Monday, December 28, 2009

Spring syllabi

At last!

The syllabi for my Spring 2010 courses have been posted on the now-up-and-openly-accessible course websites for Calc II and Topology. Supplementing these are the omnipresent links to my now-standard writing stylesheet, and for the budding topologists I've included a more-robust-than-ever introduction to LaTeX, now including a guide on inserting graphics files into LaTeX documents.

What about those syllabi? I've tried to take all of the feedback my students offered to me by e-mail (about 20 students eventually got back to me with their suggestions!) as I put them together, and I've tried to stay somewhat true to my goal of de-emphasizing grades (and thereby at least implicitly emphasizing collaboration), all while minimizing the "bureaucratic overhead" that was probably the number one complaint my 280 students had about their course this past semester, as measured by the course evaluation forms I got back a couple of weeks ago.

A few additional comments:

1. Rinse and repeat. I've backed off from explicit portfolio grading, but I've included the portfolio-esque unlimited revision-and-resubmission option in both courses: exams and projects in Calc II and everything in Topology can be revised and resubmitted as many times as the student would like to in order to craft an increasingly correct, complete, clear, and well-composed piece of mathematical writing. These systems of unlimited R 'n' R should help to cut down on the complexity of grading: if you're not happy with a grade, revise and resubmit. Again, if you'd like. And again. And again and again and again. I like this idea in theory and I'm excited to see how well it's going to work in practice.

2. Homework presentations. For the first time in two years I'll be teaching an upper-division course without homework committees, replacing the committees in Topology with "homework presentations" instead. This move too is designed to reduce "bureaucracy" by eliminating the deadlines associated with committee problem submissions and committee presentations, and it introduces a good deal of flexibility: as you can see on the syllabus, any number of students (including a single student!) may opt to join in on a particular homework problem's presentation, and there's only minimal bureaucracy involved in assessing participation in presentations. Speaking of flexibility, I believe the new syllabi offer tremendous...

3. Flexibility in assessment. Though there's no explicit mention of portfolios, in both courses I've left the exact manner of grading up to the students: in the first full week of class one of our tasks will be to decide, as a class, how each sort of assignment will be assessed, and with what weight it will affect students' grades.

4. Homework in Calc II. In Calc II, I will still be requiring homework, but at least half (probably more) of the homework will be handcrafted by yours truly, and I'll include frequent "short essay" questions demanding interpretation and explanation in addition to more traditional problems requiring no more than computation. Although I've still not made up my mind on the matter, I believe I'll grade the homework by giving each problem a number from 0 to 3:

0: no attempt, or a clearly half-assed attempt, made at solving the problem.

1: an honest attempt has been made, but the method of solution is not correct, and/or the student's work is very unclear and logically jumbled.

2: the method of solution is correct, but perhaps there are one or two minor computational errors; the writing is clear and well-composed, but insufficient explanation is given when explanation is called for.

3: solid; there are no errors (save perhaps a transcription error or a minor, minor error in computation), the student's logic is clear and straightforward, and all necessary explanations are sufficiently complete and are given in complete sentences.

It'll probably be a relatively easy matter for most students to get a 2 on a problem, but I'll warn them that getting a 3 is typically going to require careful attention to mathematical exposition as well as mathematical computation.

The syllabus for Calc II doesn't allow for unlimited revision and resubmission of homework problems, but if there's enough call for it, maybe I'll consider it.

5. The "textbook" assignment. The Topology crew is going to follow in the footsteps of this past term's 280 folks in writing a textbook for the course. As I did this past fall, I'll randomly assign topics from each "chapter" to students in the class. As in the fall, we'll meet up at "editing parties," but I'll be making these parties a bit more formal (going so far as to bring homemade goodies to them!) in order to encourage attendance. Finally, I'm taking La Donna's suggestion and writing a template the students will be asked to use in order to craft their textbook submissions; this will encourage at least basic typographical consistencies between the sections of the book.

Incidentally, La Donna and I will be meeting next week to hammer out some more edits on this past semester's textbook. I'm excited to see what we can make of it.

Before I go, I wanted to provide a link to the pedagogical Live Journal authored by my collaborator in crime, Cogswell. He and I will be sharing ideas with one another (often openly on our respective blogs) as we move forward with our courses this semester. I believe he's thinking about using the textbook assignment in his abstract algebra course, but is understandably skittish about the amount of work it might entail (he's the proud father of a brand-spankin'-new baby and has therefore clearly got other major commitments!).

More power to ya, Cogswell!

Wednesday, December 16, 2009

Course coordination

Cogswell (as he'll be known here), a colleague whom I've met in various places (while a grad student at UW-Madison he spoke in the Group Theory Seminar at UIUC, and later I ran into him while he was doing a stint as a preceptor in Harvard's Math Department), has suggested that we put our heads together as we design our courses for the upcoming semester, and as we put them into action.

A lovely idea! We've got many of the same ideas regarding teaching, and we're eager to try out some of the same methods.

I think our collaboration will be a fruitful one, and I'll look forward to trading traveler's tales with him.

Friday, December 11, 2009

Collaboration II: Electric Boogaloo

Today's collaborative extra credit session for Calc I is slightly better attended than Monday's was, with 26 people plugging away at problems while they partake of tooth-rotting holiday-themed treats, 7 more than the 19 who showed on Monday.

I'm not sure if this should be surprising: final exams end this evening, so in a way it's shocking to see so many people still engaged enough to make it to this session; on the other hand, perhaps enough people are desperate enough to do anything to add a few points to their grades that attendance is thereby boosted.

I don't sense desperation on most people's parts, though. Of course, everyone wants to get a good grade, but as a whole the students in these two sections of Calc I have done a good job in focusing their efforts on understanding and not on realizing largely artifical benchmarks of excellence. "I think our class already de-emphasizes grades," one of my students told me just a couple of hours ago as we were talking about my plans to further de-emphasize them next semester in Calc II. "I've felt all along that as long as I'm working on the homework and keeping up then I'm going to get a B."

"For the most part, that's true," I told her. "If you're doing what you need to to stay involved and engaged in class, and you're finishing the homework and doing decently on the exams, you'll get a C or a B, and most people in my classes get Cs and Bs. If you go above and beyond the basic expectations, you'll get an A, but you have to work pretty hard to get a D or an F."

I talked with her a bit about what a portfolio-based course would look like, and I admitted that I still haven't worked out all of the details for myself. "You have to turn in a grade at the end of the semester anyway, right?" she asked. "How would you do that?"

