most-struggling. When I walk into class on a given day, I'm not teaching to only the top 10% of the class, the future superstars: I'm teaching to everyone.

Ramifications?

I typically teach slowly, and with frequent appeal to intuition, rather than formalism.

I typically teach via realistic (or at least authentic) examples and applications

I typically offer frequent and varied opportunities for feedback from, and dialogue with, students.

I try to recognize that not everyone is going to be intrinsically motivated to study what I study simply for the sake of studying it. (Thus I avoid falling into the trap that snares many well-meaning mathematicians, who assume their students appreciate math's unadorned and unapplied beauty.)

Hmm.

Insomnia sucks

Why am I still up?

I went to bed about three hours ago, but woke up worrying about the current shitstorm involving SGA and ILSOC. I got up to get a glass of water and hammer out a short list of talking points I'd like to address when I meet with the SGA Academic Affairs Committee's chair tomorrow afternoon. I want the points on this list said, not mis-said. I want to be clear and forthright, I want to be honest. I want no more bullshit. I want continued and ongoing discussion between our respective groups. That's all I want. I don't think it's too much to ask for.

I realized earlier this evening that I've gotten older and wiser, and concomitantly more pragmatic and less idealistic, than I once was: fifteen years ago when I was these students' age, I was just as impetuous and hot-headed, just as incapable of seeing things in anything other than black and white. I was just as committed to lofty, unrealistic and unattainable ideals. I was much more excited by storming the castle walls than I was by sitting in the boring committee meetings taking place in the castle's keep.

Case in point: it was much more exciting to deliver my valedictory address in high school than it was to serve as a student representative on the committee to hire a new principal for my high school (a position that was no doubt granted me on account of said valedictory address).

More context? My valedictory address wasn't the standard saccharine "here we all are now and now we're off to somewhere else to bigger and better things, but wasn't it fun, y'all?" It was essentially a scathing report on what I felt were shortcomings in the public educational system I had gone through. (It's worth noting that, knowing much more about the state of K-12 education in this country now than I did then, I feel even more strongly about some of those shortcomings now.)

After delivering this opus magnum to the assembled crowd of a thousand or so students, friends, and family, I was given the chance to serve on the committee I mentioned above. Not having been overly involved in many student organizations in high school (I am what I am, and I have no regrets, but I wish I'd been more involved back then), I was unaccustomed to committee work, and I found my day-or-two-long involvement with this hiring committee to be dull, dry, and uneventful.

However, as I realize now, it was a far more effective means of enacting change than delivering a rousing and rafter-raising speech to a bunch of pimply-faced teens and their parents. As a member of that committee, I was getting involved meaningfully in the institutional process; I had a role, and I had a voice.

There was an interesting parallel that took place yesterday: on my way from a conversation with a colleague who's visiting my department to study our program's successes, I walked past a walk-out sponsored by Students for a Democratic Society (yes, they still exist). The walk-out's organizers stood on a stage at the foot of the library steps, shouting slogans with which I agree ("education is a right, and not a privilege!") and calling for laudable goals ("Affordable educations! Reasonable demands on faculty!").

I couldn't stay and listen, though I would have liked to: I was on my way to the first of three discussion sessions whose purpose is to decide on our school's QEP (Quality Enhancement Plan...Number 7 in this sampler; you'll be seeing me write much more about it in the coming months). The QEP session was not fantastically well-attended. There were perhaps twenty people present for most of the session, and most of these (12 to 15) were students involved with SGA. I was happy to see them there, but I was chagrined that there weren't more faculty present.

About ten minutes into the session, we heard shouting outside in the halls: SDS had moved their protest to the student union.

Here's where the parallel begins: the folks who had assembled in order to help identify and reify what meaningful institutional change looks like, with the ultimate goal of enacting that change (and we will, because we must!) were getting drowned out by people shouting about their desire for change. I respect the point of view the SDS students were expressing, and for the most part I agree with it. I feel, however, that they could have accomplished more by joining us in our relatively stodgy and conventional discussion than by shouting in the halls.

Maybe I'm just getting old.

Meh. I'm going to have another crack at sleep. I'll see you on the sunny side.

Open it up

Ugh.

The conflagration which sprang up a couple of weeks ago between the representatives of ILSOC and the Student Government Association, and which was later checked (see this initial post, and this, more upbeat, one) has found new life, and I hope that calm and diplomacy will prevail.

Let me simply say I hope that all parties involved truly have the best interest of the students (and the campus community as a whole) at heart. I know that I do.

I wish everyone were as open as I am.

I realized this evening as I was wandering the aisles at Ingles, picking up ingredients for risotto and mojitos, that I've never really been afraid of opening myself up, professionally speaking. I've never feared showing my true intentions, I've never feared making my methods known, never feared that people might find fault and call me on it. I've always been up to dealing fairly and openly with others. (This blog, nearly 450 posts strong and personal as hell, is a living testament to that fearless openness. I want every one of my students and colleagues to know what it is I'm thinking as I enter into my dealings with them.) This was true even before I received tenure, and it's certainly true now that tenure has been granted to me.

