Pre-summer reading and subsequent ruminations

My summer reading has begun with a bang.

I've been working my way through a book my colleague Tip lent to me, Radical equations: civil rights from Mississippi to the Algebra Project, by Robert P. Moses and Charles E. Cobb, Jr. (Boston: Beacon Press, 2001). Tip met up with Bob Moses, a noted civil rights leader, Harvard-trained mathematician, and founder of a grassroots math program (the aforementioned Algebra Project) targeting middle-grades mathematics students on the fringe, at the Mathematics and Social Justice Conference Tip attended in Brooklyn a month or so ago. Clearly Tip's thought deeply about the ideas in this book, especially as regards his nascent community-building math program and the NSF grant proposal he and I are soon to set about writing, and he was kind enough to lend his (signed!) copy of the book to me.

I'm about three fourths of the way through the book right now, and though I've found much of the text itself dry and unengaging (particularly when it gets bogged down in the toledoth of Moses's project's converted teachers; at some points it reads like a biblical genealogy: "and Norma Jean begat Shannon, who begat Tom, the Explainer-to-Children. Tom taught twenty years, and then begat Sylvia. From the Delta did Sylvia come, and her students were good in the eyes of the Sunflower County Board of Education..."), one or two of its ideas have really struck me. I've been won over in one respect in particular. To best describe my conversion, I ought to mention a conversation I had with a colleague at a school I recently (in January) visited.

We were talking of undergraduate math majors; I had mentioned that at UNCA we have something on the order of 80-90 majors at any given time, and my colleague was thoroughly impressed. He said that at his school (a school nearly twice as large as UNCA), there were perhaps half as many undergraduate majors. Ever wanting to be helpful, I made some comment along the lines of "I've got some ideas that might help you to bring those numbers up."

My friend's response was something like "I don't think we're looking to do that."

I was taken aback. Not only had I not scored a point on this man's scoreboard (he and I share a long history, and I've never wanted to let him down), but my view that more young people should be helped to embrace mathematics had been met with hostility. This man had no interest in bringing math to the masses; he preferred to let it stay the closely-guarded territory of the few and the proud.

Moses's book has helped me to make out this not-so-well-hidden trap many professional mathematicians fall into: too often we think of math as a religion, a cult of worship into which only a select few are to be inducted, and in which only an even smaller few are allowed to become high priests. We take pride in our weeding out of the undesirables, defined as anyone who doesn't have a natural knack for math, who comes pre-programmed with a love of abstraction and analysis.

I've seen our culling at work at several levels, more clearly at some institutions than at others, and I'm ashamed to say that I've even taken part in it, though unwittingly. Math professors have a tendency to emphasis the arcanity of their field, its abstruseness, its disconnect with reality. Many of us take pride in the difficulty of our field, and make every attempt to show off our own intellects by making math seem imponderable, impenetrable, dense. I saw this most clearly at Illinois, where research mathematicians would show open disdain for all but the brightest undergraduates in their classes ("most of them are dullards, I've got a few who might prove capable"), lending a hand to the top 5% while leaving the others to drown; "they're just not cut out for it."

For too long mathematicians (and scientists of other stripes) have tried to fill their ranks from the lists of the best and brightest of the American studentry (to borrow a wonderful turn of phrase from William Strunk), leaving the dregs to find work elsewhere when they prove themselves incapable of meeting the high standards set for practitioners of math research. This leaves the untouchables, consisting primarily, in this nation, of poorly (read: publicly) educated blacks, latinos, and poor whites, out in the cold.

Moses makes clear that there's something wrong here.

What's called for, in his mind, is a radically different paradigm: rather than drawing from the few whom opportunity and natural talent have buoyed to the top, we ought instead be working to ensure that everyone is given the chance to rise to the surface. With a system wherein all are given the tools needed to excel mathematically, everyone benefits: traditionally underrepresented groups obtain the opportunities they need to succeed academically, and academicians expand broadly the talent pool from which they will one day choose their colleagues and successors. This "bring 'em on, all of 'em," attitude is the core of the Algebra Project, a program committed to making sure that every 6th, 7th, and 8th grader is given the background needed to successfully navigate a college-prep math sequence in high school. As Moses sees it, not everyone will go to college, and while there, not everyone who goes will study mathematics, but everyone should be ready to do so.

It just makes sense.

