Day One, revisited, or, Patrick very quickly gets off into pedagogical theory

Back to the grind. Today hardly felt like a school day at all, as my only MWF class is an 8:00-to-8:50 section of Linear Algebra I that was over nearly as soon as it started. (At least on Tuesdays and Thursdays I won't be done in the classroom until noon-thirty.) As first-days-of-class go, it was a good one, though. I've had better, but I've had far worse.

Plus ça change, plus c'est la même chose. Both other times I've taught this course (including the first time, the iteration of the course that occasioned the founding of this blog) I've started with some variation of the same game, a simulation of a Markov process in which the students shuttle some sort of token back and forth at each iteration of the game. The first run (Fall 2006), the students themselves were the tokens as the class participated in a great big single instance of the game; I switched to pennies (and smaller groups) the next time I taught the course (Fall 2010), and I stuck with that latter version today, though leaving a bit more room than I did before for students to discover and speculate upon the patterns their own damned selves. This time around I also asked the students to take bolder and more unassisted steps toward the next conceptual mile marker, solution of the linear systems that arise from the Markov process we investigate together: not only must the students experiment and then speculate on the outcome of their experiment, they must then find the appropriate mathematical model (a simple linear system in two unknowns) and then back-solve the "run the model in reverse." All in 45 minutes' time!

All in all, it went remarkably well. No one seemed lost ("one in a row!" as my colleague Tip would say), and everyone participated actively. I'm aided this semester by the fact that I've only got 23 student in the class (yay), though they're packt like sardines in a crushd tin box (boo), sitting at single-person-sized tables (yay) bolted together and arrayed in orderly rows (boo) in such a fashion as to discourage all but the most anachronistic teaching techniques (boo hiss).

Interesting facts (yes, there is a train of thought that took me from the previous paragraph to this one): recently, while reviewing the literature on the effect of class size on learning, I discovered that (1) said literature says almost nothing about college-level instruction, most research having been done at the K-12 level, and (2) a number of studies do not, strangely enough, suggest small class size improves student learning in mathematics. It was only after a bit of reflection that I realized why this might be: such studies, while controlling for class size, do not (and, methodologically, cannot) control for instructional method. Thus what I suspect is happening in these studies is large-section lectures are being pitted against small-section lectures, lecture being, until recently, about the only viable instructional paradigm for large-section classes. Of course, it is pedagogically retarded (in the literal...well...until recently literal...sense) to assume that one's instructional method remain the same when smaller class size permits more effective application of student-centered learning strategies: pit large-section lectures against small-section IBL and you're sure to see a difference.

Maybe more about that in a post soon to come (why on Earth was Patrick researching this topic? Edge-of-your-seat action!). For now, I've got reading to do for my first meeting of HON 479 tomorrow!

Moore ain't less

Back in January at the Boston Joint Mathematics Meetings my frolleague Stanislaus told me my name had come up in a conversation about plenary speakers for this year’s Legacy of R.L. Moore conference, an annual celebration of inquiry-based learning (IBL) sponsored by the Educational Advancement Foundation and the Mathematical Association of America. I was honored: this conference is well-known and reasonably high-profile. I wasn’t sure I was the best person for the job, though, for although I practice IBL in every course I teach, I generally do so in moderation. Only rarely do I use techniques that even closely approximate all-out Moore method, as I did this past semester in my two sections of Calc III. (Not having taught in Moore’s style for several years, I was a bit rusty at it, and I think the results were mixed.)

Nevertheless, I accepted the invitation.

I warned Stanislaus that I felt like something of a charlatan, for not only did I use Moore’s method infrequently, I had never even attended the conference before this past week. Stanislaus and others on the conference’s program committee reassured me and insisted that I might have something to say about inquiry and undergraduate research, something about which I do know a bit more.