"It would be determined by looking at the products of the work you'd done throughout the semester and making sure that it demonstrates your achievement of various learning goals that we'd agreed upon in advance. Maybe we'd have said 'You need to be able to compute integrals of these types,' or maybe 'You need to show that you know some basic problem-solving techniques,' and I'd look to see that your portfolio contains assignments that show you can compute those integrals, and assignments that show you can solve some complicated problems."

I think we both ended the conversation with a better understanding of what our class would look like if I switched to portfolio-based grading, but I indicated that I'm still not sure that I'll implement that system in Calc II next semester. "I may try it out in my upper-division class," I told her, "and if it works out well there I'll contemplate using it the next time I teach a calc class of some kind."

But is this fair? I think now: one of the aspects of my own teaching I'm most critical of is the relative eagerness with which I apply techniques like inquiry-based learning and discovery learning and whatnot in my upper-level courses and eschew those same techniques in lower-level courses. To some extent this is understandable, since my lower-level courses are generally considerably larger than my upper-level ones, and such student-centered methods are much more easily implemented in smaller classes. Would portfolios present the same difficulties?

I don't think so. So why not go for it? Maybe I'm just clutching uncharacteristically conservatively at tradition, afraid to take that long, long leap all at once, preferring a few baby steps in its place.

I'll sort it out.

For now I'm going to sit back, close my eyes, and enjoy the pleasant hum of my students' voices as they puzzle through their extra credit problems.

Tuesday, December 08, 2009

Words to watch out for

Here are two tremendously off-putting words one might encounter in reading others' teaching philosophies: lecture (particularly when used without exception as a synonym for "class") and cover (or any variation thereof).

Monday, December 07, 2009

Random Walk, Take One

I've got a few photos to share from yesterday's inaugural random walk, sponsored by Algebra al Fresco. In all we had 16 different people take part (never more than 13 at a time), including three members of the general community who found out about the event through local advertisements. Good fun!

First, a demonstration of some of the equipment we used to help us plot our course:

Four-sided dice led the way at each four-way intersection; traditional six-sided dice directed us (with two values for each route) at three-ways. Only once or twice did we have to flip a coin for "either/or" choices.

It being a rather chilly morning (about 30 degrees Fahrenheit at the outset of the walk at 10:15), an early consensus decision was made to stop at Izzy's Coffee Den on Lexington Avenue, where walkers warmed up and partook of toasty caffeinated beverages:

Out on the road again, we stopped at every corner to conference on our next move. One person would roll, and another one or two, before the roll, would call the directions, pointing: "1, 2: that way...3, 6: that way...4, 5: that way..." It was a genuine group effort.

I don't know why La Donna's laughing in the picture below, but she's always laughing at me about something. ("You look goofier than me," she tells me. She's probably right.)

Pauses, though by necessity frequent, were never long, and we were soon on our way again. Below, we stride confidently down College Street, Ino (one of three of my Calc I students to show up...way to represent, y'all!) leading the way.

By my reckoning we took 38 steps, visiting 22 distinct intersections. Of these intersections, 11 were visited once, 6 twice, and 5 three times (none more frequently than this). As anticipated we never strayed off of the map of downtown Asheville I'd printed out for folks to follow as we walked. The furthest-flung intersection we ever reached from our starting point at the southeast corner of Pritchard Park was eight blocks to the east, the roundabout at the corner of College and Oak.

I'd be happy to provide readers with further statistics upon request.

All in all, it was a great social event, a good way to get some exercise on a brisk and beautiful late autumn morning, and a fair bit of nerdy fun. For sure there'll be another in the spring!

Collaboration takes coordination

We're about 50 minutes into the collaborative extra credit problem session (and potluck) I'd planned in order to give my Calc I students some sort of group-learning activity for the final exam.

So far, they're working together wonderfully.

18 students showed up by the session's beginning, and after a few minutes to organize the foodstuffs and return graded assignments, I pitched to them three problems that are considerably harder than the problems appearing on the exam itself, in the hope that their collaborative effort would make the problem-solving process a relatively easy one. I've asked them to submit their own individual solutions to these problems by the end of the session in order to receive extra credit, but they're allowed to work together in whatever way they'd like to in order to prepare those solo solutions.

Immediately the student broke into three groups, of 3, 4, and 11 students, respectively. (One student has now left, bringing the 11 to 10, and another just arrived, bringing the 3 up to another 4.) At first there was silence as the students started feeling the problems out on their own. Within about five or ten minutes, though, the chatter began, and in the murmurs they made I detected clear signs of honest collaboration: some students understood one part of a problem, and others another. They began comparing their solutions and sharing their methods. One student went to the board to log a tricky computation he'd just performed. "Just so y'all know," said another student, "he's writing up the step in Number 2 where we got stuck."

Another student has now gone to the board to indicate an approach for the third problem.

This is working well!

If only I can get them to eat some more of the lovely guacamole and pumpkin muffins that a couple of the students brought.

Sunday, December 06, 2009

The calm at the center of the storm

It's early, early, early on Sunday morning on the weekend between the last week of class and finals week. I've had a good stack of grading to get through, but I'm about two thirds of the way through that, and tomorrow morning brings the most recent Algebra al Fresco event, a random walk through downtown Asheville.

The semester's ending well. I've been worried about how my philosophical frustrations from a few weeks back might have been adversely affecting the students in my classes (especially Calc I), as I've feared they may have been buffeted by ever-shifting winds. But signs point to students' weather the storm rather well. One Calc I student spoke of being "inspired," and one 280 student said that, despite the difficulty and relative disorderliness of the textbook project, he learned "50% of his understanding of the concepts from the project." That comment, coupled with the envy my Spring 2009 students have shown for my current ones' getting a crack at this assignment, has convinced me that the same assignment, modified to fix the weaknesses it's shown this semester, should be a part of the curriculum for next term's Topology course.

I've still not put much thought into the precise structural details of that course, but it's starting to come together. It's not going to be as tightly structured, and it's going to be highly collaborative, and all assignments will allow unlimited revision and resubmission. Beyond that, who knows?

For now, it's late, and I'm off to bed. I hope to post again tomorrow with pictures of the random walk!

Wednesday, December 02, 2009

There's a first time for everything

I'm a pretty bad "sympathy crier": if I see someone close to me crying, I'm almost certain to join in.

I've had a good number of students cry in my office before (usually out of stress, sometimes out of disappointment), and there've been a few times when I've gotten in on the action.

