You complete me

I had a blast in today's Calc I classes: students were engaged and alert, and eager to follow up on some of the missteps made in The Clock Problem. The few students I've talked to who missed the boat on that project were more than happy to make up for it and grateful for being given a chance to revise and resubmit. I look forward to seeing their recovery!

I spent a few minutes at the start of both sections drawing a big fat ol' line between the following two solutions I received to the problem "Find the number of local maxima and local minima the function f(x) = |x| has":

1. "The function f(x) = |x| has no local maxima and one local minimum, at x = 0."

2. "0, 1."

Though technically correct, the second response is dramatically inferior to the first: the first makes sense even when taken completely out of context. More than simply correct, it is complete and perfectly composed. It could appear in a textbook, expressing as it does, in a clear and unambiguous fashion, a true mathematical fact. One doesn't need to know that it was written in response to a homework question, for its truth transcends that humble context. From a practical standpoint, this solution could be used as a useful study tool later on, even without the question which prompted it.

The second solution, on the other hand is practically meaningless.

I asked the students whether they would dare to hand an assignment into their instructor of their LANG 120 (first-year composition course) which did not entirely comprise complete sentences. Predictably, they shook their heads.

So why shouldn't they pay the same respect to their math instructor? After all, aren't they using writing for the same purpose here, namely, to convince, and to communicate?

Moreover, it hit me this evening what the most profound philosophical difference between the two solutions above is: while there's no "movement" in the second solution, the first moves students a great distance, from being consumers, merely reacting to a stimulus given to them by their instructor, to being authors of their own truth and knowledge. The simple act of creating a complete sentence (expressing a concomitantly complete thought) transforms the student's role from a passive one to an active one.

I'm looking forward to tomorrow. I feel like this semester is finally getting into a groove. Stay tuned!

Rediscovery

I'm about 10 hours into an estimated 15 total hours of grading this weekend, and I'd like to pause to make the following observation (I know I've made it in this blog before, but I can't seem to find it in an earlier post), which I rediscover every time I teach Calc I in a Fall semester:

Almost without exception, an older student (a junior or senior, or a nontraditional student), even one who is not a math major or even particularly mathematically inclined, will outperform even the brightest and most math-literate first-year student.

Why?

It's simple: lengthier life experience, superior time management, and stronger study skills.

By the time they've finished a year or two of college, or after they've had several years of real-world experience, students have (by necessity) developed means of managing their time. Some (most?) of my nontraditional students are holding down 20-to-40-hour-a-week jobs, taking care of families, sometimes large and extended ones, and somehow keeping up with one or more difficult college courses. (To my freshmen: if you think you're busy, think again. As busy as you feel you are, I'll tell you this: you're not. You'll find that out in a year or two, trust me.) To do all of this they've got to have some way of budgeting their time, and they've got to be serious about their studies.

They do, and they are.

This semester's no exception: I've got several nontraditional students in my morning section of Calc I, and they're performing wonderfully. Not only are they doing well on all of the assignments, but they're clearly grasping the concepts and challenging themselves to really learn them rather than simply memorize a couple of formulas and move on. I'm immensely proud of them all.

I'd like to give a little speech to my first-year students (I know a few of you are reading this blog, and I hope you'll take heed...and I hope you'll tell your first-year friends what I've said).

Ahem...

...On average, the "big kids" are doing much better than you are. This isn't surprising, because it's nothing new: they always do. It isn't because they're smarter than you are. (They're smart, but you're just as smart.) It isn't because they've seen this stuff before. (Generally speaking, you all have more, and more recent, experience with calculus than they do.)

It's because they know that it takes a little work to do well, and that it takes a little dedication. That it takes a desire to actually learn rather than to simply fill a seat and get a grade. That it means taking an active rather than a passive role in their own educations. That it means managing your time, starting early, and earnestly seeking help when it's needed (by and large they do; by and large you don't). That it means working for a few hours sometimes when you'd rather be playing instead.

They believe me when I say I want to see complete sentences, when I say I want to see your work, and when I say that radicals are better than decimals. And they give me those sentences, that work, and those radicals, not because I asked for them but because they realize why I asked for them in the first place. They produce homework and papers worth reading, worth keeping, worth studying from, worth not throwing away or losing under your bed.

I'm not saying all of this to make you feel bad, but to point out that no matter how well you're doing you can do better, and I hope that you do.

You can get started sooner on the homework, and you can take care to produce nicer drafts. You can take the lead in groups activities, and you can dare to ask and answer questions in class. (Do you notice who's doing most of this right now?) You can take the time to write in complete sentences, to double-check your graphs, and to use your notation clearly and consistently.

You can do better. You can do it. Give it a shot. I'll always be here to help you, and I'll be cheering you on every step of the way. I've got your back. Let's do it.

What a difference a day makes

A good night's sleep and a tall stack of calculus papers later, and I'm feeling a bit better.

This morning's chore was to respond to the 40 calculus papers which I'd merely glanced at last night. (Then I'd done a spot-check to see which ones would need the most ameliorative attention.)

They were, by and large, wonderful. Some of the students had a hard time explaining the importance of finding a horizontal tangent (there was occasional "cart-before-horsing," when students said something along the lines of "we need to find the point where the function is minimized because that's where the tangent is horizontal"), and there were some minor slips in notation (like when students reused the name of the function itself, C(x), for the slope of the tangent line), but all in all the papers were solid. I'm requesting that four of the students send me their papers electronically, so that I can post them on the website as models.

I'm happy.

What of the other 12? I've decided I'm simply not assigning a grade to them. I've written something along the lines of "please come and see me so I can help you hammer this out." Most of them will, I hope, and once they've worked up a more complete solution, I'll respond to it.

It's one step closer to portfolios. I'll get there soon.

Quandary

About an hour ago I said the following out loud: "The last thing I want to do with these papers is grade them, yet the grade is about the only thing many of the students want to get out of it."

"By 'grade,'" I clarified, "I mean rate. I want to respond to them. I wish I could just bring the students in and talk their papers over with them, hand them back, and be done with it, maybe after one or two more rounds of revision."

Maybe I'll do that, after all. Maybe I'll move to a portfolio-style grading system right now.

I don't know.

I'm feeling somewhat cynical about my job right now.

Is the 12-15 hours I'm likely to spend in responding to students' work this weekend worth the reward I'll get for it?

For the roughly 80% of my students who clearly take pride in their work and give it all that they can and are getting a great deal of out of the courses I'm guiding them through, sure: the reward I get from them is a feeling of fulfillment, acknowledgment that I'm doing my job well, and that they're holding up their end of the bargain.

For the other 20%? I'm not so sure about that.

I've received 52 completed "Clock Problems" (the first written assignment of the semester for the Calc I students). Of those, 40 of them clearly put a fair amount of effort into solving the central problem posed to them, applying (with varying degrees of carefulness and correctness) the methods we've developed in class together for the past two weeks. The others...not so much. Of these 12, maybe half of them have some inkling of what it will take to present a reasonable solution to the problem; the other half are clearly out in the cold and have gained little, if anything, from the work we've done in class for the past two weeks.

On these 12 solutions I'm writing "Please come and see me at once: you can do much better than this, and I want to help you do so." I plan on offering them a chance to revise, and strong encouragement to do that.

I feel like I'm talking in circles right now.

I spent the afternoon at the writing across the curriculum workshop my colleague Betty Lou (of nearby Montreat College) and I organized (it was originally scheduled to run in January but was canceled twice on account of weather). It was a wonderful little get-together bringing folks from four different campuses together under one roof. The very convening of this group of people was worthwhile, and the conversations were enlightening and engaging.

The discussions about writing-intensive courses I shared with my colleagues at the workshop helped me to realize how leviathan is the task I've set out for myself: in teaching both Calc I and Linear as though they are writing-intensive courses, I'm effectively teaching writing-intensive courses with an enrollment of 102 students right now.

It's no wonder I'm tired.

Would I be happier teaching at a school where I'd teach two courses per term, each capped at 20? I could make use of every problem-based learning, inquiry-based learning, application-based learning, discovery-learning, writing-intensive, student-centered teaching technique I'd like to, and they'd all be effective, simply because the classes would be of manageably small size. And I wouldn't drive myself insane in teaching that way.

I'd find that sort of opportunity at Bucknell, or St. Olaf, or Carleton, or Harvey Mudd. And the students I'd be teaching wouldn't all be working 20-30 hours a week, would likely be more intrinsically motivated, and wouldn't need as much coaxing and cajoling to get them on board with my methods.

Then the populist in me kicks in and says "don't the students who can't afford to go to such schools deserve a crack at learning through the same techniques?"

And I am where I am again.

I feel like I'm just talking in circles again.

I'm going to stop for now, before I begin making even less sense.

I'm going to write "please come and see me" a few more times, and then relax for the night.

Here's to a more upbeat tomorrow.

Feedback so far

It's been a few days since I put a "Suggestions" envelope outside my door for the Calc I kiddies to use as a means of providing anonymous feedback on how the class is going.