So I set to work on my talk. It took me a while to decide how to pitch it. Should I focus on the act of research itself, and the role that inquiry plays therein, or should I try to tie research back to the classroom, where we’re more used to finding IBL more explicitly articulated? I settled on the latter approach, putting together an interactive presentation that would, I hoped, call attention to the parallel learning outcomes we encounter both in classroom teaching and in authentic disciplinary research, and highlighting the ways in which IBL helps us achieve those outcomes in whatever setting we might make them.

Early on Thursday afternoon, not an hour before my talk was scheduled to start, I was chatting with another frolleague, Ephigenia, whom I’d met during my postdoc at Illinois (she’d been a graduate student then). “I’m not sure I’m going to be saying anything new to anyone here,” I admitted. After all, I was smack-dab in the middle of IBL central. That research takes inquiry, and that research is in many ways no more than an extension of an interactive inquiry-based classroom, are not new ideas.

“This is sometimes a bit of a feel-good exercise,” Ephigenia assured me. (Boy, have I been a needy Nadine!) Makes sense: many of the folks at the conference are coming from colleges where no one else practices any sort of intentional IBL, and these folks need to find some kind of community. Hey, I’m not one to pooh-pooh the role that affect plays in teaching and learning.

I went ahead with my presentation, and as far as I can tell it was pretty uniformly well-received. It’s not likely that someone’s going to come up to my face and tell me that it sucked, but I had many tell me quite the opposite. I still don’t think I said anything new, though I hope it helped to give concrete examples of inquiry activities that don’t quite fit the Moore-shaped mold (the birdhouse exercise from last fall’s precalculus classes and the conversation on claw-free graph powers that took place between me and this year’s REU students about a week ago now). I might not be justified in feeling like a fake.

So now I’ve been to the Legacy conference. Will I go again? It’s good people, and I always like an excuse to get to Austin. But this time of year’s a busy conference season, and I’m pulled a hundred different ways these days.

We’ll see.

Shameful self-promotion

One of my editors, Bethany, has urged me to blog more frequently. So here I am.

I understand her point: she’s legitimately concerned that I might not be doing as much as I could to promote my work. Student writing in the quantitative disciplines: A guide for college faculty (Jossey-Bass, 2012) has been out for a few months now, and though I’ve every reason to believe that sales are quite good, they could probably be better, given more ambitious self-marketing. I’m not sure I feel up to this, though.

It’s not that I feel that self-promotion would be beneath me, or would constitute “selling out.” That attitude would be intellectually elitist and unbecoming. Believe me, I’m not against garnering a little fame (and a somewhat smaller fortune) from the book. There’s nothing wrong with showing a little pride in one’s work. It’s just that I’m not sure this blog is the appropriate venue for that self-promotion. Others are doing it, why can’t I?

Bethany mentioned the blog my good friend Erdrick writes, and the one that Maryellen Weimer, who helped me tremendously as my consulting editor for the book, updates regularly. Erdrick’s a wonderful colleague and a superb teacher, and his blog is superlatively good. Maryellen’s blog, too, is a wonderful periodical piece, and a wide-open window on current best practices in teaching at the university level. But I don’t think it’s fair to compare these blogs with my own; they serve different roles. I’ve never meant this space to be an intentional documentation of best practices, or a how-to manual on pedagogy. Though I don’t doubt my own excellence as a teacher (I’ve a great deal of evidence to suggest that I am quite accomplished as an educator), I have been, and I remain, reluctant to take on the task of systematically codifying my thoughts on teaching.

Rather, I’ve always thought of Change of Basis as a safe place to unpack my own teaching activities (and not, though they may frequently coincide, best practices in teaching more generally) in real time, keeping tabs on what goes on in my classroom, in my REU, at my school, in my mind. It’s more made up of notes-to-self than it is directives-to-others. Though I may cite books on teaching, I don’t do so as a careful and intentional review of the literature, but rather as an indicator of what I happen to be reading at any given time. Though I might bring out all the buzzwords (problem-based learning, inquiry-based learning, Moore method, writing across the disciplines, writing-to-learn, etc.), I don’t treat them methodically but only as they come up in my own work. How else to put it? I try to teach by example and not rote lecture. My tone is more anecdotal and less comprehensive and directive. It’s “I tried this trick out, and it worked out well” and not “studies show that this trick will reach students most effectively.”