But I'd never teared up in class before, like I did today when I was telling my 280 students what a wonderful job they've done on so many different things this semester, and how proud I am of them and how much I appreciate all of the work they've put into our course.

To those students: this class is one of the most challenging in our curriculum. I've asked you to prove deep theorems, and to perform complex computations. I've asked you to write proofs, explanations, dialogues, a textbook. I've asked you to review each others' work almost weekly, offering advice to your peers in both in-class presentations and written comments on each others' papers. I've asked you to master a technical typesetting environment, which is in many ways asking you to master a new tongue. You've done all of this willingly, eagerly even, without complaint. At all of these tasks, varied and difficult, you have succeeded beyond my expectations. As individuals and as a class, you've shone. You've soared.

Let me say this again, just so we're clear: I am proud to be your professor. It's working with students like you all that makes me love my job so much and makes me realize that I would never be happier doing anything else.

Thank you. I will miss our class tremendously.

Tuesday, December 01, 2009

Stress sets in

Yeah, we're almost done.

Mercifully, the semester will soon come to its end.

Signs of stress are setting in: even the best students in my 280 class are having a hard time picking apart the problems on the latest (and last) homework set, due on Friday, the last day of class.

Granted, the problems are difficult ones, built upon a vertiginous pile of definitions we've been amassing for the past several weeks: the set of functions from one set, S, to another, T, rests upon the definition of functions in general, which rests on the definition of relations more generally still, which rests on the definition of the Cartesian product. The notation for the formermost set is puzzling, too: since when does it make sense to talk about a set raised to the power of another set? And then you claim that the cardinality of this set is given by some exponential formula involving the cardinalities of the constituent sets?! Interrobang city.

It's late. These kids have given it their all, all semester long, and they're finally getting tired. It's taken them a long time to reach this point, but they're tired. I figure I'll give them a break: we'll spend a little while in class tomorrow planing down some of the rough spots in the hardest of this homework set's problems. They've earned it.

And on Friday, and during the final exam time slot next week, it's back to the book: we've got two more chapters to write and edit, and then the blasted thing is done. A first complete draft, anyway; two of the class's students have already expressed interest in putting some more time into the project over the break, polishing up the pagination, making consistent all of the notation and terminology, and putting together nice pretty pictures illustrating the concepts more amenable to visual expression.

Are these folks awesome, or what? I'm going to try to recruit as many of them as I can to attend the MAA Southeast Sectional meeting at Elon in March...and to try to get at least one of them to speak on the project there, with the others singing the harmonies.

We'll see how that recruiting goes.

For now, it's off to bed. Another long day comes tomorrow, but Calc I should be fun: Riemann sums await!

Monday, November 30, 2009

A test with a twist

I've decided to try something new this semester for my Calc I final.

As in semesters past, the final exam will be a cumulative take-home exam, which leaves me having to do something with the class during the scheduled final exam time (the university mandates that the class meet to do something during this time, even if there is no exam).

Instead of having a review session, I'm going to try out some sort of collaborative component to the exam itself, offering the students a low-stakes place in which to try to put their heads together to solve some trickier extra credit problems.

I'm not sure exactly what it'll look like, but I think I'm going to pitch four or five problems to them and let people work together in whatever way they'd like to to try to find a solution.

We'll see how it goes.

Meanwhile, in the 280 we're spending most of our remaining time together polishing the textbook. Today all four revised chapters got another re-reading, and I'll spend much of tomorrow cleaning up their new suggested edits and getting ready for our collective editing sessions during Wednesday and Friday's classes.

The end of the semester brings another topic to the fore: course evaluations. I'll be asking my students to fill those out on various days this week (Wednesday for 280, and either Thursday or Friday for Calc I), and the task is made a bit less onerous this semester by the fact that the university is using a newly revised evaluation form which I feel is vastly superior to the old one. The new form actually asks students to respond to more meaningful, measurable, outcome-based items such as "the instructor encouraged you to develop your critical thinking skills" and the like, as opposed to the thinly-veiled popularity contest items of the old eval forms (some of which were as inane as "This course was a good course: strongly agree/agree/disagree/strongly disagree"). I'm looking forward to seeing how well it handles in practice.

Sunday, November 29, 2009

Survey says...

I've heard back from about ten students so far concerning the "survey" I sent out asking for feedback on my plans for designing next semester's courses. About half are students in Calc II next term, and about half will be in Topology.

Not too shockingly, the students from whom I've heard are among the respective courses' strongest and most dedicated, so some of their views should be considered accordingly.

Nonetheless, there are patterns emerging:

1. Like it or not, even the best students are highly motivated by getting good grades. While most of the students admitted they knew they shouldn't be compelled by the desire to get high marks, most of them owned that getting high marks eggs them on and gives them what they feel is an accurate measure of their progress. It's hard to say whether these students, among my strongest, are more or less likely to be motivated by grades than their peers who have to work a bit harder to keep up.

2. Nevertheless, I'm heartened that the students who have responded are all up for something new. Though the idea of their work being assessed in some sort of "portfolio" system is an unfamiliar one to them, they seem open to the possibility.

3. Most of them are very happy with the way their current (or past) courses with me have been run, and are up for more of the same. This is also heartening to me, as ultimately I think I do a good job in most of what I do for my classes, and much of what I do now will remain in next semester's courses, largely unchanged.

More to come, as more responses come in. For now, I'm off to bed.

Thursday, November 26, 2009

Obligatory Thanksgiving post

The fourth Thursday of November has rolled around again, and comes the day on which everyone is busy reflecting on their lives and feeling gratitude for what they there find. Obviously I've got a number of things I'm grateful for (health, relative wealth, a wonderful life-companion [and a couple of delightful furry four-legged hangers-on], a boatload of fantastic friends, and a few close family members), but I thought I'd update the ol' blog with a post concerning those aspects of my academic life for which I'm most thankful.

In no particular order, then, I'm thankful for

1. My strong students. Almost without exception the students with whom I work on a day-to-day basis are smart, hard-working, and tireless in their efforts to strengthen themselves intellectually. Having taught at schools where students' senses of entitlement often dwarf their receptiveness to new ideas, teaching at UNC Asheville is an especial joy: the majority of my students are clearly here to learn, and to learn how to learn, and they're up for being challenged. Though many (if not most) of them are frantically juggling work rosters and preschool pick-ups and and day care schedules and doctor's appointments and god knows what else, they somehow manage to find time for my class. I can't thank them enough.