So far, here's the feedback I've received:

1. Candy and soda!

2. More donuts π

Stay tuned.

Peer review review

Today's meetings of my Calc I class are devoted to giving the students a chance to conference with the folks who over the past weekend performed a peer review of their work on the first class project, The Clock Problem (an optimization problem involving an idealized cost function).

These student meetings this morning, by and large, were brief, but they seemed amicable. I emphasized, as always, the need to be respectful and helpful in providing feedback to one another (it's as pointless to say "good work!" as it is to say "this sucks!"), and I hope that emphasis paid off. We'll see how it goes in the second section, which meets in under an hour.

I'm curious to hear others' thoughts on peer review: students, if you're in my class, drop me a line in the comments section and let me know how you think this particular peer review activity has gone (has it helped? Would you have structured it in a different way?). Other folks: if you have any insights on peer review in mathematical fields, please let me know. I've done a bit of it in various courses, but am always looking for new ideas.

Humbled

It's always humbling to get good (read: "thorough" and "direct") constructive feedback.

Like the feedback I just got back on my first draft of More Than Numbers's intro from my consulting editor.

I plan on using my experience in getting this feedback as a teaching tool for all of my classes: "you think you've got work to do?"

An observation and a question

Observation: every one of the students in my morning section (8:00 a.m.) of Calc I submitted homework (which is graded) this past Friday...that's 32/32. Four students in my afternoon (12:45 p.m.) section did not submit homework...that's 30/34. Though both sections have stellar students, the homework completed by the morning section is also decidedly stronger than that completed by the afternoon section, on average: roughly 84% to roughly 70% so far.

This is not the first time I've taught two sections, one early, early morning and the other later in the day, and found that the morning section's students outperformed the later section's.

Question (two-part): Is there really something to this observation?...and...if there is, why the discrepancy?

I suspect that perhaps the folks who sign up for an early-morning class in the first place are going to be the type of folks who have the extra drive and determination to stay on top of things and see them through. Maybe there's something more, though.

Thoughts?

Why I teach the way I do

So I'm already having a pretty good day (making progress on Chapter 3 of my book, getting great ideas to use in Linear Algebra, enjoying working with a few of the Calc I students in the Math Lab), and along comes an e-mail that pretty much made my week.

To all of those who are reluctant to make the switch to student-centered, problem-based learning, behold the benefits:

"So I just sat down to finally ponder #4 and as I was drawing out a graph, or rather trying figure out which graph I should draw I had one of those major AHA!! moments. The average velocity calculation is the slope of a secant line to the graph s(t)=4.9t^2 !!! I love it when things start to fall into place, especially mathematically. Honestly, I feel elated."

Honestly, I feel elated.

Warming

I had a good time with my Calc I kiddies today, mostly because they did the lion's share of the work. (We spent the day working through several archetypal examples of tangent slope computations, observing patterns along the way.)

So far (six class periods in) several students have expressed mild apprehension regarding the structure of the course. It's noteworthy, though, that they're all enjoying the class and are clearly willing to ride it out and let themselves grow more comfortable as the semester goes on.

Most unsettling to them, I think, is the decentering of formulas that's gone on. There's no call to memorize formulas...in fact, there have been almost no formulas needing memorization. Central, instead, is concept: rather than hand them ready-made formulas for computing slopes of tangent lines to the graphs of various families of functions, I've asked them to derive those formulas from scratch from a single formula, emphasizing constantly why that formula works and where it comes from.

Most of the students are used to being told to memorize; they're used to "theorem proof theorem proof formula formula formula definition example example theorem proof" math courses.

Most of the students are not used to being told to understand; they're not used to "why is that what are we looking for how can we find it how does that formula work how can we use it to our gain" math courses.

I give them tremendous credit: they're patient, they're hard-working, and they're doing great.

They're also fun to talk to, in class and out of class. We jawed a bit today about the following pretty well-documented pedagogical fact: though it's painful (for both the professor and the pupils) to sit in a silent classroom in which everyone's staring at everyone else, when a greater lag time is allowed for between a teacher's question and a student's response, the response that comes is generally richer, more complete, and more meaningful than the one that will come from continual once-every-few-seconds prodding on the part of the professor.

As a part of that same conversation, I mused out loud about why it is that students seem more eager to write a solution on the board when I'm out of the room than when I'm sitting right there in front of them. "Is it because you feel safer making mistakes in front of each other than you do making them in front of me?" I asked.

"No," someone said. "I think it's just because we feel bad if you come back in and we haven't written anything."

There could be something to that.

In any case, I'm looking forward to following up tomorrow (limits!), and to continuing our discussion of matrix algebra in Linear.

For now, I've got to finish a few more paragraphs of my book (tonight: the unique challenges faced by writers in quantitative disciplines) before calling it a night.

Choose your own adventure

This one's for my current calculus students (others may answer in the comments hypothetically):

Given: we've now computed the slope of a tangent line to a variety of low-degree polynomials, at an arbitrary point. We've also seen a couple of applications of such slopes (minimization and velocity). One thing we've not yet done (intentionally) is made our method of computation precise.

Would you rather spend the next few days...

1. ...computing more slope formulas (for arbitrary powers and sums/differences/constant multiples of those powers...and maybe even some trig and exponential functions),

2. making our method of computing those formulas more precise, or

3. look into some more applications of slope formulas?

Keep in mind that we will do all of the above at some point soon...I just want to know where you want to go first.

What'll it be? The polls are open!

It takes two, baby

(Caveat: I've got writing process on the brain, as I just started work on Chapter 3 this afternoon, in which I hope to discuss writing as process, and how to effectively stage that process, in the quantitative disciplines.)

For future reference: when asked to write a dialogue in which you're to help a hapless friend who's hopelessly lost regarding one or another mathematical concept, it helps to involve two people in the conversation.

The second interlocutor plays a crucial role: with this person in play you, playing the role of The Expert, need not anticipate every difficulty that might arise to trouble your poor partner in conversation. This partner is the one who need only act naturally in order to bring to light the most subtle and sophisticated aspects of the problem you've been posed. That is, this partner is your foil, the dunce whom you can task with asking every question that might conceivably come up when learning a tricky technique.

With two people talking it through, what might otherwise be a boring monologue or lecture (in which posing trivial questions and then immediately dispatching them would seem stilted at worst or pedantic at best) is now a dynamic dialogue, where nearly real-world interaction is possible.

Talk with your dialogue partner as though you would talk with a real-life friend; help him as you would help a real-life friend, let his ignorance and your expertise take turns driving the conversation forward as you uncover the truth together.

Could there be a better low-stakes writing-to-learn activity?

Week One: done!

Ugh.

In Linear Algebra today I used rampant tidal locking as an explanation for the fact that this past week has easily been the longest one ever.

It's been a long week, but a good one, and I've gotten a lot done. More importantly, I feel I've set the appropriate mood in all of my classes. The Calc I kids are slowly starting to come to life, and Linear's just getting better and better.

I've now had "get to know ya" meet 'n' greets with about 20 or 25 of my new students, and these have been uniformly fun. As always my students very in every conceivable way: age, prior education, academic background, major, motivations for study. I've got young, old, quiet, bold, math-minded, math-shy, self-directed, and diffuse. A couple share my love of poetry, one my fascination with early human evolution. Some are sarcastic, like Dwight, the student in my second Calc I section who showed both bravado and cojones by merely modifying the work I'd done on the board to solve an earlier problem, instead of transcribing his own solution from scratch. Others are retiring, like Bertie, whose description of greatest common factors was practically unintelligible owing to the quiet of his voice.

So far, aside from a little bit of hesitation on the part of a couple of the more outspoken students, all of the feedback I've gotten on the structure of the courses (both highly student-centered, Calc I to an extent I've not taught it in the past) has been positive. A couple of the Calc I students have even commented appreciatively on the use of writing. One thanked me for asking them to write in order to more deeply understand the difference between two oft-confused rules for exponents (when do you add 'em, when do you multiply 'em?): "I didn't do so well the last time around in calculus, but I feel like I'm going to do a lot better this time."

One of my nontraditional students (I'm always happy to have them in my classes: they're generally so much less shy about speaking up) expressed some concern in my meeting with her on Tuesday morning. "Am I going to be prepared for Calc II by the end of this semester?" I assured her that we'll do all of the computations we'd do in a theorem-driven calculus course, and we'll learn all of the concepts we'd learn in that setting...we'll just approach them all from a much more useful angle.

"Don't worry," I said. "We'll get there."

In fact, freed from having to put together a punctilious list of soon-forgotten limit laws, we're making good time: just a week in and we've already computed the slopes of several tangent lines, having figured out that that's what we need to do in order to solve the problem posed to us on Wednesday morning.

Yup, classes are going well.