I mentioned to Bethany that one of the reasons I’d not updated lately was that I’d not had much time lately to write. Between near-constant travel to present at conferences, seminars, colloquia, and faculty development workshops; leadership on the Curriculum Review Task Force (a full-time job in itself lately), preparations for the REU (now done with its second day), several ongoing research projects (in both math and composition and rhetoric), assumption of the Honors Program directorship (my administration began officially a few days ago), and teaching a full load of courses, I’ve not had the time I once had to dedicate to this blog…and when I have the time, it’s often directed into other writing projects (notably, 3x30 and poetry).

Honestly, I’m too busy being a dedicated educator to write about being a dedicated educator.

It’s thus that I offer my apologies to you, Bethany. I’m sorry I’m not posting as often as I might, and that my posts aren’t as pointed or focused as they might be. Please know that I wish I had the time and energy to post once, twice, thrice a week, offering some digestible and downloadable 750-to-1000 words of wisdom each time. Please know that I’m not angry with you for asking more of me, and that I do understand, and appreciate, your concern. This just ain’t that kind of blog.

That said, please consider giving Student writing a read. I’m proud of it, and I feel that it’s a very good book. I feel very strongly that writing has much to offer to students and scholars in the quantitative disciplines, and that we do well to pay attention to writing’s potential. I will walk the Earth from end to end to say so, again and again. If you’d like to talk about it, let me know.

Bounce

In a recent post I fretted a bit about one of my sections of Calc III, which section seemed to me a bit underprepared for class this past Monday. I worried that their apparent lackadaisicalness (if that is indeed a word) regarding Problem Set 4, on which we were working in class on Monday would lead them to be unready for today's class, in which they would be presenting their solutions.

I stand corrected. That section bounced back, showing themselves up to the challenge. Every single student called on to present did so, and did so with aplomb. I was particularly impressed by Dionne's willingness to work all of the way through the dreaded #61, which asked for a proof that two non-parallel vectors in the plane span the entire plane. Dionne, one of our promising young majors, has some exposure to linear algebra and is currently enrolled in Foundations, so she's no stranger to the proof genre. With a little help from a couple of her colleagues, she beasted that problem.

Yes, they bounced back, but not before I exhorted them to keep up with their work outside of class. Don't just come ready for the problem you think you'll be presenting (padded with the one or two preceding problems for insurance); come ready to present any one of them...and ready yourself as soon as you can so that when you're offered time in class to hash out the details, you can do so without delay.

Good work, everyone! I have to admit to a bit of nervousness at running my first Moore-method class in four or five years, but so far you're all making the most of it. Thank you for that, and for all that you do.

Feedback, as ever, is appreciated.

Worth a repost

This morning I received a brief but touching comment on my most recent blog post: "I miss Patrick teaching." I responded to this anonymous post in a manner which I repost here because I think it's worth wider readership:

Please know that I'm organizing this class in a non-traditional manner not because I want to avoid "teaching" (though, believe me, I'm doing as much teaching, in a non-traditional sense, as I would in any other course), but because I truly feel that the Moore method is the best way to approach this material. By asking you all to explain your ideas to one another, it firms up your understanding of those ideas. By asking you to take responsibility for your work, you become the authors (quite literally) of the ideas you're presenting to one another. It's much more learner-centered, and ultimately (I believe, and the literature on pedagogy bears me out) more effective.

Thank you for your kind sentiment! I've not totally disappeared from the scene; as you've noticed, I hope, I'll take my turn "on stage" from time to time.