2. My clever colleagues. I am surrounded by a family of dedicated, deep-thinking, and tremendously supportive scholars. Each of my colleagues has her or his own unique talents, and each plays a different role in my work with them. Some motivate me as a teacher, offering new ideas to me and challenging me to perfect the ideas I've already dreamed up. Others egg me on and support me in my research, catching my mistakes, and helping me to see my ideas from new perspectives. Still others offer moral support, friends with whom I can sit and bitch about this and that whenever inevitable frustrations get me down.

3. The appreciation I get for the work that I do. I am endlessly delighted by the fact that I get to spending my life doing something that I love doing, and something at which I'm quite good...and that I get paid to do it. Moreover, hardly a day goes by without some sort of indication that I'm doing what I ought to be doing, and that I'm doing it well. Whether it's as simple as an e-mail from a student letting me know how much she enjoyed a particular class activity or as grand as an award for excellence in teaching, I'm grateful that my efforts are noticed, as it makes it that much more pleasant to keep doing what I do.

4. The academic freedom granted me. I speak here not just of the ethereal and ill-defined "academic freedom" to which the academy is ultimately dedicated, though I'm thankful for that as well, of course, but also of the particular freedom my institution and my overseers (chairs and deans and the like) grant me that allows me to apply often nontraditional and sometimes downright revolutionary techniques in my classes. I know very well that at some schools poetry in the math classroom would be anathema, and that at just as many schools I couldn't even dream, as I am now, of moving towards a portfolio-based grading system for my courses. (Hell, at many schools my calculus students would be asked to take a standardized final exam written by a committee of the department's faculty!) I am most decidedly grateful for the freedom I'm given to pursue the methods I think are the best ones for helping my students to learn.

Monday, November 23, 2009


I've got a cold.

And it's late in the day on the last day of classes before Thanksgiving Break.

Needless to say no one (me included) is motivated to do much today.


Saturday, November 21, 2009

Good advice

I've just send e-mails to all of the students currently enrolled in my Topology class next semester, and all of the students in one of my Calc II sections who've met me before, asking them for their input on course design.

We'll see what comes of it. Even if I only get about 25% or 30% response rate, I'll be happy.

Friday, November 20, 2009


Yesterday was a long day, and after working pretty much nonstop (class prep to research meeting to class to research to grading to another class to more class prep to another meeting to home to grade and grade and grade) from about 6:30 a.m. until 8:30 or so p.m., nothing could have made it seem longer than discovering at that latter hour that a couple of students had cheated on yesterday's Calc I exam.


Given that I've not much time to write right now (class beckons with a gently arching arm), I'll merely reference an older post which, written in a time of much more leisure, says all I feel like saying right now.

On the positive side, I've found from informal conversations that a number of my Calc I students continuing on to Calc II with me in Spring 2010 are open to the idea of portfolios and other outcome-based assessment methods.

Wednesday, November 18, 2009

Ma vie en film

I've finally had a chance to upload a few pictures from somewhat recent math-themed events, and I thought that by sharing them here I might offer a nice break from the heavy philosophy-laden posts I've been cranking out lately.

First, a couple of shots from the Goombay Festival held near the summer's end, at which I helped several of our students staff the Asheville Initiative for Math table. This street festival helped us to bring our Menger-making to a younger audience than we'd attracted in Pritchard Park during FractalFest '09:

The kid above wasn't nearly so gung-ho as the young (I seem to recall she was 10) woman in the following picture, who stuck around long enough to build her own level-1 sponge in its entirety:

Next, a few shots from the L-tile episode ("Build your own fractal") of Super Saturday this past semester. In the first, the kids are just beginning to get the hang of the iterative construction:

In the next shot, they're a bit more ambitious, busily working away at an L8:

And finally, here they are basking in the glory of the first L12 I've ever seen Super Saturday students (or MATH 280 students, for that matter) build:

Finally, here are a few shots of the most recent Menger-making, which took place a mere couple of weeks ago on the steps of the campus library, on a beautiful mid-autumn afternoon:

Above, Nighthawk (our school's most talented Menger-maker) sits proudly before a couple of the day's first creations. Below, Ino gets into the action:

Things picked up later in the day, with a number of non-math majors joining us:

By the time the sun was preparing to set, we'd nearly finished Algebra al Fresco's second-ever Level-2 sponge, which now, completed, rests in my office:

More photos as events warrant. Now, I'm for bed.

Monday, November 16, 2009

Help wanted: apply within

So here's the deal:

As regular readers know, I've been kicking around various ways of putting together next semester's courses, with ideas ranging from a simple tweaking of homework assignments all the way through out-and-out adoption of portfolio-based (and therefore virtually gradeless) assessment.

No matter what, there are certain elements of both courses I'll be teaching that will be essential.

For instance, Calculus II will involve the same sort of miniprojects it always has (we'll start things off, as usual, with A Confectionary Conundrum, and we'll keep the Funky Function Festival and various other small-scale applied projects and in-class activities involving delicious comestibles), and I'll still require that some sort of computational work be submitted each week...but much of that work will likely be ungraded, in order to engender a low-stakes atmosphere of exploration. But beyond these basics, I'm open to negotiating just about every other aspect of the class.

Topology, too will have its sine qua nons: however it's done, peer review must appear as a regular, fundamental, and well-structured aspect of the course; as in Calc II numerical grades will be de-emphasized in favor of ungraded, proficiency-based projects, and in-class discussion and discovery will form the basis of most of our class meetings. I will also be offering unlimited revision/resubmission of all coursework to be completed during the entire semester. Beyond this, I'm open to negotiations regarding the finer structure of the course.

I'd like to conduct such negotiations, in order to be sure that the courses I put together really do suit the academic needs and the learning styles of the students with whom I'll be working. Therefore what I'd like to do is ask those of my current and former students who will be joining me in one of these two courses next semester to help me put the courses together. (I'm ideally positioned to do this right now, as 27 of my current Calc I students and 4 of my Calc I students from Spring 2009 will be joining me in Calc II, and all but 4 of the 24 students enrolled in Topology have had me for at least one course in the past few years. I'll likely never have a better chance to engage so many students' opinions ahead of time.)

If you are currently enrolled in one of my courses, or if you have been a student in one of my courses in the past couple of years, and if you will find yourself in either Calc II or Topology next semester, you will soon be getting an e-mail from me asking you for your input on what our course should look like next semester.