And it puzzles me, and sometimes perplexes me, when others fail to offer the same openness.

As annoyed as I am with certain members of the SGA right now, in some ways I can understand their annoyance with me as well. I made a promise (of unrestrained openness) to them that I might not be able to keep (because it wasn't really my promise to make in the first place, I'm afraid), and in not keeping that promise I may have fed their perception that the faculty are not ultimately concerned with their well-being.

We do care, though. The current members of ILSOC are workaholics like me, accustomed to 60-plus-hour work weeks, unrewarding and thankless tasks which affect only incremental (and seldom truly meaningful) change, and the slow, slow inexorable grind of institutional change that takes years, if not decades, to accomplish. We do all of this on top of teaching, and I know personally that all of the current members of ILSOC are exemplary teachers who give their students their all, day in and day out. Their efforts are tireless, and their concern is real and unaffected.

We wouldn't do what we do, and for as little extrinsic reward as we do it, if we didn't care. And I hope that the students don't lost sight of that.

Okay, I can't think of anything coherent or meaningful to say to top that, so I'm off to bed. Tomorrow promises to be interesting...

What a difference a day makes

Today was much better than yesterday: much less stressful, much more fun.

I had a blast in all of my classes, working heavily with splines in both Linear and Calc I: it's lovely to find a project that's meaningful to both groups of students! The Linear students are learning how splines are actually constructed and investigating algorithms for building arbitrarily complicated cubic splines...the Calc I students too are going to be able to get in on the action when they look into the construction of some very simple quadratic splines in a week or so. It's all spiraled out of a question a Calc I student posed to me a week ago regarding the "interpolation" problem set I'd given them. How wonderful that such good ideas come from working with students!

Today I also managed to find something about which my second section students are delighted to talk: designing their own grading scales. Even the shyer students were happy to speak up when it came time to ask them whether quizzes should count for 5% or 10% of their overall grade. After ten or twelve minutes of debating the issue, both sections decided upon tentative weighting schemes for their grades:

Section 1
Homework: 35%
Quizzes: 5%
Projects: 25%
Midterms: 25% (total)
Final: 10%

Section 2
Homework: 20%
Quizzes: 10%
Projects: 25%
Midterms: 25% (total)
Final: 20%

The first section's scheme is closer to the one I would typically use (back when I assigned the weights myself), but the second section's scheme is within acceptable tolerance. I can hang.

We'll see how they work out; they were well-arrived-at (after a good deal of earnest give-and-take which considered amount work, locus of learning, revisability, ease, and so forth). I'm always impressed with the students' maturity when they're trusted to make decisions that affect them meaningfully. I'm glad that they don't often abuse the trust I give to them.

Yes, it's been a good day.

Before I call it a night, my thanks must go to my colleague Dolores (and her husband Ken), for driving all the way down from Virginia to give a lovely MATH 480 Senior Seminar talk on Catalan numbers. Thanks, Dolores! Very well received! We'll have to have you down again sometime soon.

Take two

My last post, in which I expressed a bit of anxiety over the quietness of the second section of my Calc I class, elicited a number of comments (on Facebook, sadly, and not on the blog post itself) from former students, most of whom insisted, more or less, that I'm worried about nothing much to worry about.

The gist of their comments is this: first-year students are first-year students. They're unsure of themselves, and they're scared of being wrong in front of one another. As one of my ex-students said, reflecting on his experiences during freshman year (in which he took Calc I and Calc II with me), "I didn't want to be wrong or make a mistake." Another (Linear Algebra, Fall 2006): "they're still stuck with a bit of that high school fear of judgement and embarrassment in class."

One of the upsides to the level of openness I cultivate in my classrooms is that I'm intensely aware of how all of my students are doing, and this awareness helps me to be sure I'm getting them everything they need to succeed.

On the other hand, one of the one of the downsides to the level of openness I cultivate in my classrooms is that I'm intensely aware of how all of my students are doing, making it very hard for me to leave someone behind if I sense they're struggling.

Sometimes, I've just got to move on.

This might be one of those times.

Be yourself

I can only be the person that I am.

I have to remember that.

Fresh off of a post in which I wax elegiacal over the stresslessness of grading in Calc I this semester, I find myself stressing out over the stultifying quiet of my two sections of that course. The morning section is a bit on the shy side, but they're coping and coming out of their shells a little bit. The afternoon session is borderline catatonic. I can't recall the last time I had a class this reluctant to speak up, to volunteer, to interact, to show any signs of life.

Let's break it down:

1. There are three or four students who have had calculus before and who are therefore pretty comfortable with the material but who (bless them!) hold off from blurting out the answers and volunteering to work something out on the board...at least not until someone else has had a chance. I have a feeling that these few folks, though, are beginning to grow self-conscious, since it's becoming apparent that they're about the only ones who are volunteering themselves at all. (Notably, these folks are mostly male.)