In other news, I've finished the first draft of an introduction to my book. I'm quite satisfied with it. It charts my personal history with writing in the discipline of mathematics, highlighting the ways in which I've grown as a writing instructor as I've progressed in my career and indicating the empty spaces in the writing literature which need to be filled. Tomorrow I hope to start work on either the first or the fifth chapter (these are the two whose structure I can most clearly envision right now).

Now, bed calls. Students: please let me know what you thought of this first week. Where do we go from here?

Oh, and: 400th post!

Dialectical driver's ed

How's it going?

Three days into Calc I, and it 's difficult to say which way the class is going to break this term: the students are still strongly engaged in class and contributing well to the problem-centered classes, but I think a few are hesitant about the rather non-traditional format of the class. As yet we've not done a great deal of computation, haven't derived or delved into any formulas, haven't stated any theorems. A number of the students are used to doing all of these things within the first couple of days of a math class and so they've got a sort of deer-in-headlights, "WTF?" sort of look in their eyes.

The formulas will come soon, I'm certain, and I hope that with them a sense of familiarity and comfort: we're about to enter into a discussion of tangent lines and secant lines and their slopes, a discussion will include a great deal of computation and review (and practice) with algebra.

I'm not sure quite how quickly we'll get there: it really all depends on the pace at which the students end up driving the course. As I mentioned to my first section this morning, they're sitting in the driver's seat, working the pedals, holding the wheel. Every now and then, as instructor, I'll reach over and grab the wheel with one hand to steady it or add a few degrees to a particularly too-tight turn.

So far the structure of the course is "dialectical," an exercise in turn-taking: I've proposed an initial course heading, and now we've walked for a while. Now, at the end of the first day of hardcore hiking, having surveyed the site at which we've made camp, I've worked out an itinerary for our next day's travel together. We'll see where that takes us. We'll make camp again, make some next-day plans, and set out again in the morning. I know the stops we need to make, and we'll make them all, but it's up to the students to figure out which we hit when, and in what order.

We'll figure it out.

Meanwhile, I am loving Linear. The class is huge (the largest I've yet taught at UNCA), but has great cohesion. The students are eager to work and are making great progress, and I feel that I'm doing a much better job of structuring the discovery process than I did the last time I taught the class. I've put a lot more thought into precisely which skills are needed to solve which problems, and we're making a (long) beeline toward our first major goal: analyzing the long-term behavior of a simple Markov process (like the game we played on the first day).

Today we figured out how row operations mirror the operations needed to solve a system of linear equations. Next, after posing and solving a couple of realistic problems requiring the solution of a linear system, we'll motivate and move toward matrix multiplication. From there, it's on to matrix inverses, invertibility, determinants, eigenvalues, and, at last, an analysis of Markov processes like those we began with. At every stop we'll solve another real-world problem or two, get some practice with computation, and write a bit about what each concept really means.

I figure it should take us something like 5-7 weeks to get where we're going. After that, we'll talk about linear transformations, vector spaces, changes of basis, etc.

Exciting!

Meanwhile, I'm reading some fascinating works on writing and rhetoric in preparation for various parts of my book. Most recently I've begun Deirdre McCloskey's The rhetoric of economics (2nd edition, 1998, Madison: The University of Wisconsin Press), a delightful rhetorical analysis of economic discourse in which she dissects the ways in which economists convince one another and others of the truth of their assertions. According to McCloskey, much of what many economists consider unassailable scientific truth is really rhetorical "smoke and mirrors."

Not that there's anything wrong with "rhetoric," a word which McCloskey takes pains to save from those who would use it only in a pejorative sense: rhetoric is ubiquitous and unavoidable, and McCloskey merely asks that we be intentional with the rhetoric we use. We all use metaphors and other rhetorical devices; it's simply important that we be aware of the devices we employ. As one might put it, extending the tried-and-true "lens" metaphor, it's no crime to have poor eyesight, as long as you can admit that you need to put your glasses on in order to see properly.

Her opening chapter, which begins with a close reading of passages from canonical articles in economics, has made me think about the way in which I begin some of my own mathematical papers: what sense of authority am I conveying though my choice of words? What world am I building? What agent is acting? What am I asserting about the truth? I'd like to look into my own writing to do a careful reflective analysis.

But with all that I've got on my plate right now, such an analysis has got to wait for a little while. I hope to finish off the first draft of the introduction to More Than Numbers tomorrow, and set to work on the slightly slanted history of WAC, WID, and WTL which will make up much of Chapter 1.

Much to do, much to do, and so little time...but so much fun!

Day One, Part III

Well, Linear went wonderfully, too.

I am tremendously happy about the way this first day went, and in all three classes I think I owe the smoothness and success to several different sources:

1. The students (duh). They were almost without exception engaged, focused, and excited to be where they were. They clearly came ready to learn, and I hope I didn't disappoint.

2. I came ready to teach, not to recite a bunch of bureaucratic nonsense from the syllabus. I truly feel that the act of delaying the handout of the syllabus by 40 minutes (at the end of the first period instead of at the beginning) does more than any other single act to place the class's focus on the proper point (learning) and not on the improper ones (grades, lateness and attendance policies, etc.).

3. Student-centered, student-driven first day activities. Both courses began with activities whose pace and ultimate outcome were determined by the students. I provided an overarching structure, but the students led in the act of discovery. Of course, this is what happens all semester long in my classes, but it's important to do this on Day One. Do NOT put it off. The more you're able to get the students working together, helping each other, and sharing ideas on the board on the very first day of class, the more comfortable they'll be doing these things all semester long.

4. Knowing their names. It cannot be overstated: no single act more quickly earns the students' respect and admiration (by showing them respect and admiration) than learning their names as soon as you can, as many as possible on the first day.

It's going to be a great semester. Dare I say...best ever? We shall see.

Students: thank you for making today a wonderful one. (And a welcome to all of you who are joining one of my courses for the first time! Post a comment, if you'd like!)

Day One, Part II

Two down, one to go. My second section of Calc I was more wide awake, but a bit more unfocused. They did a great job in the "memory dump" of review topics from algebra, trig, and precalc, however.

It's going to be a good semester, that's for sure.

Now I've got to prepare for my 35-student (!) Linear Algebra I section.

Day One, Part I

One down, two to go. Every person on my roll for my first section of Calc I was in attendance, and no one not on my roll was there. This has got to be a first.

They were lively, fun, and relaxed. They took the focused freewrite with which I started things off in stride. They got into groups with no hesitation, and they showed no trepidation about coming up to the board. They'll be great.

It's going to be a good semester!

On your marks, get set...

It's been a while, huh?

As you might suspect, I've had a lot going on, keeping me from posting here for the past, oh...two months? Ouch.

What's going on?

The REU's over, but the work has really just begun: the students this year were fantastic. They were smart, fun, and funny, and they worked tremendously hard, producing a ton of interesting mathematics. (We should get at least five papers out of it in the next couple of months, and maybe more beyond that in the follow-up.) And this year it was "buy eight weeks, get one free": six of the eight students ended up traveling to MathFest in Pittsburgh, PA to present on their summer research.

The REU took up much of my time, but I've had a number of other things going on, as well, among them: (1) working on student learning outcomes for both the Mathematics Department and the Writing Intensive program, (2) putting together the ILS Oversight Committee's report to the Academic Policy Committee (still not submitted, but finally almost finished), (3) planning my contribution to the short course I was helping run at MathFest, (4) working with my College of Charleston peeps on our paper on the rhetoric of mathematical writing, (5) looking ahead to my round table discussion of program assessment for this year's CWPA (coming up in a few weeks), (6) laying the groundwork for the book I'm now slated to write for Jossey-Bass (w00t), and (7) trying to get ready for my Fall classes, which start on Monday.

(1) Student learning outcomes: although I see the purpose, ultimately, of programmatic assessment, too highly institutionalized assessment tends to become bureaucratic and corporate. I don't want to say more about my role in this university-wide process this past summer other than that it was frustrating at times, had its minor joys, and though it was ultimately fulfilling in that I feel like some good will come of it, I worry about the intentions some people in administration may have for codifying the university's student learning outcomes as rigidly as is being done. 'Nuff said.

(2) The ILSOC report to the APC: this is more of the same. I see the purpose, and the writing of this document is a worthwhile activity, but one which runs the risk of being overly parliamentary and pro se. 'Nuff said.

(3) MathFest short course on Sage (an open-source computer algebra system): I actually wrote to my colleague Oscar (one of the co-organizers of the short course) in mid-July, a few weeks before the course was to take place, expressing my insecurities about putting together an hour or two of meaningful material.

"Nonsense," was the substance of his reply; "you'll do fine. Write about what you know." So I threw together several worksheets full of Sage code purporting to implement a number of higher-level graph theoretical algorithms and called it good. The worksheets actually went over really well in the short course and filled a rather comfy niche in the program. I felt good about what I'd done.

The moral of the story: just do it.

(MathFest, by the way, was a blast. Pittsburgh's a nice city with lots of pretty runs and nice architecture, the REU students were fun to hang out with, as always, and the MathFest program was replete with fun stuff to see and do. A good time had by all.)