To elaborate briefly: I know I'm a good lecturer, and I know that I explain things well. But seeing something done and doing it yourself are two different things, and you stand to gain much more from actually solving the problems yourself and explaining your solutions to each other than you do listening to me do it for you. It's a bit more work on your part, to be sure, but the time you spend on that work will be time well spent. Meanwhile, please know I'm still doing a lot of work behind the scenes, arranging problems in a manner I think is effective to help you work your way through the new ideas, including the definitions and theorems I think are most critical to us in our work, and working with you in class as you develop your solutions.

This I promise you: my explanations are still here for you if you need them, and I will be delighted to help you work your way through any problem you might struggle with. All I'm asking is that you give it all you've got to come up with solutions on your own first. Believe me, you'll get much more out of it that way.

So, let's stay the course, y'all. I'm enjoying class so far, and so far you're doing a marvelous job. Keep it up!

A tale of two sections

Today it was evident that over the weekend, most folks in one of my sections of Calc III took the time to work through the problem set they'll be presenting on Wednesday. It was equally evident that most folks in the other section didn't.

We'll see how things go on Wednesday. Most of the problems are pretty straightforward, but there are a few that might give pause. If Wednesday goes as I suspect it might, a few folks might learn the hard way that though it always pays (no matter the class) to keep on top of the work, but in a course structured as ours is, it pays double.

Moore is more

Three weeks into the semester, my Moore-method Calc III class has made it through three problem sets (50 problems), treating a substantive review of topics from Calc I and II and five or six sections of the textbook. It's been a few years since I've taught a course in this fashion, so there's been a bit of adjustment as I've gotten back into it.

So far, so good. The students are getting much better at explaining their solutions in front of a large audience (one section has 27 students, and the other 35), and they're becoming more relaxed, visibly. Yesterday's second section was particularly laid back, assiduously focused on finishing their tasks but willing to joke around and have fun in order to set the solvers at ease.

I've been very impressed with students' ability to be wrong in front of each other, and similarly impressed with the audience's willingness to ask questions. They're getting better at asking each other for clarification or elaboration, and not turning to me to ask. I'm letting minor errors slide, perhaps adding a little "does everyone agree?" if the solver's slipped up somewhere. Generally this has been enough to prompt one or two to express disagreement.

How's it helping the students? Hard to say. Several have said they get a lot of the course's design, though one or two have admitted "it's not what I'm used to, and I'm having a hard time adjusting." I've reminded them a couple of times now that in this sort of course they're expected to take on a bit more responsibility than they might in a more traditional course, preparing well and keeping up without my continual exhortation for them to do so.

I'm going to poll them more formally on the course structure at the end of the coming week, after we finish off the fourth set of problems. We'll see where we are.

Meanwhile, if anyone in the class is reading this and would like to comment, please feel free to do so, anonymously if you'd like.

Swimming!

Two class-days into the semester, and things are going swimmingly. I'm putting the course together with a modified Moore method, cycling (roughly) through the following steps:

1. handing out problem sets,
2. giving the students time in and outside of class to work through solutions in groups,
3. asking the students to present their solutions in class,
4. asking the students to write up solutions to selected problems as homework, and
5. quizzes the students on completed problem sets.

As of now I've met with the first section of the course twice and the second section once, but every meeting has been lively and energetic, and I've needed absolutely no coaxing to get students to work in groups and to come to the board. The atmosphere in the course has been supportive and friendly. I'm delighted that this morning's class gave us an opportunity to note that, as I'd promised, no lightning bolts came down from the sky when a student made a mistake at the board.

It's been a long time since I taught course using anything close to the Moore method (my special topics course in graph theory, run in Spring 2008), and I notice that I've grown considerably as a teacher since then. I'm more confident, and that confidence has enabled me to feel less awkward taking a peripheral role. In particular, I find that I'm much more able to sit in silence than I was in the past. Silence in a crowded classroom is disconcerting, and it's all one can do to keep from saying something after ten or twelve seconds of quiet have elapsed. I've grown accustomed to such silences, though, just as I've grown accustomed to (or, more accurately, enamored of) the thrumming of three dozen voices trading tricks as the students work in groups in class.