2. There are another seven or eight students (most of whom are female) who clearly know what's going on most of the time but who are painfully shy about it. Generally they telegraph signs of understanding to me (smiles, nods, even a little laughter), indicating that they're grasping what's going on...and they almost unfailingly come to the right conclusion on any in-class exercise we perform (and typically well before anyone else in the class), but they're beyond reluctant when it comes to sharing their ideas, even when those ideas are spot on.

3. There are about ten more folks who may not get at the right answer right away, but are happy to actively work to get at it, and these folks work really well with one another on group activities. Here they'll speak up, and they'll share ideas. There are signs of life, but not very vivid ones. I'm not particularly concerned about these people, as quiet as they are. (These people are of mixed gender.)

4. The remaining ten or so (also of mixed gender)...I just can't read. They neither volunteer to help out at the board nor interact much in groups. They let themselves get moved forward on in-class activities, but I can't tell if they're moving along with understanding or if they're only being pushed forward by their friends. As big as the class is (34 students right now) I don't have enough time to circulate around the room and police every moment of every group interaction to see how they're faring on a day-to-day basis. Truth be told, I'm worried about them.

And overall, I'm frustrated. I'm trying not to be; I'm trying to remember that just as I can only be the person that I am, so it goes for these students as well: many of them are just (a) shy, (b) uncertain, (c) and timorous (if not terrified) when it comes to math. It's the perfect storm of disaffected students. While I've always prided myself on being able to instill self-confidence and self-assertion, even in the most mathphobic of my students, and I've always prided myself on being able to bring students out from their shells, to help them become more outspoken, engaged, and involved...have I met my match in this class? Is this nut just too tough to crack?

I'm going to write notes to a few of the students I'm most worried about tonight, just to see if they can help me to figure out just what it is I can do to help them out. I'm also going to have a brief conversation in class tomorrow about what I'm hoping to see in the second half of the semester...and about what they hope to get out of it.

I want to make it work for all of us. Please help me to do that.

Thoughts?

Stresslessness

Deciding no longer to grade textbook problems in Calc I is one of the best decisions I've ever made.

Why?

1. Textbook problems (even those which are more "conceptual") are largely rote and computational; students get little real understanding from them. There's something to be said for the mechanical fluency to be gained from chugging through a few dozen such exercises, so I keep assigning them as "recommended practice." The students are far better off working through the more carefully-designed (though harder-to-grade) conceptual problems I write myself. Though solving the problems is a struggle, the students are wise enough to know that it's a worthwhile struggle. (Saith one of them at the end of his response to this week's problem set, in which the students were asked to ply their calc skills to craft a reasonable interpolative model: "I loved this problem! I was thinking to myself how I might be able to make an equation for reality as a whole -- I believe this problem begins to open the door...this is exactly the stuff I came to school to understand!")

2. Unable to simply look up the homework problems' answers to be found in the solutions manuals in the Math Lab, the students have to give legitimate attempts at their own solutions. Therefore they (even the strongest students) are likely to make more mistakes, but they'll learn from making those mistakes. I'd rather have a stack of 10/15s in which the students are struggling, straining, and coming very near (but just short of) the target than a stack of 15/15s containing nothing but look-alike plasticky responses.

3. Grading the homework is more fun! This is in part because of #2: I'm not forced to read through a few dozen halfheartedly (and poorly) transcribed solutions manual responses. It's also in part because the problems I'm posing to the students are open-ended enough to elicit thoughtful and creative responses from the students. They'll often come up with ideas I hadn't thought of, and I learn from them as much as they learn from me. They're clever, these kids.

Grading is a labor of love, but it's a lot more fun (and stress-free) this term than it's been in a long, long time.

Latest comment in the Calc I "suggestions box"

This morning brought the third "suggestion" in the envelope hanging from the bulletin board outside my office: "Make it easier! :)"

That is all.

Oh, and in completely unrelated news, the broad area of our QEP was announced two nights ago (I'm amazed that I haven't commented on it yet): "Undergraduate experiences that foster the use of open inquiry, critical thinking, creative expression, and effective communication."

At this point the area is simply meaninglessly broad. I look forward to its focusing throughout the next several months.

The importance of being earnest

Doomsday averted.

This morning's Writing Intensive Subcommittee meeting was wonderfully productive (we plowed through a TON of tasks), and the pre-meeting conversation I had with the Student Government Association (SGA) rep who's been in conversation with me for the past few weeks (let's call him Kenyon) was even more productive.

I think we're on the same page now. I had a bit of a "come to Jesus meetin'" with this young man, and we were both very honest and open about our concerns. I realized early on that he's more the messenger than the source, and that his intentions are good ones. "We have to be open with each other," I insisted. "We can't sneak around, we can't take part in romantic revolutions are crusades. We have a number of common concerns, I'm sure, and we can work on them together...if we choose to talk to one another about them." He agreed.

He sat in on our meeting, and though I think he's got a thing or two to learn about note-taking, I admire him for following fairly well the course of a rather convoluted proceedings. We hit everything under the sun, from the minutiae of WI proposal wording to the philosophical underpinnings of the WI mission itself...and faculty development and assessment in between. A real trouper!