Moses's work has definitely helped me to come around to this point of view. It's never been so clear to me that I as a mathematician have a good deal of work to do to ensure that I'm doing what I can to grant everyone the chance to experience mathematics, and to succeed at it. Work needs done at all levels, K-12, undergraduate, and higher. And the work done at each stage needs to be interwoven with work done at other stages: vertical integration is called for. I hope Tip and I will be able to capture that spirit effectively in the proposal we put together this summer.

I've also begun the book the Project NExT reading circle has chosen as its first focus of discussion, Ken Bain's What the best teachers do (Cambridge: Harvard University Press, 2004). I'm (we're) one chapter in, and so far I'm unimpressed. It strikes me as a poorly-assembled pile of truisms and platitudes, absent the concreteness and careful analysis of a seasoned student of the scholarship of teaching and learning (SoTL). Not that every SoTL text has to build up a rock-solid wall of data and facts, or deluge the reader with a statistical breakdown of every study on teaching efficacy performed since 1970...but this book just seems "lightweight" to me. So far it's a great "rah-rah" feel-good page-turner, but its place might be on the nightstand of a newbie college prof just out of grad school who needs a little cheerleading and from-the-sidelines inspiration. I wasn't a huge fan of Maryellen Weimer's book, but I found it far more useful and engaging than Bain's, at least to date.

I ought also say a word or two about preplanning I've begun for Fall's classes. Francine's agreed to help me go through the notes and homework problems we used in 280 this past semester, retooling them, getting them good. I've already written a couple new in-class exercises dealing with writing mathematics.

The one I'm happiest about was inspired by a conversation I had with my colleague Lulabelle from the Sociology Department (I'm on a team of folks helping her out with a pilot assessment program for writing across the curriculum). She indicated that as a part of the work that'd need doing for this grant we're collaborating on, I'd have to to be able to train my colleagues how to "read" mathematics. I got to thinking about how I would best do this, and realized that it's likely easiest simply to highlight the linguistic analogues mathematics shares with "natural" human languages: syntax, grammar, orthography. Not only would an exercise indicating these analogues help my colleagues; it would help my 280 students, too.

The exercise consists of a take-home portion and an in-class portion. Each part comprises three written passages of varying levels of quality; the take-home passages are in "English" and discuss the chemical element boron. Students (and my colleagues) should have no trouble in ranking these passages from worst to best, and in explaining their reasoning for the ranking. The in-class passages are in "math," each giving a "proof" of the fact that the sum of two odd numbers is even. Having given them the chance to warm up in "English," I now ask the students to rank the proofs from worst to best, and to justify their rankings. This is a more difficult task, but once it's done my students (and colleagues) should be able to see more clearly that good writing in math is only a half-step away from good writing in any other discipline.

Okay, I've prattled on long enough. I'll end this for now. As usual, feel free to check in with your comments, always appreciated!

My 100th post

How 'bout that? It's taken me a while to make that first hundred, as rarely as I've been posting this semester. It's happened that most of the time when I've thought, "huh, that's an interesting thought. I might could write about that," I've ended up being too busy to post it before forgetting about it again. (Me? Busy?)

I've come to realize that for the most part, bloggers are either

1. college freshpeople who have more time on their hands than they know what to do with, writing about why Green Day is the greatest band in history (hint: they're not), or

2. pseudo-intelligent ex-English majors working in the food-service industry, writing about the brilliant conversation on Sartre they shared with the checkout guy at the Piggly Wiggly, and who think that now that they're blogging everyone's gonna find out what sort of genius they possess and that they're sure as hell gonna land that six-figure book advance.

Considering these options, it's probably best that I don't often have time to blog.

Nevertheless, I do corner a few seconds here and there, and sometimes those few seconds come at a time when I happen to be thinking about my teaching, specifically or generally.

Like now.

I just spent an hour or so hanging out in the comments section of one of my favorite blogs (Waiter Rant). Recently he (the anonymous New York-based blogger going by the name "Waiter") devoted a couple of posts to "assholes": one post listed 50 signs that you might be an "asshole customer"; a second, 50 signs that "your server might be an asshole."

This makes me think of an exercise I recently read in Maryellen Weimer's Learner-centered teaching, a work I referenced a few posts back, and which has given me a number of neat ideas to try out in my own classes.

Saith Prof. Weimer: think about starting the semester off with a brainstorming activity in which your students finish open-ended sentences like "I find that I learn well in a classroom where..." or "I find it annoying when the professor...". Let them discuss the matter, arrive at a consensus. This exercise promotes reflection on the learning process and on creating environments conducive to learning, and can serve as a prelude to a "classroom contract" in which the instructor agrees to work to construct an environment where the students' admitted concerns are addressed, and in response, the instructor can offer up a short list of behaviors s/he finds annoying in students and ask that the students do their best to avoid said behaviors.