I'm open to suggestions regarding just about everything:

1. The nature of homework (graded? ungraded? from the book? invented by me? some combination thereof?)

2. The structure of in-class activities (handout-based? note-based? suited to small groups? suited to the individual? some combination thereof?)

3. The nature of written assignments (papers? original research articles? expository articles? reflection papers? textbooks?)

4. Grading schemes (numerical? low-stakes? portfolios? how heavily is each component of the class weighted?)

5. Topics to be addressed (this applies more to the folks in Topology than those in Calc II; sadly, there's a pretty clear-cut list of topics we'll need to get through in the latter course, since it's not an elective) and the order in which we address them

As you respond to me, I'll ask you to think about what works for you: what do you need to get out of our course, and how can we put together a course which best helps you to learn?

I can't promise that I'll incorporate every suggestion, as I'm sure the range of opinions will be incredibly broad. Moreover, whatever structure results will almost certainly be guided in part by my own recent shifts in pedagogical theory, and there are a few things that will simply not fly. (For instance, I'm firmly and fundamentally opposed to the very idea of grading curves, and have been so opposed for quite some time.) Nevertheless, I'll try to take into account every word of input I receive and cobble together course plans for both of my classes next semester.

So, if you've had me for a class before and if you'll be having me again in Spring 2010, expect to get an e-mail from me soon. If you'd like to get a head start and don't want to wait for an e-mail, please feel free to respond to this post in the comments section (anonymously is fine). Let's get a discussion going.

I sincerely hope that you'll help me out here. Your education is more important to you than it is to anyone else, and I hope that you'll help me help you by taking a hand in scripting the acts of your education in which I will play a role.


Registration for first-year students began today at 7:00 a.m., and I find myself sitting in my office squealing with glee at seeing the names of the students with whom I'll be working in my Calculus II classes...I've got great carry-over from this semester's Calc I sections, and a handful I taught last Spring in Calc I.

I love these people!

For the record, y'all: Calc II is my absolute favoritest class to teach. I love teaching it so much that I make up words like "favoritest" to describe the experience.

Stay tuned!

Sunday, November 15, 2009

Philosophical frustrations

This semester's been a trying one for me, and it was only this morning that I figured out just why this has been the case: I'm currently in the middle of the most fundamental shift in my teaching philosophy since graduate school.

As regular readers will know, I've been questioning basal aspects of course design, including assessment, grading, and basic course organization.

Most of my upper-level courses are already taught in a fashion that's quite squarely in line with my newly emerging philosophy on pedagogy, but the realization that many aspects of my first-year course organization run contrary to this philosophy has caused a good deal of dissonance.

I'm frustrated by this philosophical shift, as it's caused me to reassess, mid-semester, the way in which I've put things together, and to plan ahead, looking forward to next semester already.

The frustration is not fruitless, and I'm not regretting it: for the most part I think the changes are good ones. However, they've meant some awkward adjustments for my students in Calc I, and I apologize if these adjustments have thrown anyone off. If any of my Calc I students are reading this entry, I'd like to tell you how much I appreciate your patience and understanding, and the enormous amount of work you all put forth to ensure that our classes as strong ones. I've enjoyed working with you all immensely, and I hope that I'll see many of you again next semester in my fully-redesigned (and silky-smooth!) Calc II.

If you are reading this and you have any comments or suggestions regarding what you think might make Calc II a good experience for you, please let me know.

Thursday, November 12, 2009

Hall conversations

I spent a lot of time in the Math Lab with my students today, and in the hall outside of it.

Most of that time was spent with my Calc I students, helping them out with the volume maximization problem they're working on right now. It's a barely-unprettified problem demanding a good deal of careful computation and innovative use of derivatives for optimization. They've got to find the least costly means of constructing a collection of dumpsters designed to hold 2000 cubic meters of material, knowing that the dumpsters' shape has to fall within certain parameters. It's a tough nut to crack, whose precise solution involves techniques from Calc III. They students are either loving it or hating it, for the most part. One thing's for sure: they're spending more time on it than they'd ever spend on a set of textbook problems, and they're learning a lot. As I was leaving campus just after 5:00 p.m., two or three of the groups banded together to throw an extemporaneous "dumpster party" in the classroom in which we meet. I almost wish I could have stayed.

I also spent an hour or so talking to a couple of my 280 folks about combinatorial and topological graph theory, and just straight-up topology. I taught Uriah and La Donna how to decompose a torus into a disc with identifications, and showed how this could be used to easily find an embedding of a complete graph on 5 vertices in the torus. It was good fun.

At one point soon after that I mentioned to several current and former students who were there assembled that I'd love to put together an informal reading group (much like the RAP [Research Among Peers] groups we ran at UIUC while I was a postdoc there), and they were all game. I mentioned Herb Wilf's generatingfunctionology, a freely-available text of which I've never read more than a few chapters and into which I'd love to get deeper. I think it would be accessible to some of our stronger students, and they could help the not-so-strong ones along. It could be a fantastic learning experience for us all.

Might could be we could swing that in the Spring.

I also talked to a couple students in the hall outside the Math Lab about my plans for Topology next semester (one's registered already, and the other plans to as soon as she can tomorrow morning). They're both regular readers of this blog, so both were familiar with my portfolio plans, and I asked how they think it'd fly.

Sidney (a student in MATH 280 in Spring 2009) is all for it. "It'd definitely motivate me. What motivates me is proficiency, and you'd be measuring proficiency at achieving learning goals for the course. I'm all about that."

La Donna (a current MATH 280 student) thought it might work well, but was a bit more reserved in her acceptance of the idea. "I have to admit that I'm a little motivated by grades," she said. "A good grade is a signal to me that I'm doing well and getting it."

I suggested that perhaps, as I've posited elsewhere recently, she's motivated by grades because she's been systematically trained to be motivated by grades. She admitted this possibility.

"In any case," I told them, "whether I grade by portfolio or not, whether I hand out numerical grades or not, I know for sure I'm going to permit unlimited revisions. I'm going to let people revise and resubmit, revise and resubmit, and so on, until they're one hundred percent satisfied that they've made their work as good as it can get." Both were excited about this idea.

They're both passionate students, and a blast to have in class. En route to lunch with our speaker the other day Sidney admitted that his mind had been blown on the last day of 280 last Spring when we'd talked about the existence of infinitely many different sizes of infinity.

Crazy shit.

I'm delighted that that delighted him. It's nice to have students like him, and like La Donna. And like Cornelius and Uri and Uriah and Tedd, all of whom were there to voice strong support for the idea of an informal research reading group.

They've got my back as much as I've got theirs.