We agreed by the end of the meeting that it would be worthwhile to establish some sort of "liaisonship" between SGA and ILSOC (or some of the willing subcommittees thereof) in order to open, maintain, and benefit from a dialogue between faculty and students on issues pertaining to ILS and other academic affairs which affect us all. He invited me to attend this evening's meeting of the Academic Affairs Committee of SGA (on which he serves, and for which he's been running his little end-runs).

So I went. I'm glad that I did.

The meeting, held in the SGA offices in the student union, was attended by six members of SGA and me. It was unassuming and informal, as Kenyon assured me it would be. The students took turns reporting on their progress on their individual "homework assignments" from the previous week. One had been sent to data-mine various sets of statistics concerning the ILS Clusters, in the hopes of finding correlation between students' choice of topical clusters and their majors. (Undoubtedly such correlation exists...and as it happens this is one of the students' primary concerns, to which I'll return in a bit.) Another reported on the SACS (Southern Association of Colleges and Schools, our accreditation agency) meeting he had attended. I commended him for his ability to bust out all of the buzzwords.

Kenyon discussed his meeting with me and his attendance of the WI meeting, and I took a moment to explain our feelings about establishing a "liaisonship" between SGA and ILSOC. I'm not sure that everyone at the table was sold about the efficacy of establishing such a dialogue, but one of the student leaders (the Vice President, Samantha), was totally on board. She pleaded vigorously and eloquently for our case, and I'm glad that she did. I think her advocacy helped the case considerably.

As the meeting went on I became a little more annoyed by the continued reluctance of the students simply to come out and say what concerns they were having, and it soon came to the fore why it is they have this reluctance. Historically, it appears, every time they've brought complaints to faculty, they've either (a) been blown off or (b) been told that they need "data" to back up their claims.

So they've been gathering data.

I let them know that they were likely talking to the wrong faculty back then, but that they should feel safe in talking to me and to the other members of ILSOC. At this time, ILSOC consists of very active, engaged, and motivated faculty who are heavily invested in meaningful interdisciplinary learning and fully committed to the spirit of the ILS program. They won't need to sell their story; we've bought it already, and we're eager to talk. I assured the students that we are the people they need to talk to. Samantha was resold, and once again Kenyon put himself forward as a liaison.

After the meeting I lingered a bit and talked some more with Samantha. At long last I learned a bit more about their specific grievances. For instance, it became evident that their primary concern with the LSIC Colloquia is strongly related to our own: uniformity of the quality of instruction across all colloquia. Knowing this, we can talk about it openly and brainstorm ideas together.

It also became evident that the students' primary concern with ILS Clusters is that students aren't getting enough incentive to be daring in their choice of clusters: because they're so pressed to finish their majors with the number of credit hours they're given (lest they pay egregious overage charges), they find they have to select topical clusters focused on topics cognate to their majors. These students would like to see greater incentives (more flexible rules for "double-dipping" courses? Elimination or lowering of overage fees?) offered to students to try out clusters more distant from the safety of their majors, thereby engaging in a richer interdisciplinary learning experience.

Honestly, this perspective is far more mature than that of many members of the faculty, who simply want to scrap the clusters altogether. (Admittedly, this contingent of curmudgeonly academic extremists is getting smaller each year, as the fogyish stalwarts retire one by one.) I was thoroughly impressed with the students' position, and I told them so. We can definitely work with them on this.

Ultimately, as I said above, I'm glad that I went to the meeting. A lot was accomplished, and I'm excited to see the directions in which this heads.

Before I go I should mention one more (not wholly unrelated) incident. Late this afternoon, while sitting in my office, I overheard a couple of my Linear students (sitting in the near end of the Math Lab) quietly voicing their frustrations about not being able to keep up in class because of the way we'd plowed through the definition and derivation of eigenvalues and eigenvectors. I guess we'd been moving a bit too fast. Although I felt a bit uneasy about admitting to my eavesdropping, I sneaked across the hall and joined in the conversation.

"You've got to let me know," I told them, "if you're having trouble with something."

"I'm just not one of those people who can pick it up really fast," one of them said. "I have to think about it and let it sink in before I understand it."

Although at first the conversation was a bit strained and awkward (I had been eavesdropping, after all), after a bit it warmed up. I agreed to keep tabs on the pace, and to throw in a few more examples and explanations here when needed. They agreed to let me know if things get moving too quickly again.

"I understand that you can't change the way you teach the class for just one person," the more outspoken student said.

"True," I admited. "That's what makes it hard, hearing, as I do, from all of you all of the time. It's awfully hard to teach a course at any sort of pace when I don't want to leave anyone behind. On the other hand, though, I can take your perspective into account and use it, along with everyone else's, to come up with a sort of 'normalized' perspective. If I only hear from the people who are chugging on ahead, I can't help but think everyone's all right with the way things are going. I need to hear from folks like you."

I know it took courage for them to have that conversation with me, and I'm really impressed with that courage. I told them how much I appreciated their earnestness and forthrightness. I think that conversation, a difficult one for both sides, was a fruitful one.

Students, please remember this: there's no shame in taking a little more time to learn something than some of your peers. It's okay to be confused. If anything, there's shame in not owning up to your confusion in the first place.