(4) Math rhetoric: as I mentioned in my last post, oh so long ago, Bella and Damian came up from the coast to spend a couple of days with me, hammering out a plan to finish the paper we'd begun (with Nicola, who couldn't make it) on the rhetoric of the writing my REU students had produced in 2008 and 2009. They had a lovely time while here, visiting with and interviewing the REU students from this past year, working on our draft at the time, and just hanging out. I'm happy to say that as of last night we have a draft that's nearly ready to submit to the comp/rhet journal Across the Disciplines. It's the first of what we hope will be...well...at least more than one article on the rhetoric of mathematics and its learning, and the role writing plays in inducting students into the scholarly mathematical community.

(5) CWPA: I'm already looking forward to the Wildacres retreat! It's going to a be a full house this year, several dozen of us crowding into the nooks and niches of the Wildacres North Lodge to join in round table discussions of assessment with various forms and functions. I plan on presenting a brief "natural history" of the assessment that's gone on in our WI program during the past three years, exposing to public view the layers of rock comprising the "pilot" assessment Lulabelle headed up in AY 2007-8, the refinement she and I performed in the following year, the plans for assessment she and I laid out in the lead-up to the university's reaffirmation of accreditation (notice I haven't mentioned SACS yet in this post...), and the modifications those plans underwent as I limned the WI program's assessment of student learning outcomes this past summer.

It's sure to be riveting.

Actually, I'm more looking forward to hanging out with Damian and Bella, and with them planning the next phase of our study of the REU students' writing.

(6) Book: Oh yeah...I'm now under contract to write a text (working title: More Than Numbers) on the role of writing (specifically, writing to learn and writing in the disciplines) in mathematics and the mathematical sciences. Jossey-Bass received well the proposal I pitched to them several months back (in March, I think?), and after some fine-tuning of the project, I was brought on board. I'm excited, but at the same time terrified. I'm confident it'll all work out well, though. I've already done a ton of enlightening reading (if you get a chance, check out Scott L. Montgomery's The scientific voice...it's a fascinating read that, along with several other sources, has helped me put together a theory of the schizophrenic nature of undergraduate mathematical writing), and I have some great ideas. Furthermore, my consulting editor for the project is someone a few of whose books I've read and whom I respect very much; I'm sure she'll be very helpful to me.

I will say this, however: if between now and next March I start randomly babbling to you about reader response theory or the role of writing in the mathematical finance classroom, just smile and nod your head.

(7) Fall 2010: yup, we get underway in two days. Yikes. I'm actually really excited: I've got the first week mapped out in all three of my courses, and I've plotted out what I think are clear and navigable problem-centered paths for both of the lower-level courses (Calc I and Linear Algebra I) to follow.

Calc I is going to begin with a brief review, and then the students will work toward developing the skills needed to solve an optimization problem from economics. They'll need to "invent" the method of secants and tangents, the definition of a limit, and the definition of the derivative; and they'll need to learn how to compute some simple derivatives in order to solve this problem. Once all of that groundwork is laid, we'll spend some time working with the textbook and picking up some useful formulas for limits and derivatives, but everything we do is going to be application-driven and learner-centered. I hope to not have to lecture for more than 10 minutes at a time for the whole damned semester.

Linear's going to be the same: Day One features a variation on the "Markov Dance" that started the semester off the last time I taught this course, in the semi-mythical beginnings of this blog. The limiting behavior of this system is going to be the motivating problem for everything we do in the first half of the course, as we work our way through everything we'll need to do to get to the fun and useful stuff (eigenvalues, diagonalization, and Markov processes): solving linear systems, inversion of matrices, solvability criteria, and finally eigenvalues. I'm going to downplay theory and upplay (what a cool word!) functionality. I hope to get to the fun stuff by the sixth or seventh week, at which time we can spend almost all of our time on applications. It's going to be fun.

I should mention that I've modified my grading schemata somewhat, but not substantially, from last semester: students in both of the above courses will still have an opportunity to perform potentially unlimited revision on most things they hand in, but I'm tempering the grading scale so that it's no longer possible to earn full points back with each revision. I think the system I've set up is a generous and reasonable one, though, and I believe it'll still serve to motivate students to learn rather than to grub for a good grade.

Meanwhile, the Senior Seminar's speaker schedule is chocked full, with four visitors (all versed in speaking to undergraduate audiences) coming to share their knowledge and spend a little time with the students. It's going to be a good semester in that course.

Okay, that's all for now...I'm going to try to post more regularly during the coming semester, although I can't promise it'll be more than anxiety-tinged stream-of-consciousness rambling about this or that idea for the book.

Fare thee well!

Weather's great, wish you were here.

I haven't had much time to write, but I thought I'd send a postcard from this edge.

We've just finished three weeks of the 2010 REU, and things are chugging along smoothly.

Incredibly smoothly.

Wow.

The students' first presentations (yesterday) were masterful, already conference-worthy: their talk mechanics were spot-on, they displayed superb mastery of the concepts and computations they presented, and their slide designs were eminently professional. In every way their presentations were superior than many of those I've seen given at major conferences by seasoned mathematicians.

They're ready for the "minisymposium" we're playing host to next week, to which we've invited the faculty and students from two other area REUs.

I hope that I can take some of the credit for the students' success to this point: I've been yet more intentional in my design of the program's activities than I've ever been in the past, providing even more and more timely instruction on writing, presenting, use of scholarly sources, reading of math papers, etc. than in past iterations of the program.

And I've had help in this, too, that I've not had before: Bella and Damian, two of my Charleston-based partners in composition-theoretic crime, came up the mountain a little over a week ago to talk to the kids about writing and math. The visit was twofold...perhaps even threefold: (1) to give the students the rhetorician's point of view on mathematical writing, (2) to interview (though this term was never used) about their past experiences with academic writing in general and mathematical academic writing in particular, and (3) to get our shit together and make some progress on the paper we began back in February when I joined them for a working weekend down in South Carolina. (To this last point, briefly: I was tasked with coauthoring sections on the students' mastery of visual rhetoric, specifically with regard to the visual effects on the reader of white space and mathematical formulas; and on the students' use of sources. My analysis of the latter was very enlightening to me, and helped me quantify the fact that this year's students are well ahead of their predecessors when it comes to making effective use of the literature they find in the course of their investigations.)

The two-hour-long "interview" with the students was wonderful. In the discussions that arose the students revealed rich and robust histories with writing, including a good deal of relevant and highly intentional instruction I might not have expected. These students are very well-prepared (which helps to explain the success they've had so far in this summer's program!). The interview also gave us a great deal of data we can use as a foundation for our next article, in which we hope (already!) to trace the development of students as communicators of mathematics through the course of a summer research program one of whose foci is on writing in the discipline. Bella's already kicking around the idea of coming back toward the end of the program to interview the students again and see if anything's moved.

Mathematically, the students are fantastic. They're very self-directed and intrinsically motivated. They're displaying superlative creativity and originality of thought, to a greater extent than did many students in previous years of the program. They're making terrific use of the sources they find, plying them effectively to serve multiple purposes: to find other sources, to support their own propositions and theorems, and to situate their work in the context of what's come before. And every one of them has a clear sense of purpose in his or her project: no one of them is "stuck," as I've seen happen in the past.

More important, perhaps: they're getting along exceedingly well with each other. From Day One they've bonded remarkably well. They do almost everything together, not because they feel like they have to or because they're afraid to do thing alone (both of these have been moving forces in previous iterations of the program), but because they want to. They clearly like each other. A lot. It's good to see.

It comes across in simple things, too: yesterday, during the students' first round of talks, this year's group proved themselves to be the most attentive and responsive audience members we've yet had in the program. They listened with clear and unfeigned interest, they gave encouraging nonverbal feedback (nods and smiles) at appropriate moments, and they asked a number of good questions of their peers at the end of every talk. Midway through the talks Tip (my sole faculty partner in running the program this year) and I marveled at the fluency of the students' presentation skills and remarked at how though the students had all chosen to work on individual projects (something we initially bemoaned) it was evident that to a greater extent than in previous years the students are collaborating with one another behind the scenes, sharing ideas, talking about LaTeX and Beamer, and helping each other through rough spots.

Ladies and gentlemen, we have established an effective learning community.

What else is up?

I've only been to one meeting of the Learning Circle (having missed last week's to work the run-off election) I chose to take part in this summer, but I have a good feeling about the direction it seems to be taking.

I've received good feedback on the proposal I sent to Jossey-Bass for a text on writing in mathematics.

I've finally submitted a reasonable draft of an assessment plan for the university's Writing Intensive student learning outcomes to the powers that be. I doubt that's the last I'll hear of that, though.

It's been a busy summer, but a good one. I'm excited to see what the coming week will bring.

For now, I'm going to kick back and watch Ghana get the better of the U.S. soccer team.