Both I and my sole colleague in this semester's Learning Circle (shout out, Darlene!) agreed that this activity would probably seem condescending in an upper division class, but it might be a useful one to pull on first-years at the semester's outset.

Why not try it now? I'll share with you a list of my own pedagogical pet peeves, and in response, I hope you can feel free to share yours with me. I'm not claiming that any of my current students are guilty of any particular charge, but you might just recognize yourself in one or two of them. If you do, I hope that you'll do what you can to rein it in. As you'll know if you've been in one of my classes, I'm an easy-going guy, and I'm not likely to tear you a new one if you occasionally step out of line, let your cell phone ring because you sincerely forgot to set it to vibrate, can't seem to stay awake because you were up all night cramming for your Organic midterm, come in unprepared every now and then...I'll let it go, because I know we all have days like that, and I'm not an ogre.

And I like my students. I really like you guys. I have to say that in the almost-decade I've been teaching at the college level, of the roughly 700-800 students I've had in my charge at one time or another, I've personally liked about 99.5% of them. There have been a small few who've rubbed me the wrong way, a couple here and there that've gotten my cheese for one reason or another, but at the end of the day, I can literally count on one hand the number of students I've had whom I just couldn't stand. Really. You wanna know how many? Two. For real. Just two, and neither at UNCA. One at Vanderbilt University (initials RG), and one at the University of Illinois (initials KC). That first was a real piece of work. Remind me to tell you about his golf game up in Kentucky sometime.

If you find yourself identifying with one of the annoyers in the list below, please remember that it's the annoying habit I despise, not the person performing it. Chances are really good that I like you, and I want to continue to work with you as best I can. Just cut the crap, and we'll get along fine.

With no further ado, let me present you with

8 Annoying Student Habits
(I honestly couldn't think of any more. See how easy-going I am?)

1. I'm annoyed by endless complaints about how long it takes one to do one's homework (in my class or someone else's). Complaining about it doesn't finish it, it doesn't make it any easier, and it's not going to earn points from your professor (me included). If I think an extension is warranted (and often one is), I'll figure that out for myself, I don't need your help. Note: freshpeople are most often guilty of this behavior, as they've generally got a pretty poor sense of how much homework is "appropriate." By the way, I'll almost guarantee you that I spend at least twice as much time (often much more) in thinking up, designing, writing, photocopying, posting, grading, commenting on, and returning any single assignment or exam than you do in completing it. (If you ever wanna know how long a particular assignment took me to process, I'd be glad to give you an estimate, it's probably longer than you think.) Please keep that in mind before lodging a complaint.

2. It annoys me when students ask in class about course information that's available on the website. This isn't a big issue, but it's an annoying one nonetheless. I keep a pretty well-stocked website (this too takes a lot of time to maintain properly); if something's not listed/available from the course website, chances are it's not all that important. So if you've missed a couple of days of class and you need to find out what homework was assigned while you were gone, please don't ask me to spend three minutes at the beginning of class tracking that information down for you.

3. In the same vein, if you miss a few class periods, please don't expect me to give you a "synopsis" of the classes you missed. If you had a valid excuse for being gone, I might very well be able to spare 10-15 minutes to brief you on what went down while you were away, but I'm much more likely to actually give you this time if you've taken time beforehand to prepare for this briefing by reading the material we covered in your absence ahead of time.

4. Please don't complain about having to work in a group. I don't care if you don't like to work in groups. You know what? Not all of us do. I include myself in that list. I've always been one of those folks who wants to do everything for himself because he's not quite sure anyone else is going to do it as well as he will. You know what else? At some point in life, you're going to have to work in groups. It's called "committee work," another term for "hell." The experience in group work you gain now, in the relatively low-stakes, comfortable, safe environment of your classroom, the better you'll be at it in the future.

5. I've never been a huge fan of going over homework problems in class if doing so is not an integral part of the course's design (as is the case in my current 280 and 368 courses), especially if the students are not the ones doing the "going over" (see previous parenthetical comment). Some profs like to devote a good chunk of time to going over homework problems, while I, most of the time, don't. Occasionally I'll find it worth the class's while to go over the odd problem, but I'd rather you not ask me at the beginning of every class, "can we go over Problem 346?"