I guess the moral of the story (by now a twice-told tale) is: if you've got a tale to tell, tell it. Someone will be willing to listen.

How adults do it

I'm pissed.

One of my colleagues on ILS (Integrative Liberal Studies) Oversight Committee with me met with a student representative from SGA (the Student Government Association) about two weeks ago. He met with us because he "had some questions about the clusters [topically clustered courses] as they're implemented" as part of the ILS program. Lexi (my ILSOC colleague) and I left the roughly half-hour meeting suspecting that the kid was looking for some dirt.

Since then it's become apparent, from the minutes of their meetings and various other campus goings-on, that SGA's sending its student representatives to a number of faculty and staff on campus trying to get information on the following components of the ILS program: Humanities, the LSIC colloquia, the ILS Clusters, and the Writing and Diversity Intensive programs. The minutes I've read make it clear that the students are hoping to bring about change in these components of the program. The minutes don't make clear exactly what issue SGA has with these components.

This is what's pissing me off.

The child Lexi and I met with (I'm sorry for the condescension, but that's how he acted, and not at all like adult he was trying to be) had an obligation to speak up and be forthcoming about his purpose for meeting with us. He had an obligation to let us know that the student body has concerns about various aspects of the ILS program, and that they would like to dialogue with us about possible ways to ameliorate any shortcomings they think need addressing.

He was not at all forthcoming about his motives, even as Lexi and I pointed the way to every bit of information about the ILS program that SGA could ever want to read...even as I invited him to the next meeting of the WI Subcommittee (we meet tomorrow morning; I suspect I'll have something more to say after that meeting) and said, in nearly these words precisely: "we have nothing to hide. Every step of the process is transparent. I hope that you'll take a look."

Apparently there is to be no return of the favor.

Pardon my French, but let's cut the juvenile, passive-aggressive cloak 'n' dagger bullshit. You're adults now...or at least that's how we'd prefer to deal with you. Institutional change is not best affected by sneaking around, "gathering evidence," and dramatically confronting your opponent with that evidence in a highly public place. Rather, such change is affected by openness, plain-dealing, and compromise. If you have a problem with the system as it stands, tell us. Tell us, so that we may meet and discuss it. So that we may meet and hash out a plan to deal with your concerns. So that we may make our concerns and motives plainer to you, in the hope that you can see our points of view as we begin to see yours.

Why all of the secrecy and sneaking around? Maybe they're suffering from a romanticized notion of "fighting the good fight" or "leading a revolution"? Committee meetings might be boring as hell, and change at the university level may be glacial in its slowness, but the compromises hammered out in such meetings are far more lasting than ramrodded fiats and ultimatums.

Look, I have problems with the ILS program. It's not perfect, and I can admit that. It's a system designed by dozens of people, overseen by other dozens, and implemented by literally hundreds. Though I feel that for the most part it's working quite well (better than comparable programs at other universities), it's got its flaws.

What do I do about those flaws? I meet with my colleagues, I brainstorm ideas of ways we might address those flaws, I work with my colleagues to workshop a few of those ideas into more robust plans of action, and I help to implement those plans. In other words, I work with everyone else on campus together in order to fix the flaws I and others might see in the system.

That's how adults do it.

To be continued, I'm sure.

Philosophy 101

Well, it's that time again...for one reason or another, I've found it necessary to update my "teaching philosophy," that nebulous document no one's really sure how to write.

I find that the older I get, the less patience I have for philosophies which read like litanies of pedagogical tricks, no matter how clever those tricks are. I see no reason anymore to brag about "use of technology" or "co-curricular activities" or even "inquiry-based learning" in my statement of teaching philosophy. It's not the place.

For what it's worth (I've nothing to hide), here's my current philosophy, version 2010.1.2 (or thereabouts):

My philosophy of teaching, like my teaching itself, has undergone many changes in the years past. Like those of many novice teachers, my earliest philosophies were tailor-made to fit one or another job description and often relied on catch-phrases like “use of technology” and “collaborative learning.” Embarrassingly recently my teaching philosophy read like a behaviorist’s manifesto, a long list of actions typically taken by me or by my students, actions which merely indicated a certain philosophy at work without getting at the heart of that philosophy. As I’ve grown as a teacher (and scholar of teaching and learning) I’ve been better able to tease out from those actions their essential qualities in order to understand why it is I do what I do, and what it is I hope my students will do with me when we work together in and outside of the classroom.

As a result my philosophy has become more streamlined and systematic, and while it is on its face less “practical,” it still has profound practical implications when put into action. It is no longer so describable by a few pages filled with phrases like “co-curricular activities,” “writing-to-learn,” or even the loftier “inquiry-based learning.” Though all of these feature prominently in my teaching, none is the prime mover, none is the ultimate reason why I do what I do.

If not these things, then what is it that guides my teaching?

