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?

________________
* ...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.