I'm confident. It's going to be a good semester.

On deck

It's a new year, and we're only a few days away from a new semester (beginning Monday, January 9th). There are big, big things in the works: my book comes out in a couple of months, the work of the Curriculum Review Task Force should be coming to a head this term (with concrete recommendations to the Faculty Senate due by April), and...well...other news items about which I'll be able to say more in a few weeks' time.

What's in store, teaching-wise? I've got three preps this term, one for two (large) sections of a course I've not taught in almost six years (Calculus III), one for a single (small) section of a course I've never taught (a Masters of Liberal Arts [MLA] course on the cognitive psychology behind mathematics), and one for my section (of two) of our senior seminar.

In the seven years I've been at UNCA we've not taught two concurrent sections of that last class, so I'm not sure exactly how we're going to manage it. I've yet to talk to my colleague Timon, who'll be teaching the other section. I imagine we might hold many activities together, splitting when it comes time for the students to present. There's simply no way we'll get through 26 student presentations in the five or six weeks we can offer them to speak. We'd have to have an unprecedented four talks per period to make it work, and that's simply unworkable. Thus splitting into separate sections for that part of the course, though not ideal, is about the best we'll be able to do.

Meanwhile, I've got plans for the other courses. Calc III, which I've not taught since Summer 2006 (the last summer I wasn't running the REU), I'll be running with a modified Moore method: one day each week will be devoted to discussion of new definitions and discoveries, a second day to small group work on the current problem set, and the third to problem presentations. Both sections of this class are big enough (roughly 30 students apiece) that I'll probably ask students to "present" simultaneously (two to four at a time) whenever feasible. This'll ensure that we make it through problem sets in a somewhat timely fashion, and that each individual student gets more opportunities to present. I've already worked with about half of the students in both sections, which familiarity will help me ease into the new term.

I can't say the same for my MLA course. It's been nearly a decade since I taught a graduate-level course (a special topics course on Coxeter groups and related groups which I led at UIUC back in Spring 2004), and this course differs dramatically from that one. We'll be exploring the learning and cognition of mathematics, and I plan to inject a good deal of philosophy and sociology into the mix as well, drawing on a number of sources to paint a picture of mathematics most people never see. We'll begin with Stanislas Dehaene's marvelous book Number sense: How the mind creates mathematics, about which I've blogged a bit before (see the "Dehaene" tag at the right), surveying the psychology of mathematical discovery, before moving onto Imre Lakatos's Proof and refutation, a philosophical treatise designed to lay bare the workings of what might be called the "mathematical method."

I'm not sure what to expect from this course. I suspect there'll be a week or two of me feeling out the students (currently there are seven students enrolled) to see where their interests and aptitudes lie. Likely none of them are straight-up mathematicians; I'll be curious to learn what they're hoping to get from the class, and I'm certain they'll help me give it more direction.

Ah, well...one week to go. Before then I'm off to Boston for this year's JMM, at which several UNCA students (and a few past REU students) are presenting. I'm particularly excited to see how far Ned's and Ino's work on nutritional data has come since their presentation at Kennesaw State in November. (They're presenting in a special session on mathematics and sustainability.)

Further bulletins as events warrant, likely soon.

State of mind

A few months ago Zima, one of my grad school colleagues who now teaches at Kennesaw State University in Kennesaw, Georgia asked me to present at an undergraduate research conference for which she'd just received MAA funding. (Said conference is this coming weekend; it's the one Ino and Ned are presenting their findings at.) I'll be giving a run-of-the-mill plenary talk on some of the graph theory I did with a couple of REU students this past summer, and I'll be presenting in a workshop on inquiry-based learning (IBL) at the outset of the conference.