My primary goal is to address my students’ affective needs as well as their cognitive ones. Put simply, I believe, and the literature on teaching and learning bears me out, that how my students feel about what they do is as important as what they do in the first place. Students who feel confident about their abilities will pursue greater challenges and aspire toward greater goals than students who lack that confidence. Moreover, confident students will strive toward their goals far more effectively than will unconfident ones. Therefore, instilling a sense of security and confidence in my classes, and in all other interactions with my students, is of paramount importance.

To help my students feel secure and confident, I aim to create a safe learning environment characterized by openness, honesty, and friendliness. Such an environment cannot help but lead to a shared sense of respect and understanding. Such an environment relies on a commitment to clarity and transparency, and this I keep in place by maintaining open lines of communication. I go to great lengths to make sure that my students are always able to get in touch with me in a timely manner, and that the concerns they raise in their correspondence will be met with legitimate concern and care. In this way open communication fosters a deep sense of mutual trust.

Once I have my students’ trust, and once they are convinced that they have mine, we can work more effectively together. But working effectively requires that we have a shared sense of purpose, and I cannot presuppose that my students will come to class with the same purpose I will. Some work is needed on all of our parts to align our purposes. I spend a great deal of time early in the semester learning as much as I can about my students and their academic and life goals, so that I can better make the case that what we will learn together will be useful to them: every aspect of every subject I teach I try to imbue with relevance and applicability. Here my aim is to instill in my students an intrinsic desire to learn, rather than an extrinsic one, for their learning experience will not fail to be a richer one if they see how what we study is inherently useful to them. Put another way, it’s better that my students see how useful a subject is, in actual practice, than that they merely be told of its usefulness, in theory.

Once students are intrinsically motivated to learn, it’s up to me to place before them challenging opportunities for deep learning. These opportunities are often driven and directed by the students themselves. While it is difficult to characterize broadly the activities in which my students take part, they are as a rule

• active and not passive,
• guided by discovery and not prescription,
• concept-driven and not computational, and
• authentic (that is, "real-world") and not artificial.

In every one of my courses, from the first day of class, my students do rather than see. They are encouraged to cooperate and collaborate, and competition of every sort (including for grades) is minimized. I prod them to be skeptical and to ask probative questions, like “why should I care?” and “why is that true?,” and I encourage them to answer these questions themselves before looking to me for a response. With a bit of practice, they end up learning more from each another than they learn from me.

If I am successful in my efforts, my students soon become (often very literally!) the authors of their own knowledge. They allow themselves to become the experts and are no longer beholden to an intermediary who stands between them and their engagement of new ideas.

Of course, no two students are alike: some develop more quickly than others, some are more or less astute, observant, or mathematically apt. Moreover, students at varying stages in their academic careers exhibit a broad variation in maturity and intellectual development. When put into practice, my philosophy must take these variations into account, and I do this by adopting a sort of “dialectical” approach to teaching, engaging in frequent conversations with my students about the pace with which we proceed and the direction in which we travel. What do my students need from me, as individuals and as a class, this week, on this day, at this moment? I can never plan more than a class or two in advance, knowing that on any given day we might linger longer than I’d anticipated on a surprisingly challenging concept, or that an interesting conversation will spiral outward into an engaging and enlightening example.

For the same reason, no two iterations of the same course will look at all alike, and no amount of experience or preparation will fully ready me for the next time I teach a course. Herein is the true challenge my philosophy must face: meaningful teaching is time-consuming and work-intensive. It requires constant vigilance and refinement, because midcourse adjustments are almost unending. It requires humility, and a rather thick skin, because mistakes are often made, and it’s often hard to not take them personally. Finally, it requires seemingly limitless patience and flexibility, because to teach well I must be ready to work effectively with every sort of learner I can imagine…and a few I cannot.

Why work so hard?

I can think of no better way to affect the world in a positive manner than to teach, and to teaching meaningfully. I can think of no better way to spend my time. Indeed, I feel blessed that I get paid to do something which I do well and which I love to do anyway. When I reflect upon my experiences with my students, I realize that I truly am one of the luckiest people on Earth.

Workshop Idea #483 (not really)

I promised that I'd soon share the faculty development workshop idea which came to me during the Tuesday evening session at this year's CWPA.

Scene: lunch. We're at the outset of a half-day workshop featuring a short keynote speech. The speaker finishes her half-hour spiel, the congregants put down their half-empty bags of chips and wipe their mayo-covered hands awkwardly on their blue jeans or tablecloths.

"All right, everyone, groups of three." At each table the six people sitting there split into two groups. While this goes on one of the organizers walks about the room with a bucket full of folded pages. One at a time one person from each group of three picks a page from the bucket. On it is given a writing-related scenario of some sort, a short case study.

"At the end of the final stage of a multistage assignment three students come to you and tell you they can't finish their final draft because etc. ..."

"Your two colleagues who are helping you complete a reading of your department's senior portfolios fall into a heated argument about an apparently irreconcilable difference in grading philosophy. They ask you to mediate etc. ..."

I'm not claiming to have a dozen of these at the front of my mind; a book like Chris Anson's WAC Casebook would make a great source for these scenarios.