Day Two is in the can

We've wrapped up two days of the 2010 REU, and things are running smoothly so far.

The students are gelling tremendously quickly into a friendly and supportive learning community. As early as the welcome barbecue a few nights back it was obvious these folks were going to get along really well, and the last couple of days have borne that observation out. "I was a little worried before coming here," one of the students told me at the barbecue, "but everyone's so nice that I know I'm not going to have trouble working with them. It's going to be easy to ask them for help if I need it."

Although I can't yet claim to know them all incredibly well as individuals, each is beginning to make known her or his particular strengths and weaknesses. Three of them are particularly eager to demonstrate their skills and hesitate not at all in traipsing to the board to show off their solutions. A couple of the others are more retiring, willing to whisper the answers (almost always correct) from where they sit but less enthusiastic about taking to the board. The remaining three are quieter still and it often takes a little prodding to get them to share their ideas. Notably, to an extent not seen since the first year of the program, one of the students is clearly emerging as the "social director." She's the most outgoing of the group, though a couple of her peers aren't far behind when it comes to eagerness and enthusiasm.

I should note that the students needed little convincing to complete the first day's freewriting exercise, which was designed to tease out of the students their personal goals for the overall program. I've posted the resulting list of student goals here; happily, most of the learning goals coincide very well with my own.

The students are making quick progress through the long list of graph theoretical definitions I bludgeoned them with yesterday afternoon, and we're already over halfway through the graph theory notes I hope to complete before calling it quits. We've also hit on several topics from dynamical systems and geometry, and tomorrow brings more graph theory, dynamics, LaTeX, and Mathematica. It's going to be a full day, but I think we'll get through it.

Further bulletins as events warrant. For now, bed calls. My bedtime reading? The basis for the Learning Circle in which I'm taking part this summer, Helping college students find purpose: the campus guide to meaning-making, by Robert J. Nash and Michele C. Murray (2010, San Francisco: Jossey-Bass). We'll see what comes of it.

REU ramp-up

As of 11:00 or so last night, all of the 2010 REU students are in town.

They're gelling as a group really well this year, much as they did two years ago. Last year's folks took a little while to come out of their shells, although once they did so, they got along famously and I joy in seeing their friendships continue via Facebook.

This year's crew is ready to get underway, I think. They're as excited as I am to get started tomorrow morning.

As I've mentioned before, I've made some changes, mostly minor, and all for the best, I hope, in the program this year.

What's new?

1. New focus on specific aspects of communication. As I indicated in my last post, I'll be requiring students to practice "elevator talks" and provide at least one visual in their every-other-weekly presentations. My hope is that these new requirements will lead to greater intentionality on the part of the students as they craft their talks and papers.

2. Deepening of engagement with source material. Students will now be required to perform a thorough literature search as a natural adjunct of their respective research projects, and this literature search will be accompanied by the creation of a simple annotated bibliography. This exercise should help them in developing meaningful "history" sections in their papers.

3. Alternation of presentations and papers. The students will be completing only four (not six) drafts of their papers throughout the summer, at the end of the second, fourth, sixth, and eighth weeks. They will be presenting only every other week, rather than every week, at the end of the third, fifth, seventh, and eighth weeks. We will still have a mandatory meeting on every Friday, but on alternating Fridays this time will be used for conferencing on papers, peer review, practice of elevator talks, and so forth.

4. Emphasis on collaboration. In addition to more frequent peer review (I plan to have them perform three peer reviews, as opposed to the one done in the past years), I'll be emphasizing the collaborative and communal nature of mathematical research by encouraging them to work together on their projects. As exciting as it is to have eight students working on eight different projects, the students stand to gain more by working with one another, and my attention can be much better focused if I only need to stay on top of three or four projects rather than eight.

I've got a good feeling about it. They're a bright bunch, they're getting along well already, and I think it's going to be a particularly good summer.

Sampler

During the past week I've attended, presented at, participated in, or facilitated two conferences and three faculty development workshops, on topics ranging from pure mathematics (group theory and graph theory) to the pedagogy of writing and advising first-year students who are brand new to a liberal arts university. I've written dozens of pages of notes, collected several handouts, worksheets, resource lists, and sets of slides. I've talked with, listened to, or sent e-mails to dozens of colleagues. In between doing all of the above I've been spending most of my time reading up on the history and philosophy of science and its teaching and on the role played by gender in mathematics achievement.

Below I've collected a number of quotes, factoids, observations, and questions dealing with all that I've been doing for the past week or so. Every one of these items deserves further follow-up; maybe I'll get around to addressing some of them over the summer, maybe not. I just want to get them out there for now.

1. Contextualization (and resocialization) of science. First, more from Thomas S. Kuhn's The structure of scientific revolutions (pp. 136-137), on the apparent linearity and cumulative nature of science and the related transmission of scientific knowledge:

Textbooks thus begin by truncating the scientist's sense of his [sic, here and following] discipline's history and then proceed to supply a substitute for what they have eliminated. Characteristically, textbooks of science contain just a bit of history, either in an introductory chapter or, more often, in scattered references to the great heroes of an earlier age. From such references both students and professionals come to feel like participants in a long-standing historical tradition. Yet the textbook-derived tradition in which scientists come to sense their participation is on that, in fact, never existed. For reasons that are both obvious and highly functional, science textbooks (and too many of the older histories of science) refer only to that part of the work of past scientists that can easily be viewed as contributions to the statement solution of the texts' paradigm problems. Partly by selection and partly by distortion, the scientists of earlier ages are implicitly represented as having worked upon the same set of fixed problems and in accordance with the same set of fixed canons that the most recent revolution in scientific theory and method has made seem scientific. No wonder that textbooks and the historical tradition they imply have to be rewritten after each scientific revolution. And no wonder that, as they are rewritten, science once again comes to seem largely cumulative.

My question: does it have to be this way?

My answer: no. But an elaboration of that answer will have to wait for now. I have a good deal more to say in reflecting on the social constructivist point of view of science, first (or at least first explicitly) elaborated in Kuhn's work, as in the following passage (p. 42):

Though there are obviously rules to which all practitioners of a scientific specialty adhere at a given time, those rules may not by themselves specify all that the practice of those specialists has in common. Normal science is a highly determined activity, but it need not be entirely determined by rules. That is why, at the start of this essay, I introduced shared paradigms rather than shared rules, assumptions, and points of view as the source of coherence for normal research traditions.

The nature of the rules to which specialists adhere is fluid and dynamic, susceptible to the exigencies of the day-to-day applications of those rules. Rules, as applied, evolve, and they evolve in accordance with their usefulness as judged by practitioners of the specialized discipline concerned with those rules. More than anything else reading Kuhn makes me aware of the need to be more intentional about including opportunities for my students to explore, discover, interpret, investigate, and describe the concepts we consider in any given class; they must be made, to the greatest extent possible, to feel like they as much authors of scientific discovery as I am. (Particular attention to the role of youth and neophycy in scientific "advancement" is critical as well.)

2. Invention versus discovery. I've often asked my Calc I students to think about the difference (if there is one) between invention and discovery when they take part in the Newton v. Leibniz project. Kuhn will serve as an excellent source for those students interested in learning more about the distinction between the two notions: pp. 51-53 contain a discussion of this distinction, highlighting invention as the adjustment that goes on in a paradigm's conception of science in the wake of a discovery: one may be the first to objectively observe a physical phenomenon, say, but until the significance of that phenomenon is understood and elaborated, and the discovery's relevance is described, one cannot be said to have invented a thing. In a sense invention is the recognition of the importance and relevance of a discovery via its incorporation into the normal scientific tradition that operates within a given scientific paradigm.

What might students have to say about this?

3. Revision (in writing) as revolution. The parallels between Kuhn's portrayal of scientific revolution (especially as it compares to political revolution) and revision of one's writing as a process are too great to be ignored: revision occurs concomitantly with the recognition of the inadequacy of what one's written to account fully for one's perception of the subject of the writing. That is, revision is undertaken in response to a perceived discrepancy between the author's intent and the author's ideas as communicated expressly on the page. Compare (p. 91): "In much the same way, scientific revolutions are inaugurated by a growing sense, again often restricted to a narrow subdivision of the scientific community, that an existing paradigm has ceased to function adequately in the exploration of an aspect of nature to which that paradigm itself had previously led the way."

To continue the parallel between these "revolutions" would force us ultimately to recognize what students of writing are often loath to admit (and what teachers of writing already know very well): writing is a social process, as much, if not more so, than is scientific discovery.

I should note that in the following pages in Kuhn's text (pp. 92 ff.) he most clearly articulates the role of social forces in shaping scientific revolutions. When competing paradigms come up against one another, adherents to one or the other must be prepared to argue in favor of their particular paradigm.

4. Portfolios (again). Enough about Kuhn (for now, at least). Let's get to some observations on the International Writing Across the Curriculum Conference, the first two days of which I was able to attend at the end of last week.