I offered Zima a title that's so generic I really could talk about anything: "Guided discovery in the mathematics classroom." I feel confined by this generality. Indeed, when I actually sat down a week or so ago to try to figure out what in the hell I needed to say about IBL, PBL (problem-based learning), Moore method, etc., I had a hard time coming up with much to say other than expressing my feeling that all too often these techniques are too "formalized." That is, I get the sense sometimes that the people who apply these techniques look on them as an all-or-nothing process: "if it ain't straight-up Moore method, it ain't anything at all" or "I use guided discovery every single day to address every one of my students' learning outcomes." So I put together a half-hour laundry list of things to say along these lines: be open to using guided discovery in moderation; it's not the be-all-end-all any more than any other pedagogical paradigm may be.

Then just now, while lying in bed unable to sleep (though admittedly probably needing to), I realized that I can say more, for I realized of a sudden why I've had such a hard time trying to come up with something practical (and original...I suspect that the folks I'll be addressing in this workshop are going to make up a choir to whom I won't really need to preach) to say about guided discovery: in my mind, guided discovery is not so much a pedagogical process as it is a state of mind.

I find more and more that in teaching it's not so much what I do with my students as how I do it that matters most. Put another, perhaps more practical way, effective teaching comprises a gestalt-like complex of actions and not a single action individually. Guided discovery is what might be called an emergent operation which cannot be broken down into its constituent parts without losing much of its energy and effectiveness. So it is that I don't necessarily engage my students in singular activities, each of which forces students to lead themselves to original, new-to-them, conclusions, so much as I try to treat them in every way, in everything I do, as co-learners, co-discoverers, seekers of authentic knowledge.

Practically, this realization makes it possible to grow opportunities for genuine discovery in the most infertile academic soil, for every simple textbook problem becomes, if viewed from the right angle (like anamorphic art) a chance for authentic "research-like" engagement. Guided discovery is an "angle" from which these problems may be viewed.

I'll try to say a bit about this on Friday when I'm leading my portion of the IBL workshop. Are these views original? Meh...perhaps not. But they're more original, and, more important, more meaningful, than whatever else I will have to say. We'll see how they're received.

Partners in crime

I know a number of people (many former and current students, a good number of colleagues, and assorted folks I've never met) read this blog, but not many often comment publicly. Quite often, though, I get comments about it on Facebook or in my in-box.

This past week I got a note from a fellow who's teaching precalculus at an inner-city high school in Boston, using inquiry-based learning. He wrote me asking about the methods I'm using in my own precalc classes right now and shared some of his own (he's asked me not to post his notes, as they're very much works in progress). I'm very impressed! His notes are clever and engaging, offering students a scaffolding students can use to climb from the barest basics up to properties of advanced functions, logs and exponents, and trigonometry. You can read about his exploits here.

By comparison, my methods that are considerably less purely inquiry-based...he's doing pretty much straight-up Moore method with high-school students! Inventive and impactful.

Running in reverse

Recently I've had a chance to feed my undying love of linguistics. I've been reading up on the history of English and its antecedents (like Angl0-Saxon) and victims in the clash of tongues that's taken place on the British Isles since the early common era (like Cornish and Manx). The text I'm currently reading is Dick Leith's A social history of English (London: Routledge & Kegan Paul, 1983), an interesting book offering a glimpse of English's development as a social, as well as a purely linguistic, phenomenon.

More interesting than Leith's treatment of English per se are some of the observations he makes about the codification of language, and the role of "authority" in the preservation and propagation of language across time and space. A central thesis of his book is that all too often we forget that language is very much dynamic: it is ever in flux, constantly changing...and that in the end that change is not driven by grammarians or the intellectual or economic elite so much as it is by the ways in which every member of society chooses to use the language.