Each team now has half an hour to come up with a proper response to the issue raised in their scenario. At the end of that half-hour, the teams will take turns, using five minutes to explain and interpret their scenario (or even to act it out!), and another five to resolve it. Each resolution will be followed by a brief discussion, and after four or five resolutions there will be a break during which those who've not yet presented may reflect on what they plan to say.

Could be fun, relevant, and meaningful, all at once.

I'd like to do this, I think.

Bragging on my students is a full-time job

Before I scooted off to the 2010 Carolinas Writing Program Administrators conference (also known as "the best damned conference on the planet") at Wildacres the past few days I entrusted my Linear students with a task to perform in my absence on Monday, and all signs show that they pulled it off wonderfully.

Their job was to meet as a class, brainstorm topics from linear algebra which would be placed under broader "general headings" (also brainstormed), and assign themselves, in whatever way they felt effective and appropriate, to teams each of which would be tasked with writing a "section" of a review manual on one of the general headings decided upon earlier.

Apparently Ino, never one to let chaos cramp her style, enlisted Iris's help in leading the class through the brainstorming exercise, and they had it all done in short order. She later e-mailed me the list of headings, each with three or four students assigned to it.

I'm delighted that I was able to trust my students to meet without me and get the job done, and I'd like to think that the effort they showed in finishing this job was payback to me for showing them that trust in the first place. I think there's something to be said for the benefits that mutual trust can bestow both on teacher and on students. I trust this particular batch of students fully. They're great.

As I hinted above, I got back in town early this morning (in plenty of time for my 8:00 a.m. Calc I class) from CWPA. As ever, it was a fantastically productive experience. This year's get-together wisely eschewed rigid structure, offering instead ample opportunities for participants to meet in whatever groups and subgroups they felt they needed to.

I spent Tuesday morning talking assessment with folks from Elon, Mars Hill, and Charleston Southern. Our conversation helped me to tease out the issue at the root of what was puzzling me most about our current writing-intensive assessment plan. Namely, what do we do with the assessment data once we've got them? There must be more in store for them than a place in a forgotten file cabinet or a SACS reviewer's dossier. Yet for assessment data to mean much more, they must be highly esteemed by the faculty to whom they're given.

I realized about halfway through the day that faculty need to be helped to see the intrinsic value of both the assessment data and the assessment process itself. If faculty can learn to see assessment as a formative, reflective, and dialogic process involving many inter-interested parties, rather than as a summative, unidirectional, and punitive process instituted from above, they may come to understand its usefulness. As I said to a few of my colleagues at CWPA, it would be great if we could get them to the point where they'd say, "hot damn, data!"

I can dream, can't I?

Although I'd like to think that faculty development workshops can solve all problems, it would be great to come up with a more exciting means of changing faculty perceptions regarding assessment...any thoughts?

Speaking of workshops, I've got a great idea for what I think could be a fun faculty writing workshop, but I'll save that for another post. Now, it's bedtime: tomorrow's another long one.

Live, from Wildacres, it's 2010's CWPA!

Yes, it's that time of year again, folks. I find myself up in the woods-covered mountains of Western North Carolina, spittin' distance from the Blue Ridge Parkway, hanging out with a gaggle of composition theorists and rhetoricians with only nominal control of their drinking impulses.

Seriously, these are wonderful people, and as was the case with the previous two years of this shindig (my first time was in 2008), by the dawn of the first full day (now) I've already had a dozen wonderful and insightful conversations. I've gotten a few pointers for my book from the point of view of my friends in writing centers and first-year composition programs, having asked several of them pointedly "so...what would you want to get out of such a book?" I've gotten a few nibbles of interest for the poetry conference, including a few from the writers' workshop that's going on across from us in the other lodge. Mostly, though, I've been talking assessment. (w00t.)

Assessment is the theme of this year's get-together, and we started things off last night with a keynote presentation from none other than Chris Anson, writing assessor extraordinaire. Of course, ever in teaching mode, all through the conversation he instigated I thought not only of the programmatic assessment we're undertaking with the Writing Intensive program but also my own assessment of my students' performance. Am I assessing what I claim to value as learning outcomes for my courses? Am I applying suitable methods in order to help my students achieve those outcomes, both at the micro (assignment) and macro (course) levels? And are my outcomes measurable, reasonable, and meaningful ones in the first place?

I think that the answer to all three of those questions, fortunately, is "yes." I feel confident that I'm doing the right thing, more or less, by this point in my career.

But I could be doing better, and after last night's conversations I am more firmly convinced than ever before that portfolios are the right way to go.

I was complaining to my colleagues Cammie and Nico (both of whom teach rhetoric at Western Carolina University) about how in mathematics assessment of student mastery is all too-often tied to completion of a particular course with a suitably high grade, with little behind that grade other than similarly high performance on exams and quizzes which essentially test rote memorization and unthinking application of various formulas and algorithms. For context: this followed a rather lengthy conversation stemming from Chris's presentation that began with his assertion that it was difficult to assess a student's knowledge of the works of Shakespeare by asking whether the student got a B or better in a course on Shakespeare's writing. Nora (from UNC-Charlotte) countered that such a measure could be an effective one, depending on what exactly you were measuring: it all depends on how it is that B was arrived at.