That's a comforting feeling.

Tuesday, November 10, 2009

This and that

This week's gotten off to a good start, though Tuesday already feels like Thursday, and Friday will feel long overdue once it's come.

Today I played host to one of my colleagues from Samford University. Having driven seven hours from Birmingham, Alabama, Colin spent last night and today with me and my colleagues here, giving a great talk, chatting with me about REUs and the Sectional MAA, and meeting with various faculty and students from the department.

His talk was fantastic, offering the audience a unique blend of real analysis, linear algebra, and introductory proof techniques. There were about a dozen students present, and many of them are currently enrolled in...well...Real Analysis, Linear Algebra, and Foundations. For the analysts there were metrics, and orthogonal families of functions, and convergence; for the linear algebraists there were opportunities to apply eigenvalues to compute the closed forms for the terms of the Fibonacci sequence. For my MATH 280 students there were both implicit and explicit references to a number of the core concepts from the course: bijections, the pigeonhole principle, induction, proofs by contradiction, and equivalence classes and partitions. The talk was challenging but, I hope, accessible, and there were knowing smiles on a number of the students' faces as Colin reached his deftly delivered denouement.

In the afternoon, after his talk, Colin spent a few hours with me in my office talking about the design and execution of REUs, as he's hoping to submit a proposal to start one up at his own institution. I think I was able to give him some pointers and step through the process I followed as I put my own program together, but I couldn't answer every question. I honestly don't know what in particular about our program, aside from hard work and dedication on the part of the participating faculty and students, has made it so successful.

Colin will be heading home tomorrow; I've already been invited to join him at Samford in April, where he'll return the favor of hospitality he granted him during his stay here.

What else is new?

I realized yesterday that I was so busy bitching about grading over the weekend that I neglected to mention even once that on this past Thursday Algebra al Fresco sponsored the building of our second full Level-2 Menger sponge. (Pictures soon, I promise!) This one came together on the quad, on the steps leading up to the library. Working from 10:45 in the morning until nearly 7:00 that night, last Thursday several different students joined me in making the monster which now rests on a card table in my office, right where this past summer's sponge sat for a few weeks before moving on to the Engineering Department to get shellacked for display (so I'm's yet to reappear).

A single student, Nighthawk, was singlehandedly responsible for about half of the cube's construction. The guy's a born folder. By 5:00, when I had to head home, Nighthawk and my current Calc I student Lambert, having overseen the splicing of 16 of the 20 Level-1s needed to complete the Level-2, decided they'd not rest that night unless they'd finished the sponge, and so they worked away in the Math Lab for a few more hours, wrapping up over eight hours after construction had begun.

Nighthawk swears that he'll be able to set the unofficial world record for solo construction of a Level-2 sponge (current record: 15 hours). I believe he'll be able to do so, maybe after a few practice runs. Speedy construction poses an interesting operations research problem, actually: imagine a team of four builders working together to complete a Level-2 sponge. How best to use their time? All four should start out building Level-0s, and at a certain point one or two should switch to sewing together the Level-1s, and at a later point still one of these should switch over to the making of the Level-2, all while their two friends keep plugging away at the basic building blocks.

But when should the switches occur in order to minimize construction time?

And is there a more efficient means of splicing the lower-level cubes to form the higher levels? (There surely is...the question is more "what is the most efficient method?")

As I said above, I'll soon post some pictures of the construction. Most of it took place on an unseasonably warm and sunny day on the library steps. It was a pleasant Thursday.

What else is new?

Perhaps an update on the Fall 2009 Calc I Homework Debacle is in order.

After a good deal of thought, I decided to make all homework for my Calc I students optional for the remainder of the semester. It's simply not worth my time to grade half-hearted attempts at homework completed (or, more to the point, incompleted) by undermotivated students who are more often than not cribbing their answers from the solutions manual. To those (who I suspect will make up the majority of the class) who still wish to complete the homework, I promised to continue providing the same robust feedback and the same careful attention I've always given. (Not once have I begrudged granting such feedback and attention to deserving students; I'm frustrated only when a dozen hours of my time spent grading sloppy work remains unreciprocated and undervalued.) To these students I also promised to "lock in" their current homework grades, ensuring them that their grades will not fall but can only see improvement between now and the semester's end.

I can't stay mad at these students: for the most part they're hard-working, well-intentioned, bright, and fun to work with. As I said to them in class, I'm not frustrated with them so much as I am frustrated with the process. And as I said to one or two of them in the cozy confines of my office, I'm not disappointed that they come to me seeking ways to maximize their grades, I'm just disappointed that they and I have been caged in a system in which they feel it's necessary that they maximize their grades in the first place.

The students' relatively strong performance on the applications handouts from two weeks back has convinced me that such assignments may be able to form the backbone of a yet more student-centered Calc II course. Next semester's homework schedule might look something like this (assuming a four-day class meeting on MTWF):

Week 1, Tuesday: suggested textbook problems from Section x

Week 1, Wednesday: suggested textbook problems from Section x+1

Week 1, Friday: suggested textbook problems from Section x+2; due for feedback only: textbook problems from previous week; due for a grade, or for inclusion in a student's portfolio: applications handout regarding Sections x-3 through x-1

Week 2, Monday: applications handout regarding Sections x through x+2

And so on.

There's that "p" word again: "portfolio." I've thought a bit more about portfolios, and about what might go in them. Whereas, as I've said before recently, students might be able to demonstrate their achievement of very skills-oriented learning goals (like mastery of derivatives or integrals, for example) through including in their portfolios more traditional exams or quizzes, suitably suggestive applications handouts could provide students with relatively uncomplicated low-stakes writing assignments through which they might demonstrate achievement of some of the harder-to-get-at goals, such as maintenance of skepticism and application of problem-solving methodologies.

Speaking of skepticism, it delighted me to no end to hear Uriah, one of my Foundations students, talk about the ways in which our class has begun to change his perspective on mathematics. "You just can't take anything for granted," he said as we sat at the dinner table with our guest speaker. "I want to question everything, and prove everything to make sure it's true."

His comments reminded me of the Calc I learning goal I recently discussed on this blog: "Demonstrate (through informed question-asking) a healthy skepticism regarding mathematical and scientific arguments." His comments assured me that he, like a number of his peers, is getting a lot from our class.