These are points that even the most perspicacious language-lovers among us tend to overlook. The reminders Leith offers have made me think of new (to me, at least) and "subversive" paradigms for poetry (a post on that soon, perhaps)...but they've also recalled for me the central role every member of a learning community plays in that community's advancement of knowledge, while issuing a reminder as to just how dangerous it can be to trust blindly in the authority of a textbook.

The following passage from Leith (p. 68) struck me (cf. the comments some of my precalculus students made on their last exam):

Unfortunately, many people tend to treat dictionaries with reverence: rather than being seen as a record of usage, the are often regarded as the arbiter of it, a source of enlightenment for the ignorant non-specialist. In fact, the traditional arrangement of words in dictionaries gives people a strange idea about language. The alphabetic arrangement disassociates a word from the company it keeps, presenting it as a unit isolated from context and words of similar meaning. More important, many dictionaries give the impression that words have only one meaning, to be found on the right-hand side of the page. Even the fullest dictionary, the Oxford English Dictionary (OED), which shows the whole range of meanings by citing examples of a word in use at different periods in its history, puts the meanings first, then lists the examples, thereby obscuring the process involved in deriving the meanings; for we learn the meanings of new words most efficiently by hearing them in a wide range of contexts....It is not surprising, therefore, that people often misunderstand them.

How often too we ask our math students to use their textbooks in the same way they'd use a dictionary, placing theorems and proofs before (or, more often than not, simply in lieu of) the intuition and arguments that led to those theorems and proofs in the first place? How do our textbooks obscure the many long hours of exploration and discovery that went into the derivation of the theorems that pepper the textbooks' pages? Without access to the discoverer's process of discovery, the reader is apt to feel as though a given fact or formula arises ex nihilo, and that they, the uninitiated, are not privy to its inner workings.

Food for thought. By me, it's better to let the students stumble around a bit, piecing things together for themselves as they author their own textbooks. That's just what I'll be doing when we talk about general rational functions in Precalc tomorrow...strap yourselves in!

Out of the wilderness

This morning I finished reviewing and responding to my MATH 461 class's homework set, the one featuring several very open-ended problems related to Fibonacci sequences, Euclid's Algorithm, and greatest common divisors. Their work was fantastic, and the sense I got from most students' solutions was that they'd achieved genuine understanding, something I honestly don't see present in most responses to the cut-and-dried prove-this-theorem sorts of questions one might ask in an upper-level mathematics course.

Moreover, the students made remarkable progress in proving some nontrivial mathematical results they themselves got to formulate. Several presented solid proofs of one direction of the equivalence I mentioned in my last post, and though no one successfully proved the converse (I was only able to do it myself last night, after several false starts), a few students made earnest attempts at so doing and a few finished just shy of the mark.

Furthermore, two of the students noticed that, though I'd not intended it, the fourth problem on the homework set had close ties to the previous three (all of which were similar). This last problem asked them to characterize the numbers which caused "worst-case" performance in Euclid's Algorithm when divided into 99. A bit of thought (after examining a mess of data) will convince you that the worst case is achieved when the numbers you select give the most "Fibonacci-like" sequence of quotients when divided into 99, numbers for which most of the "q" values stemming from Euclid's Algorithm are 1, so that the corresponding remainders remain as large as possible. Thus, these two students pointed out, Euclid's Algorithm should perform most poorly when you use to divide one Fibonacci term into its successor. One student even presented several pages of numerical evidence for this worst-case behavior, building off of the generalized Fibonacci sequences we'd just worked with above. It was splendid.

Finally, one student made an observation regarding the frequencies with which each "run time" occurred when Euclid's Algorithm is applied, noting that when plotted, these frequencies traced out a very normal-looking curve. "What might happen with other values for b, besides 99?" he asked. No doubt there's some nontrivial number theory lurking just below the surface: primality plays a role, for sure, and I'm sure Euler's φ comes into play.