6. In classes where the solutions manual is broadly available, it annoys me to no end when students submit homework which was clearly copied from the manual. The manual can be a useful tool, if used properly, but it's worse than useless if the only purpose it serves for one is as a crib sheet. In the end, it's usually the student's loss, for a few extra points on the homework will be more than counterbalanced by the smack in the face the hapless student'll get come exam time when the solutions manual is unavailable for consultation.

7. Obvious obliviousness on the students' parts annoys me. If you're not gonna mind what I'm sayin' at all, then go home. If you're going to be your group's fifth wheel, go home. If you just can't be bothered to stay awake, go home. If you'd rather sit back and check out the box scores (Spring 2006, Calc II, Section 1?) in the sports section than focus on what the rest of the class is doing, go home.

8. Hateful speech. I hope this goes without saying, but for Pete's sake, people: please don't be crackin' "jokes" or whippin' off "smart" remarks about others' color, gender, ethnicity, nationality, religion, sexual preference, disabilities, intelligence, and so forth, whether it's in general or specific terms. There's really no room for that kind of thing anywhere in this world, and there's sure as heck no room for it in my class.

***

That's it, for now. Honestly. That's all I can think of off the top of my head. I'm probably in the minority, but little things like inadvertent cell phone rings and discreet lunch-eating don't get to me much. I don't even mind class clowning, if it's not too rambunctious or mean-spirited. It's just the big things, really.

So how 'bout it, Studenten? What professorly habits annoy you? What things have your profs done in the past (no names needed!) that you really could have done without? I'm truly curious.

(Re)start your engines...

All righty, then.

Tomorrow we recommence, revving up for the straightaway dash to the end of the semester.

This is as good a time as any to take stock of where we are in the semester, content-wise. Accordingly, I'm going to ask folks in each of my three classes to spend around half of their respective class periods tomorrow in reviewing what we've done so far: what have we learned? What techniques have we developed? How does it all fit together?

I've been doing a good deal of reading on pedagogy over the break, from the text for this semester's Learning Circle, Maryellen Weimer's Learner-centered teaching: five key changes to practice (Jossey-Bass, San Francisco, 2002), and Alife Kohn's No contest: the case against competition (Houghton-Mifflin Company, Boston, 1986). The latter does not deal strictly with pedagogical theory, but I came to it through Weimer's text, and I've found its insights useful in designing new classroom concepts.

A digest of ideas:

1. "Our classrooms are now rule-bound economies that set the parameters and conditions for virtually everything that happens there" (Weimer, p. 96; emphasis mine). A page later: "our classrooms are now token economies where nobody does anything if there are not some points proffered" (p. 97, again my emphasis). This economic image is an oft-used and apt metaphor for the give-and-take between the student and the professor, and I've come across it in one text after another. Surely some such variety of exchange is inherent in whatever classroom structure one could imagine, but my question is: must the classroom economy always be a capitalist one?

Given the research that Kohn lays out (suggesting that competition in the classroom and elsewhere is generally detrimental to both group and individual achievement), doesn't it make more sense that the classroom economy be one in which cooperative values serve as the "gold standard" for the course's currency? To carry the metaphor one step further, what if we redesign the economy so that it takes on a more "communist" hue?

For instance, I can envision, in a sufficiently small course (no more than, say 7 or 8 students), an untimed, class exam. Either in lieu of or in addition to a stand-alone individual exam, the entire class would be asked to complete a few problems as a unit, the professor sitting by as an observer and as a "clarifier," roles she or he typically already plays in proctoring an ordinary final exam. All students participate in generating solutions, offering ideas, helping to synthesize ideas already put forth. At the outset of the exercise, a single student could be chosen as a scribe in order to create a single solution to the problems presented, and perhaps no solution could be submitted which had not been "ratified" by every person present.

Yes, yes: there are problems with this idea. For instance, there would almost inevitably be "slackers," those who would get the same grade as everyone else without having participated at all, whether out of lack of knowledge or out of shyness. The more outgoing students would also have a tendency to monopolize the discussion.

A compromise between this innovation and the "traditional" exam format might look something like Weimer's study group exams, presented on pages 89-90 of her text. I think Weimer may have turned me off of this idea with her heavy-handed treatment of the "best" students who chose not to participate in the group exam (p. 90).