It was interesting to note that during that earlier conversation Cammie and Nico were musing about how nice it must be in mathematics, where quantifiable outcomes lie so thick you can't but trip over them.

No, portfolios are the way to go. I'm already so repulsed by assigning numerical values to single iterations of students' work that I don't know how much longer I can continue to do it. It almost made me physically ill yesterday to put numbers on the "final" drafts of my Linear students' papers on the geometry of linear systems.

What needs to be done next, if I plan on taking big steps in that direction? I need to convince my students that it's worthwhile and doable (I don't think this will be too hard a sell). I need to firm up each course's learning outcomes (which I already have for most of them) so that they're clear enough to explain to students and solid enough to be measurable. I need to make sure every assignment or activity I craft is explicit in its intentions (I'm already doing this). I need to more cleanly codify the way in which the students' portfolios would be put together, added to, and ultimately assessed.

As I mentioned in the previous paragraph, much of this I've already done. I just have to be more purposeful about it, take a deep breath, and jump.

Okay, the first bell rang about ten minutes ago...I should get ready for breakfast.

More to come!

No regrets

I've said a lot lately about the way Calc I has been going this term, and I've said relatively little about Linear, perhaps because I feel that course has felt fewer obstacles along the way so far. I honestly feel that Linear has been going more smoothly than just about any course I've ever taught. (Fall 2006 Calc II and Fall 2009 Foundations are possible exceptions.) And I'm having a blast in it.

What's made it work so well? The high quality of the students, their outgoing nature, their friendliness, their willingness (nay, eagerness) to work together both in and outside of class...and, I'll own up to it, the course plan I've laid out is working very well.

I'm never planning too far ahead in that course. Rather, I'm responding to the way the students handle each new activity I give for them. If they need more time, we slow down; if they're bored, we speed up. More importantly, perhaps, no activity follows another without a reason for doing so. We introduced inverses because we needed them to solve a particular problem, and we introduced the determinant of a 2 x 2 matrix for the same reason. We defined matrix multiplication the way we did because it made sense to do so, not because the textbook told us to.

Moreover, I've avoided technicalities where I feel those technicalities tend to swamp out understanding and intuition. For instance, without knowing it, per se, the students have now worked with bases, matrix linearity and singularity, and Markov processes, generally without explicit mention of those terms. They don't yet know what a vector space is, nor a linear transformation, yet they do know how to apply the techniques of linear algebra to solve nontrivial problems in graph theory and geometry, and they have robust intuitive understanding of those problems, as well as the nature of linear equations and their solutions. I remain convinced that now, as we're finally getting around to proving conditions for singularity of a matrix (still without using that term), the students' understanding of those conditions is so much deeper than would be the understanding of a typical student by this point in the semester.

I do not regret the emphases I've chosen to give in this class. I hate to brag, but I've got to say that though we've not "covered" a number of the terms and techniques (for everyone's sake, do not focus your attention on the mechanics of row-reduction and matrix inversion for two or three weeks, people!), I'd bet that the students have a much richer understanding of linear algebra than would students in most Linear courses by this point in the term, and I'd also be willing to bet that that understanding will last, too, and not disappear immediately after this semester's over.

Any takers?

Mystery...solved?

I have a conjectural hypothesis regarding the stronger overall performance by my 8:00 a.m. section. It hit me this morning as I was loitering in the Math Lab about a half hour after class.

Around 9:30 or so, there were seven or eight members of that morning section hanging out in the Math Lab. Several of them stuck around for a couple of hours. This was by no means exceptional behavior: several of them (some of the strongest students in the class, in fact) often spend a couple of hours in the Math Lab after class. Almost every day.

My hypothesis is that after they finish class at 8:00 a.m., many of the students simply have nothing else to do for the next few hours, so, having already dragged themselves out of bed, they figure they might as well get some work done. To the Math Lab they go!

There could be something to this...

Clarification

As I was falling asleep at faaaaaaaaar too late an hour last night (technically this morning...aaaaah, bowling night!), I felt a twinge of anxiety over something I'd written in my blog yesterday. I want to speak briefly to that something.

I realized that both in class yesterday (twice!) and then in my last post here I really gave it to that one response: the "0, 1" answer to the homework question on last week's problem sets. True, the answer is a pretty awful one, but what I should have made clear is that though the answer is awful, that says nothing about the talent or intelligence of its author.

I realize that most of you (my Calc I kiddoes) have probably never been asked to write in complete sentences in mathematics before, and most of you are trained to think of "writing" and "math" as two ends of an academic spectrum. (I'm addressing these issues in Chapter 2 of my book.) Therefore I understand that there's a good deal of inertia you're trying to overcome as you're moving toward writing more robust responses to my homework questions.

I appreciate all that you're doing to get things moving in the right direction. All of you, from the most experienced "math writers" to the least, are intelligent people, and I want to make sure you all know that there's a difference between a lousy answer and a lousy mathematician. We all give the former from time to time, but that doesn't mean that any one of us is the latter.

See you in class!