And speaking of getting a lot from our class, I'm getting more and more excited about the textbook as it begins to come together, and as several of the students are expressing increasing interest in ensuring that it's executed as cleanly, completely, and correctly as possible. "I intend to share it with future 'generations' of students who come through this course, so please keep in mind as you write it that you ought to be writing to help them." It's got tremendous potential, and I hope to share it was as wide an audience as I can. You can bet I'll bragging on it at the Southeast Sectional Meeting of the MAA in March.

Okay, I'm clocking out for the night. I'll leave with a notice of publication: I found out a week or two ago that my article on using poetry in the mathematics classroom, complete with poems by several wonderful students whose work first appeared here and here, has now appeared in The WAC Journal. Let the celebration commence.

Sunday, November 08, 2009

Night and day

After all of the smack talk about grading I've laid down in my past few posts, I have to admit that I really enjoyed responding to the students' work on which I was working today.

Today's task was to give feedback to the students on the "application miniprojects" on which we'd worked in class for two days late in the last week of October. I'd made up three handouts, each of which led the students through an application of derivatives involving differential equations. One concerned terminal velocity, another population dynamics, and a third capacitance, current, and charge in a simple circuit. I made only the slightest effort to clean up the computational details, making sure to leave some messiness for the students to deal with as they solved the problems placed before them. (I wanted them to see some at least marginally unprettified problems stemming from realistic applications.)

Working in groups in class, the students were asked to complete one handout apiece and then put together a fairly extemporaneous informal presentation on their solution. Those presentations were solid, especially considering the students hadn't prepared much at all. They were then asked, for this past Friday, to complete two of the three handouts neatly as part of their homework for the week.

With no solutions manual to fall back on (they'd only whatever notes they'd scrawled during their peers' presentations to help them out), the students' completed handouts offered authentic examples of their work. They made mistakes, obviously, but the mistakes were real and understandable ones, not like the odd transcription errors that show up when a student is sloppily copying straight from a manual or from a friend's superior solution. (A tip to those of you who rely too heavily on the manual: when you begin a problem on your own and get stuck, ending your work in a messy pile of erroneous figures...yet somehow in the next line the correct answer magically appears after a logical lacuna the size of Texas, I'm liable to suspect that you didn't do the whole problem yourself.)

When the students failed in these handouts, it was because they honestly miscomputed a derivative, and didn't simply miscopy it. Or it was because the wording of their interpretations were clumsy, and not because the interpretations were offered in the stilted technical language peculiar to textbook authors.

In short, without the solutions manual, they really honestly had to do this homework. It was a refreshing experience to respond to them.

I'm going to ask them how they felt about it. Similar assignments could serve as a stepping stone towards a more outcome-based course, something I could reasonably put together for next Spring's Calc II courses. I envision suggested (but optional) textbook problems for computational practice, coupled with weekly handouts challenging the students to apply the principles discussed in class. These handouts could be the basis for in-class presentations, just as were the handouts from two weeks ago.

We'll see. I'm going to get the students' take on these handouts soon.

Further bulletins as events warrant.


"Why do homework?" I ask myself, after a long and frustrating day (yesterday) spent plowing through somewhat lackluster and clearly lackadaisically-done homework sets completed by my Calc I students.

Why, indeed?

For the opportunity for practice it offers in applying important concepts.

For the chance to experiment with relatively unfamiliar computations.

For the offer of exploration it gives.

Not for a grade.

So why grade it?

Because, like it or not, students are motivated extrinsically by receiving highly idiosyncratic, often arbitrary, and sometimes meaningless numerical scores on their papers...the bigger the numbers, the closer to the onset of the alphabet the letter they can receive for those numbers at the semester's end.

About those letters, at the risk of sounding crude, who really gives a flying fuck?

I don't.

Nor should the students.

I wrote "like it or not" above almost cavalierly, as though I myself am a victim of circumstance, that I play no role in establishing the primacy of those numbers, the hegemony of grades.

Of course, that's nonsense: it's clear from the comments I receive on student evaluations and the feedback I get from them after class that I play a major role in their academic developments. I'm proud of that.

But I can't be proud of building up and bolstering the hegemony of grades.

This shit has got to change.

Those grades have got to go.

Not the homework: the homework should stay. As should the feedback provided on it. But the homework itself should be the end, and not the number scrawled at its top.

The same goes for quizzes, exams, team projects: they all should stay, sans numerical rankings.

That much is clear.

But it's just as clear that making the transition from a graded to a gradeless introductory mathematics course is going to be a tough task, and I'm not sure it's one I'll be able to tackle between now and January's start of a new semester (and a new Calc II course).

I am, however, willing to try. I've just got to wrap my head around this portfolio idea.

Anyone else up for it?

Saturday, November 07, 2009

A simple observation

Boy, it's been a hell of a week, hasn't it?

Subterranean Homework Blues: an open letter to Calc I students

Dear Calc I folks,

Well, it's another exciting homework-filled Saturday night, and I'm about 4/5 of the way through my Calc I students' textbook problems. (I've yet to get into the similarly-sized pile of differential equations applications.)

More than half of my sixty-odd students have clearly spent a fair amount of time during this past busy busy busy week putting together honest and authentic solutions to the assigned problems. They've shown their work, and though they've made occasional mistakes, they've carried those mistakes through to a "wrong, but consistent" end. They may not earn perfect marks, but their work is, as I said above, honest and authentic: they've gotten out of the homework what I'd hoped they would.

The other dozen or so folks who submitted solutions...not so much.

Believe it or not, my young friends, I don't assign homework in order to give myself something to do on the weekend. I'm only human: there are definitely other things I'd rather be doing at 9:28 on Saturday night than working my way through a four-inch-thick stack of calculus papers, especially when a dozen or so of those papers are little more than sloppily copied versions of the solutions manual so readily available in the Math Lab.

Believe this, too: I assign the homework for you. Not for me. Presumably, if you're in my class, you're in it because you want to get something out of it. Maybe it's been your passion to be a physicist, or an engineer. Maybe (I hope, I hope!) you've always wanted to take up serious study of mathematics. Maybe you're not sure what you want to do, but you thought you'd give math a chance and try Calc I on for size.

Whatever your reasons for being with me for 200 or so minutes out of every week, the homework I assign is meant to help you out. It's meant to give you a forum in which you can apply the ideas we discuss in class in order to refine them, explore them, and take them out for a test drive. It's meant as a place in which you can practice. It's meant as a place in which you can learn.

It's not meant to be an eight-hour time-sink for either of us.