In talking with my colleague Nero (currently at the University of Hawai'i, Hilo) I found myself suddenly able to articulate, far more clearly than I've ever been able to before, what exactly a portfolio means of assessment might look like in a mathematics course. One or two communication outcomes would join one or two affective or metacognitive outcomes, and these would join two or three content-centered outcomes as a basis for the course's assessment. (I already generate such outcomes for all of my courses.)

Throughout the semester the students would be given a variety of assignments, successful completion of each of which would demonstrate achievement at one or more of the outcomes on the list described above. These assignments could include more traditional problem sets (though likely not sets of problems pulled from a textbook), written components of projects I already assign (like Newton v. Leibniz, Confectionary Conundrum, etc.), reflection or response papers in which students explore their personal and emotional engagement with mathematics, and so forth.

Students will have a chance to perform unlimited revision on many of these assignments, so that if a student isn't happy with a given iteration of a given assignment, she can revise her work to improve upon it. (As regular readers know, I'm still working on adjusting my revision policies.)

In the last week or so of the semester the students will be asked to select four or five assignments from among those completed during the semester to represent their mastery of as many of the course learning outcomes as possible. They will then write a brief (no more than five or six pages) paper in which they articulate explicitly the role served by each of the assignments they have chosen to include in the portfolio: why include this piece? Mastery of which outcome does it purport to demonstrate?

I'd like to recruit one or more of my colleagues to help me assess completed portfolios the first time around; there ought to be some sort of validation process.

Thoughts?

5. Intentionalize, intentionalize, intentionalize! This coming summer's REU students will receive yet more intentional instruction in writing than any previous year has received. At least two of my College of Charleston colleagues will be coming up to help impart their wisdom on rhetoric and composition. I'll be giving the students more models of professional writing than they've been exposed to in the past. And, most notably, I will obey the exhortation of the four presenters from Virginia Commonwealth University and place more emphasis on the "middle" stage of student research writing.

What do I mean by this? Much like the faculty at VCU (as described by the four presenters mentioned above), I find I've been very intentional about helping my REU students find sources at the outset of their research program, and I've been very intentional about helping them through draft after draft of their week-to-week research reports once those reports have assumed a certain level of coherence. But, like the aforementioned faculty, I've been somewhat remiss in offering the students explicit instruction in the middle stages of the process: how does one evaluate sources? How does one compare them? How does one decide on the relevance of a particular source to one's own researches?

I've decided that I'm going to require the REU students to follow their initial literature searches (which most of them do) with the construction of an annotated bibliography in which they highlight the important contributions of each source, summarize the relevance of each source to their own work, and prioritize the source, ranking it alongside the other sources they've found in terms of its strength of contribution, its clarity, and its relevance to their particular research project.

Will this make more work for the students? You bet it will. But since I'm only going to be asking the students to produce a draft of their report every other (rather than every) week, I feel it's a fair amount of work to ask of them.

It occurred to me this morning, in sitting in on a faculty development workshop focused on our LSIC courses, that the same sort of exercise should be required of students in our MATH 480 course in order that that course warrant its Information Literacy Intensive designation. Just two years ago I suggested that the department begin requiring students in MATH 480 to produce an expository paper; this suggestion met with almost no resistance. I hope this new suggestion will go over equally well.

6. Other thoughts for the REU. What else will I be asking this year's students to do? Nothing excessive, I believe. It seems to me that I should require the following of the program's participants:

History and context. Every draft (not just the last) of every student paper this summer will be required to have a section describing the history and context of the topic the student is investigating. This section, like the rest of the paper, may be rather sparse and tentative at first, but like the rest of the paper it will become more full and flourishing as the summer goes on. I believe it's important, though, that from the very onset of the program the students become accustomed to contextualizing their work and establishing its place in the field.

Visuals. One of the VCU folks mentioned above presented a metaphorical means of constructing an annotated bibliography and literature review, comparing the process of finding, evaluating, prioritizing, and applying sources to planning a conference, at which participants must be placed at various tables, grouped in various sessions, and so forth, according to interests, purposes, and points of view. The most striking aspect of this presentation to me was the insistence on a visual representation: the presenter required her students to come up with a visual means of portraying their evidence. I am going to start requiring each REU student to include at least one visual representation of her or his work in the bi-weekly presentations they'll be delivering. That visual may be the same from week to week, but if the visual remains unchanged I will ask the student to justify her or his reason for retaining the same visual. This, I hope, will encourage students to reflect upon the way in which they are representing their work through nonverbal means; this reflection could lead to further discovery and, of course, refinement of the visual rhetoric the students use in describing their work.

Elevator talks. Even the strongest undergraduate research students have trouble articulating their work clearly and concisely. I'm going to begin asking every student to open her or his presentation with a no-more-than-one-minute "elevator" version of the presentation. What is the main focus or question of your research? What method or methods are you using to try to study that focus or answer that question? How does your work fit in with others' work on the same topic? I hope this additional intentionality will help students develop the ability to communicate their work in the hurly-burly world of conferences and cocktail parties.

7. QEP. As many of my colleagues in the Southeast part of the country know, QEP stands for "Quality Enhancement Plan," and is the means by which the Southern Association of Colleges and Schools (SACS, the accreditation agency for an enormous number of institutions of higher learning in the Southeast) asks the colleges and universities it oversees to plan and implement institution-wide changes to enhance student learning.

I'll have a lot more to say about this in the coming weeks, months, and, if all goes well, years, but I'll simply say now that I am more committed than ever before to making writing the focus of UNC Asheville's QEP. I will do all that I can to lobby for this position.

8. Inkshedding. Perhaps the most delightful thing I took away from this past Wednesday's workshop on writing instruction of ESL students (ably facilitated by my colleagues Hannah and Tabitha of UNC-Chapel Hill and NC State University, respectively...thank you both so much for coming out!) is a new form of low-stakes writing to which I'd not before been exposed. "Inkshedding" is much like a collaborative form of freewriting. As they would be in a freewrite, participants (in groups of three or four) are asked to write on a given topic for a set amount of time (three minutes, say) or until they have written all they would like to on the topic at hand. When finished, each participant places his writing in the center of the circle and waits for someone else to do the same. The papers are then exchanged, and each person reads what the other has written and then responds in writing on the first writer's paper. Once done responding, the second person places the paper in the center again and takes another. And so on. In theory, the process could continue endlessly, readers writing in response to others' responses to their own responses, and so forth.

Beyond its obvious pedagogical usefulness, I think this would be a fantastic way to construct collaborative poems, or at least generate ideas and images for rich poems or other pieces of fiction. I'm eager to find a few folks who are willing to try this out. If you're game, let me know!

9. Gender matters. I'm currently reading a book that I picked up (in the simply marvelous bookstore Caveat Emptor) in Bloomington, the site of the IWAC conference last week, Mathematics and gender, edited by Elizabeth Fennema and Gilah C. Leder (1990, New York: Teachers College Press). This collection purports to analyze the different ways in which gender influences math performance, success in math coursework, and affective responses to mathematics and its study. Unsurprisingly, men and women differ with regard to their experience with math, and factors such as confidence, perception of utility, sex-role congruency (the "math is for men" stereotype), fear of success, and attribution of performance to one or another cause (effort, ability, or outside forces such as sheer luck) all strongly, and differently by gender, affect an individual's mathematical understanding and performance.

I've yet to read much in this book that's given me reason to adjust the way in which I teach math, aside, perhaps, from Lindsay A. Tartre's study (Chapter 3, "Spatial skills, gender, and mathematics") suggesting that in women there is a far stronger correlation between spatial skills and mathematical performance. Might I do well to place particular emphasis on visual representations of problems when working one-on-one with a female student? I already attempt to adapt my explanations to whatever mode it is in which I know a given student most clearly understands mathematical ideas.

It's something to think about. I may have more to say about this book as I get into the later chapters, which deal with the role of the teacher and the classroom dynamic in assisting or impeding students' mathematical understanding.

That's enough for now. I realize that this is one of the longest posts I've written in a long time. Believe me, I've tried to keep it short! I hope to be able to elaborate on one or more of the above issues in later posts, especially as I begin to implement some of the proposed changes to my REU and to my regular courses.

To be, as ever, continued!

Snippets

From Leon Botstein's Foreward to Writing to learn mathematics and science (eds. Paul Connolly and Teresa Vilardi, 1989, New York and London: Teachers College Press, p. xiv): "The use of ordinary language in the teaching of science and mathematics enables the teacher to connect what otherwise might seem an arcane and distinct set of languages, thought processes, insights, facts, and understandings to experience, in the everyday sense that Dewey realized could constitute the basis for motivating learning, memory, and long-term comprehension."

From Paul Connolly's "Writing and the ecology of learning" (op. cit.), p. 7:

[In the all-too-typical science classroom] the important feature of education becomes saying the right words, not learning how to use one's own words. In such circumstances the "language" of science remains for many students a set of foreign words, dead as Latin, to be memorized from a book. It is not the constructive speech of a vital culture. Students then regard "definition" as a chain of words that bonds one to "truth" and "reality"; if one link is forgotten, the whole chain of understanding is broken. They have...no notion that meaning is recomposed in each new personal performance on the public instrument of language.