2. An idea transversing both Chapters 2 and 5 of Weimer ("The balance of power" and "The responsibility for learning") is the following: grant the students the opportunity at the semester's outset to, within reason, decide the distribution of point values for various types of assignments. This student-led distribution could occur on the first day of class, students breaking into small groups to meet one another and discuss the pros and cons of weighting this sort of assignment that much, and so forth. After giving each small group the change to come up with some rough guidelines, the class could be reconvened as a whole, and ideas shared. A consensus can then be approached: how much will this be worth? Once point values are arrived at, we'd record the result and all stick to the deal.

Obviously there should be some initial parameters outside of which the students would not be allowed to deviate. For instance, in Calc I class, I would ask that each of homework, quizzes, projects, and exams count for some percentage of the class's points, and I would likely set some minimum values (HW must be worth at least 10%, quizzes at least 10%, and so forth). But from there, the students would be on their own. I'd even let them throw in extra requirements, like attendance, if they saw fit to include them.

This arrangement has the benefit of providing students a chance to take control of the grading system to some extent, and thus while it gives them greater power (and less excuse for complaining should they not keep up!), it also invests them with commensurate responsibility.

3. Through Kohn's text I've found some interesting tidbits on pedagogical competition, from other sources: Morton Deutsch, in Education and distributive justice: a social-psychological perspective, Yale University Press, New Haven, 1985, writes: "If educational measurement is not mainly in the form of a contest, why are students often asked to reveal their knowledge and skills in carefully regulated test situations designed to be as uniform as possible in time, atmosphere and conditions for all students?" (p. 394, from Note 48, Chapter 2 of Kohn). Good question. As a fairly non-competitive soul myself, I hate in-class exams and see little purpose to them in the long run. It was this line, in part, that made me think up the class exam scheme in (2) above.

Also, Kohn says on one of the works of the brothers David and Roger Johnson ("The socialization and achievement crisis: are cooperative learning experiences the solution?," Applied Social Psychology Annual 4, L. Bickman ed., Sage, Beverly Hills, 1983): "In fact, even the widely held assumption that 'students learn more or better in homogeneous groups...is simply not true.' A review of hundreds of studies fails to support this assumption even with respect to higher-level students" (Note 28, Chapter 3 of Kohn). There's some ammo for the folks who take flak for "making the smart students work with the dumber ones."

All in all, I'm enjoying both books. Weimer, though I'm not always agreeing with her and I find her tone a bit condescending at times, has given me a good deal of practical ideas, while Kohn's work has been a great fount of references to other authors who purport to prove claims I've heard bandied about before but have never been able to track to the source.

Priming the pump

Here we are again!

It's the last day of the old year, and I'm really starting to get things in order for the start of classes, coming up in a little over two weeks.

I've got (and have had for a couple of weeks now) the syllabus for Calc I put together and posted on-line. I've taught that class often enough that I'm sure I could do it with my eyes closed and both arms held behind my back...which is exactly why I need to challenge myself to do it differently, better, this time around. Not that I've taught it poorly in the past, but I believe that now I'm capable of running this course so much better still that it'll make my previous efforts look like those of a first-year grad student. (I ain't knockin' on first-year grad students, some of them are hella good teachers; what they lack is experience.)

What'll be different about this coming semester? I plan on teaching this course in much the same way I've taught the last four sections of Calc II I've had: lots of application-oriented projects (which are, for the first time ever, built into the syllabus), structured team activities, including the ever-popular team quizzes, carrying over from last semester's MATH 365 course.

Then there's 280, our "Foundations" (read: "Proofs") course. To be honest, I haven't given it much thought, though that'll change in the next couple of weeks.

For 368, the course with the hifalutin' name "Theory of Numbers" (it's "number theory," people! "Number theory"!), I'm envisioning something much more akin to a seminar than a lecture. I may just have to take a page from Maryellen Weimer's playbook and let the students come up with their own course, selecting the assignments they'd like to complete from among a smorgasbord I place before them.

There is one goal I want to lay before them and make a sort of lodestone for the semester: what's the largest number you can prove is prime? I might make it a contest between the members of the class, to see who can come up with the biggest provably prime number before the semester is out. This'll spur them into reading about all sorts of primality tests, involving everything from basic modular arithmetic and Fermat's Little Theorem, through quadratic reciprocity and Dirichlet characters, all the way up to Dirichlet's theorem on prime congruences, and the Riemann Hypothesis itself!

Obviously this is a bit to bite off, let alone chew. But I have a feeling we'll get farther if I let them lead the race than if I serve as a pace car.

I'm off for now...I hope to get a working syllabus up for the other two courses before the week is out and I head down to New Orleans.