Yeah, I'd estimate that I spend something on the order of four to eight hours per weekend grading homework, and I suspect that the most diligent of you spend roughly that same amount of time per week on the homework and on going over class notes, putting together projects, and preparing yourself for the time we share together in the classroom. For these people, the homework serves a real purpose (see above), and it's to these people my grading is dedicated.

To the rest of you, I have the following thoughts.

First, to those of you who take your answers straight from the solutions manual: please give these exercises a shot. The homework is worthless, both for you and for me, if you aren't really doing it yourself. If you've fallen behind in your work a little, now's a good time to catch up again: the sections we're working through right now are pretty straightforward, interesting, and useful ones, and students generally find that they're quite fun. Give them a shot, huh? I promise you you'll get something out of it.

Second, to those of you who've clearly (as evidenced, for instance, by your exam scores) got a grip on the course material but who for some reason just can't seem to find the time to do the homework: wake up. Classes come a lot harder than ours, and you're not going to be able to coast through them not doing the work. You might be able to get by on minimal effort now, but minimal effort will only take you so far.

Finally, to those of you who feel as though you're putting your head into a wall every time you open your textbook, please, please, please come and see me. A little struggle is good: without at least a little bit of struggle, you're not making progress, and you're not learning. But a lot of struggle is bad news: it's distressing and debilitating, and it can sap your confidence like nothing else. (A propos of very little, I hope soon to post on my thoughts on one of Alfie Kohn's essays I just finished reading, on self-esteem.)

The same offer I make to you all: come and see me. Talk to me. Ask me questions. I'm open, I'm approachable, and I'm friendly. As one of my 280 students said to me by e-mail today, I'm human. I want to see you succeed. Hell, I want to see you soar. I know that not every one of you is going to be a math major (though I hope a good number of you will!), but whatever your goals, I want to help you achieve them.

With that, my friends, I'm going to get back to grading. Wish me luck.

At the corner of stress and frustration

There was something in the air today.

Everyone (and I mean everyone) I dealt with today seemed beaten, defeated, on the verge of tears.

Including me.

What have the students got to be down about?

There's a bad case of Multiple Exam Syndrome going around campus.

Tuition money's scarce.

Time is scarcer.

And relations are a bitch, aren't they?

They're tricky, they're terrifying...but they're downright beautiful once you start to get the hang of them.

My 280 students did really well on the first go-through of Exam 2, getting caught up on 2 of the 5 questions, both of the bugbears having to do with equivalence relations. Those that took them nice 'n' slow and wrote out everything precisely and explicitly had no difficulties; those who just kind of threw some stuff down on the page fared more poorly.

I get the feeling that several of them were about ready to kill me by about 2:30 this afternoon.

One student caught me in transit as we headed to and from our respective classes, claiming dibs on my time once she got out of her Calc III class. "I've got three advising appointments between now and 5:00," I told her.


"Come on by, and I'll see what we can do."

"I don't know why you keep saying this exam is easy," she told me. "I'm finding it really hard, and you calling it easy makes me that much more frustrated."

"Maybe it's not so much easy as it is basic...or elementary. Meaning that you don't have to use complicated concepts to finish it...everything goes back to the definitions."

She stared at me somewhat icily.

Minutes later, another student cornered me in my office and confessed she'd not started the exam until that morning. She's a senior with a full course load, and had three papers due that week. I thought she was going to cry, and I knew for damn sure that if she started to cry, I'd start crying, too. We shared a candid conversation about how much was expected of us, and I'd like to think that we both left the office feeling a little better about where we are right now. (I did.)

Another student still admitted that she's simply no longer motivated about the class. She's a non-major who's recently come to the conclusion that, once she drops her math minor, she has no reason whatever for taking the class, except for the Writing-Intensive credit she'd be able to get from a major course anyway. She's enjoyed taking math classes, but when faced with the likelihood of being here for more than four years just to finish up her major (minors notwithstanding), she's finding it hard to get into the mathematical swing of things. "I hate to drop all of this on you," she told me.

"I really appreciate your honesty," I said. "I've always liked the fact that you're not going to bullshit me or hand me a line."

Things started to pick up a bit once Calc III got out (several of my 280 students are in that class) and the students drifted on over, one at a time, almost continually until just shy of 5:30. I noticed that the students weren't doing nearly as poorly as they thought they were doing, and I began handing out a few hints here and there to encourage them to keep moving in promising directions. One by one, the exams came in, most of them complete.

"I really, really understand relations now," one student confessed to me. He was fairly glowing with the excitement of understanding. "And that's really cool, because I didn't understand them at all before this exam."

Sweet. What I hope most for my exams is that they'll prove to be effective means of strengthening student understanding, offering yet one more chance for students to explore, to analyze, and to synthesize. Exams in upper-division classes are more for me than merely assessment tools; they're means for making better thinkers of my students.

In that regard, I think this exam succeeded.

As of 12:45 a.m., seven hours later and 40 minutes ago, I finished grading the exams.

They did all right.

Actually, they did quite a bit better than all right. Granted, I gave them a couple of different extra credit opportunities, but even so the class average was higher than I'd expected, roughly 79.1%, before the revisions I'll allow until next Friday. Aside from the two tricky problems dealing with equivalence relations, the exam proved to be a walk in the park.

They'll get the hang of it. It's all good.

And now I must away to bed; I've got to be up in about 5 hours in order to get to Super Saturday tomorrow (on the syllabus: Euclid versus Lobachevsky!) and to get a head start on grading Calc homework.

Thursday, October 29, 2009

Editing party

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

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

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

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

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

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

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

Tomorrow: donuts!

Tuesday, October 27, 2009

School daze, part II

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

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

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

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


"Can you write it down please?"

"Do I have to write it?"

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

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

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

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

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

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

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

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

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

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

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

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

I hope she doesn't burn out.

And she's at a good school.

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

Decisions, decisions.

School daze

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

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

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

I'll be sure to report back.

Monday, October 26, 2009

Back to the basics

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

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

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

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

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

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

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

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

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

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

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

4. Be able to perform and properly interpret derivatives.

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

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

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

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

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

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


I don't have time to write anything meaningful.

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

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

We're all tired.

We're frustrated.

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

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

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

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

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

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

Thursday, October 22, 2009

Another thought while reading Kohn

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

Will the (Learning) Circle be unbroken

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

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

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

Which to choose?

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

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

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

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

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

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

And so forth.

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

I wanna try it!

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

Tuesday, October 20, 2009

The little gifts

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

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

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

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

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

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

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

It's the little gifts.

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

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

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

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

What little gifts will tomorrow bring?