From Sheila Tobias's "Writing to learn in science and mathematics" (op. cit.), p. 49 (cf. my Charleston colleagues' and my recognition of the importance of visual rhetoric in the legibility of mathematical writing):

Even a cursory examination of math and science textbooks reveals that these books are not meant to be read as we understand the act of reading....As I write elsewhere, clarity in books in other subjects is achieved through repetition, using different words to restate a single idea, slowing the pace, using a spiral kind of organization that keeps coming back to the same idea at different levels, using topic and summary sentences to nail down what the paragraph contains, and always foreshadowing the point to be made later on. In math and science texts, we find, instead, pages of information with virtually no repetition, no varying of pace, few topic and concluding sentences, as few words as possible, written with the expectation that the reader will not proceed to the next sentence or point without having thoroughly mastered the one at hand.

From Thomas Kuhn's The structure of scientific revolutions (1962, Chicago: The University of Chicago Press), p.11: "Men [sic] whose research is base on shared paradigms are committed to the same rules and standards for scientific practice. That commitment and the apparent consensus it produces are prerequisites for normal science, i.e., for the genesis and continuation of a particular research tradition."

And, p. 20, on the evolution of scientific genres: "No long [after the establishment of a paradigm] will [a scientist's] researches usually be embodied in books addressed, like Franklin's Experiments...on Electricity or Darwin's Origin of Species, to anyone who might be interested in the subject matter of the field. Instead they will usually appear as brief articles addressed only to professional colleagues, the men [sic] whose knowledge of a shared paradigm can be assumed and who prove to be the only ones able to read the papers addressed to them."

Finally, p. 21: "Although it has become customary, and is surely proper, to deplore the widening gulf that separates the professional scientist from his [sic] colleagues in other fields, too little attention is paid to the essential relationship between that gulf and the mechanisms intrinsic to scientific advance."

Gems

Though not all of them are directly pedagogical, I've done a few things today about which a few words might be here said. They're definitely uncleaned jewels, diamonds in the rough. Extracting these dusty gems from the cracked earth in which they still lie, polishing them, cutting them, and showing them off will give me something to do for the next few months.

Let's see...

...I hurriedly hit "publish" after blogging just an hour or so ago, elliptically, about dinner this evening with a candidate for the University Writing Center Director's position. Obviously it would be inappropriate for me to say more than I already have about the candidate or even our conversations, but suffice it to say the conversations I had with the candidate and with my own colleagues on the search committee taught me much about the university's functions.

This afternoon I spent a few hours in researching the history of the writing across the curriculum movement and the writing-to-learn movement, and in the process put together a few ideas for first-day low-stakes writing activities I might try out in my Calc I course this coming fall. I hope to do a few focused freewrites, the first of which will be designed to let the students write a little bit of a mathematical autobiography, while the subsequent ones will ask them to bring forth from the cobwebbed corners of their brains whatever it is they remember from the last math course they took, whatever and wherever and whenever it was. These latter freewrites can then funnel into a class-wide discussion.

I also spent a couple of hours reading the several remaining LANG 120 essays I've been assigned as a first-round judge in this year's First Year Writing Contest. Not long after telling Maggie that I'd been disappointed that none of them had "popped" for me (there had been several fairly good ones, but none that were clear front-runners), I read three papers, back-to-back, which were strong in every way: compositionally, grammatically, citationally. Their authors had rich vocabularies, a knack for imagery and inventive, challenging sentence structure, and a strong compositional thread along which the reader could pull her/himself from beginning to end. I was happy to find these!

This morning's run gave me the perfect chance to do a bit of self-analysis and self-assessment. I found myself asking, at this point at which the close of the academic year offers the closest thing I'll ever get to the end of one chapter and the beginning of a new one, what it is I like about myself, and what it is I don't like so much: the former, ideally, I can play up and the latter, play down and try to work away...or at least admit and accept with greater awareness.

I don't know what this says about the way in which my brain makes sense of the world around me, but I found that when I wrote them down at the end of my run most of the likes and don't likes fell into dialectical pairs, nestled together like fraternal twins in some sort of psychic womb. I'm not sure, as open as I am, that I'm up for sharing every one of these pairs, but here's one which deals more directly with my role as an educator:

"I like that I'm a good problem-solver. I don't like that I often slip too quickly into problem-solving mode."

I've dealt recently with this issue in my personal life, in incidents in which I've started trying to puzzle out often inchoate and inappropriate solutions to problems about which whatever friend with whom I'm speaking really just wants to bitch. The mathematical, and, some might point out, stereotypically masculine, part of me wants to simply get at the problem and root it from the ground. This isn't always the appropriate course of action: sometimes all my interlocutor wants is for me to shut up and commiserate or empathize. (Another one of my "like/don't like" pairs, closely related to the first: "I like that I'm a good listener. I don't like that I seem to think that gives me license to respond, when often all my partner in conversation wants is someone to listen.")

Pedagogically, I see the effects of premature problem-solving first-hand in many of the interactions I and my colleagues share with our students on a daily basis. (I thought of the following at dinner this evening while we were all discussing the ideal meeting between a writing consultant and a writing center client.)

Imagine that a calculus student has a problem with a run-of-the-mill textbook problem, and he comes to his teacher for help in solving it. He's bravely stepped into his teacher's office, interrupted his teacher's work, and sat himself down heavily in the chair across from the teacher's desk.

At this point the weakest teacher (often eager simply to get back to her work) will simply work the problem out for the student. (Novice teachers are often prone to this.)

The stronger teacher will take the time to ask the student what work he's done so far in trying to solve the problem on his own and try to build on what's already there. (This is where I am most of the time.)

The strongest teacher will ask the student what problem it is that the student's trying to solve in the first place. "Is there a problem at all? What, really, are you being asked to do? Can you articulate it for me, maybe in your own words?"

For a confident student with reasonably strong mathematical skills, a student who's generally on target most of the time and who only occasionally really just needs a little nudge in the right direction, the second teacher's action will have served the intended purpose admirably: after an initial survey of the student's own work, it might take little more than one more step worked out jointly for the student to see where to go next on his own.

Yet the second teacher has instantly leaped into problem-solving mode, presupposing there is a problem in the first place, presupposing that the student knows what in the hell he's being asked to do. Does he? Should the student be a weaker one, a less confident one than his peers, perhaps he's not even sure what's being demanded of him. In jumping at once into problem-solving mode the teacher has missed an opportunity to help the student to develop a perhaps-yet-more-important skill than problem-solving: problem-posing. One cannot hope to become an effective learner if one is adept only at answering questions; one needs also to know how to ask them in the first place.

I'm sorry if this train of thought appears to be riding on rails laid out by a track layer on LSD. I'm really just writing this out in order to better understand it myself; I'm actively (even as I make this next keystroke) engaging in writing-to-learn. (Kids, let this be a lesson to you!)

I'm afraid that's about all I've got right now. It's been a busy Sunday, and it's late. I've got more meetings with our University Writing Center candidate tomorrow, and an early meeting with one of my undergraduate research students, so I've got to hit the hay. Perhaps I'll soon say more about other relevant "like/don't like" pairs. We'll see.

Should one ever chance...

...to be asked to serve as the "outsider" on a search committee, or as a member of a search committee not in one's area, accept! I've learned a tremendous amount about the workings of my own university (regarding funding, internal operations, interdisciplinary initiatives, distance learning projects, etc.) from my experience serving on the search committee for our new University Writing Center Director.

Rocks and gravel

I've just finished reading Jonathan Kozol's Ordinary resurrections, begun just a few days back, and I feel, as I usually do on reaching the end of one of his books, an overwhelming sense of tiredness bound together with a bright and unbreakable thread of hope and strength.

It's a good book on which to end this academic year, surely the most tiring yet of my career (although, as I've noted on this blog before, not the busiest). Reading it at the close of the year gives me at once a sense of closure (which I've sorely needed) and a feeling of renewed energy. The book ends on the same note, as Kozol describes a gift of art Pineapple, one of the children of whom he speaks about most often in the book, gave to him: "an imitation stained-glass window that she made from tissue paper, brightly colored with green paint and with a wash of light-blue ink...When I asked her recently if it was supposed to be a rising sun or a setting sun, she seemed at first to not remember what I meant...'You decide,' she told me uselessly...at the risk of being sentimental about somebody whose sunny disposition brings a lot of joy into a world that has too many cloudy afternoons, I like to think it's rising" (p. 339).

Is it?

Kozol often admits in this book and others he's written since that he's getting older and feeling weaker. Although not yet as frail as he is physically (he's over seventy now, if I'm not wrong, and was sixty-four when he wrote the book I've just finished), I feel, on certain days at least, that I can empathize: each passing year steals away a little bit more of my relevance, my credibility, and my coolness.

