Workshop Idea #483 (not really)

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

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

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

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

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

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

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

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

I'd like to do this, I think.

Bragging on my students is a full-time job

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

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

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

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

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

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

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

I can dream, can't I?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

More to come!

No regrets

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

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

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

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

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

Any takers?

Mystery...solved?

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

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

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

There could be something to this...

Clarification

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

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

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

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

See you in class!

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!