"You may have them eating out of your hand now," my department's chair sometimes warns me, a hint of glee in his voice, "since you're not all that much older than they are. But just wait until you're my age." I'm warned that I won't be seen so much as the cool older brother but rather the stern-but-caring father. Just this past semester I encountered a student out of whom my most earnest attempts at cool cajolery could coax nothing. Magda, scarred, I suspect, by recent unpleasant experiences with mathematics, was timid in her dealing with me, reluctant to take part in any activities in class, and unresponsive to personal offerings of assistance I sent to her by e-mail. "I have to be honest that I've gotten a sense of 'defeat' from you for much of this semester," I wrote her. "I know you'd mentioned earlier this semester that you'd considered math as a major, and I hope that that's not an idea you've abandoned entirely."

I urged her to reply, but I never heard back from her.

This incident can't help but make me think of a noontime meeting several of my colleagues in the department and I shared with a pair of textbook company representatives. While my colleagues and I pored through elaborate boxed lunches bought from an off-campus catering company, the two textbook reps gushed for several minutes over the features of their company's latest Stewart-clone calculus textbook.

They tried to make their case, I'll give them that. I'd indicated that I honestly don't use whatever calculus textbook I've been assigned as much more than a source of examples and exercises (generally the exposition in such textbooks is godawful, and the organization is unmotivated, at best), supplementing the textbook substantially with descriptions, worksheets, activities, and projects of my own devising, including a number of nontraditional writing assignments. "We're very proud that [their text] contains [some unsubstantiated and moderately large number of] pre-written group projects." I nodded, unimpressed. (So does every Stewart-clone; generally these projects are as unmotivated as the integument of the text itself.)

About fifteen minutes into their demonstration, and after about ten minutes of fiddling with recalcitrant teleconferencing software, the reps got the lead textbook author himself on the line. He sounded tired, as though he'd made these dogs and ponies to dance two or three times already earlier that morning (or at least that week). He showed us a number of the features of the on-line text, most excited about the numerous Mathematica-driven animations he and his colleagues had developed for the textbook. (Such animations are not in themselves bad things; however I fear that without empowering the students to learn how to create their own animations, the animations alone do little more than provide an alternative visualization tool.)

After trotting the poor schlub of an author around the rink for a few more laps, the reps resumed their own presentation, and reached the part of their sell about which I was reminded above. "The beauty of the on-line scoring capabilities," one of them said proudly, "is that once a student has submitted her homework and the homework is graded, the computer can generate a personalized letter indicating to the student what sorts of problems she got wrong, telling her where she needs to focus her study. This can be done automatically, for a class of two hundred students. They'll think that you cared enough to write a personalized note to every one of them about their work."

I don't want to give the impression that this past year was all about weariness and defeat. To the contrary, I feel I've had some tremendous successes in the classroom.

While the textbook exercise fell on its face this past semester in Topology, it went over fantastically with the MATH 280 folks in the fall, and I can't think of many moments in my career at which I was prouder than I was when Ulrich and Uriah spoke on the project at the Elon conference (including in their presentation the phrase "writing-to-learn," and defining the phrase correctly!).

While I feel I lost the "faith" of some of my favorite students, like Magda, above, and like Tish, who became openly disillusioned about topology by the semester's end, I feel like that loss was only temporary, and that through open communication and understanding I've won that faith back.

While I feel my attempt at crafting a more meaningful system of feedback met with mixed success, the success I've earned is great enough to encourage me to recraft the system and try it again next term, rather than simply to abandon it.

While I've had my share of dealings with troubled and troubling students this past year, I've also had my share of dealings with marvelous ones. Jacobina's transformation from a hesitant and nearly math-phobic student at the outset of Calc I to one of the most outspoken, confident, and competent scholars in the stronger of my two sections of Calc II was nothing short of astounding. Words cannot describe the pride I have of her. Equally exciting is the undergraduate research Tonio and Siegfried have done, and the potential Ino and Iris have to do similar research this summer.

These small (and not-so-small) victories and many more like them are the rocks and gravel out of which, as the old blues standard goes, a solid road is built. Where's the road headed?

I've got back-to-back conferences coming up in less than a week (the International Writing Across the Curriculum Conference in Indiana and the AMS Eastern Sectional Meeting in New Jersey), then a half-week of Integrative Liberal Studies workshops in which I'm playing some role or another. A week after those come to and end the REU students start arriving. I have high hopes for this year's program; they're clearly a bright bunch with a lot of strengths. I've got a book proposal in at a good publisher, on which I'm still waiting to hear a decision. I've got great ideas for my courses next fall. It's going to be a good year.

The road is a long one, but the sun's shining yet.

I think, indeed, it is rising.

Time to put the "I" back into "integral"?

From Jonathan Kozol's Ordinary resurrections (2000, New York: Crown, pp. 54-55):

I used to edit out these questions that concern my private life when I was writing about children. I think, in part, I did this to avoid the risk of "complicating" things too much by intermingling the details of my life in Massachusetts with the more important details of the stories and impressions that the children chose to share with me. I think that this was probably connected also with the old idea that I, like many other writers, used to have of trying to remove ourselves from any situation we described in order to convince the reader, or ourselves, that we had not become entangled in these situations in a way that might affect our objectivity or have some power to affect the way children chose to speak to us.

I think most of us recognize, however, that we do become entangled and that we're never really neutral in the way that we conduct a conversation with a child.

Couldn't have said it better myself, only I'd simply omit the last three words of the above quote. It's time we face up to the social forces that shape our understanding of the world around us, whether we're talking about our understanding of spirituality and religion (as Kozol is above) or our understanding of science and mathematics (as I often do).

A modest proposal (one having nothing to do with cannibalism)

Hmm.

In the past week or so I've had several really good conversations about this past semester's revisions policy with students in both Topology and Calc II. A few things are clear from these conversations:

1. A large number of students benefited from the policy of allowing unlimited revisions (on exams in Calc II and on homework in Topology). Moreover, they benefited in exactly the ways I would hope that they'd have benefited: they profess to being much more eager to look over their past mistakes, to understand what they did wrong, and to attempt (sometimes over and over again) to correct their errors.

2. A much smaller number of students played the system like a two-dollar fiddle, not taking adequate measures to prepare for the exams (or the homework), putting off studying various topics until after the exam's been given, graded, and returned (or until after time's been found to corner one of the class's brighter students and ask said student for a transcription of her or his homework).

3. Something needs to be done to counteract the inevitable effect of procrastination, so that I'm not socked with a hundred revised exams and homework sets to grade at the same time final exams come due. (I apologize if the mass of grading I had to do during the past week as more and more revisions came in made me a bit more snippy of late.)

One of my Topology students made the following suggestion (thanks, Karl!): include with each successive revision an "expiration date," which, once past, precludes further revision. For example, I might allow a student a week to complete each iteration of a revised assignment: say I hand back a graded assignment on Monday, May 10th; the next revision, should a student wish to undertake said revision, would be due on Monday, May 17th. If that date passes without revision, the student's out of luck. If the student does perform some revision and receives the paper back on Wednesday, May 19th, then he'll have until Wednesday, May 26th to effect further revision, and so on. It would be the student's responsibility to be present in class to pick up graded revisions.

This would definitely help address the third observation above. Moreover, I think it would encourage more students to take advantage of the revisions: instead of putting them off until the inevitably busy end of the semester, students would be pressed to perform revisions right away and receive the benefits thereof.

The second observation above remains unaddressed. What to do about those who play the system? I will say that to some extent the chickens came home to roost when the final exam was graded: it was clear to me from performance on the final exam which students cruised on through with little concern for actual understanding of the concepts treated in the course. However, perhaps it might be appropriate to provide some sort of system of diminishing returns, at least for the lower-level courses (like Precalc, Calc I or Calc II) in which I might allow unlimited revisions: the first round of revisions will earn 1/2 credit back, the next 1/3, and so forth, until after two or three rounds there's really no purpose, from a "credit" standpoint, to revise. I'd like to think the best students would continue to revise anyway, striving for a perfect theoretical mark and a more and more perfect understanding.

Anyway...that's where I stand right now.

FYI: I just kicked off my summer leisure-reading season by starting Jonathan Kozol's Ordinary resurrections: children in the years of hope, and although I'm only about thirty pages in (and enjoying it immensely), it's making me more and more aware of the need for "ethnographic" methods in the "hard" sciences. I suspect this coming summer will see me write a number of posts on this blog in which I discuss the humanistic side of mathematics, in which I experiment with what Laurel Richardson would call "creative analytic practices." Stay tuned!

Axe

The state legislature, like its sister bodies in other states, is currently considering further massive cuts to education.

I hate to sound cliché, but isn't it odd that we live in a society willing to spend billions on bombing sovereign nations into the Stone Age but unwilling to scrape together enough money to offer its most intelligent young members a halfway decent education?

To be continued...