All in all, I get the feeling that the students got far more out of this set of exercises than most (any?) I've ever assigned in my career. I'm going to see that all of my homework sets for the rest of the semester are in a similar vein.

Into the woods

It wasn't until I wrote yesterday's blog post that I realized the extent to which I'm pushing inquiry-based learning in both courses I'm teaching this term. In both Precalc and Abstract Algebra I, the majority of the homework problems students are being asked to complete are what can legitimately be called research problems, and I'm posing them as such, guiding the students through an initial "data collection" stage, leading them then to a "conjecturing" stage, and from here to a point where they should be ready to offer at least a partial proof. The questions I'm asking are very open-ended, and in a few cases already this semester I'm not even sure I know the answer.

Example: I've got the Abstract students making conjectures about the relative primeness of consecutive terms in generalized Fibonacci sequences: for what natural-number pairs (α,β) is it the case that any two consecutive terms in the sequence defined by s₀ = s₁ = 1, s_n = αs_n-1 + βs_n-2 are relatively prime? I admitted up front that I don't know the answer to this (though I have some guess as to what might be true), but I asked students to try out several cases, formulate a conjecture based on the data they gather, and try to prove their hypothesis.

What fun! I'm having fun, anyway. And what a way to learn! I have no doubt that the students are apt to become more talented mathematicians (and more generally, problem-solvers) when asked questions like this than when asked to complete cut-and-dried textbook proofs for which the answer is already told to them.

No regrets

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

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

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

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

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

Any takers?

Why I teach the way I do

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

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

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

Honestly, I feel elated.

About last night...

Having a few spare minutes before I have to get underway with this long working weekend, and feeling guilty as hell about my long internet absence in this space, I thought I'd take a few moments to follow up on one or two of the brief comments I made in my peremptory post last night before leaving for Charleston.

Indeed, I am in Charleston. I'll be here for a couple of days, during which time I'll be working away with the three folks from The College of Charleston whom I met at the CWPA conference back in September 2009. (Nicola's already got an alias; I'll call the other two Damian and Bella.) I'm not sure how much I've said about this project: we'll be digging into the weekly written reports penned by the REU students during the 2008 and 2009 programs, analyzing them from a rhetorical point of view, using them as mileposts to help us chart the development of the students as professional writers of mathematics as the program progressed. In order to help perceive this development I've refined the list of rough criteria I developed a while back on this blog. We'll see how that goes.

I'm tremendously excited about all of this; I feel like I'm taking my scholarship of academic writing to a new level.

Speaking of which, I finished up and sent off the book proposal I've been planning for several months now, flinging it across the country on diaphanous electronic wings. Its working title is "More than numbers: writing-to-learn and writing in the disciplines in the mathematical sciences," and I've submitted the proposal to Jossey-Bass via an editor I was directed to by my wonderfully supportive grad school colleague, Erdrick (thanks, man!), who himself has a text published through Wiley (Jossey-Bass's parent company). I'll let folks know how things progress on that front.

About the allusion I made last night to advancements in IBL: I've reached that point in the term at which I'm leading the students in the tedious work of integrating rational functions via the method of partial fractions, and, just as I've done for the past few years when teaching this topic, I'm using the step-by-step worksheets I've developed to help the students guide themselves through the algebraically intense process of partial fractions.

I always enjoy myself at this point, since I'm doing next to nothing as far as lecturing goes, and it's up to the students to chart their own courses. And more than ever before, the students are having a blast.

"Are we going to be doing more work in groups today?" one of my favorite students in the class asked eagerly at the start of class on Wednesday.

"Yup," I said.

"Awesome!" They couldn't wait to get into sets of three or four and dive right in.

Yesterday I hit the pause button and said, "so let me ask this: these worksheets, this step-by-step deal, with me saying a few words before letting you all take the reins, whaddaya'll think? Are you all getting a lot out of this? Is this something that's helping you learn?" The response I got was a more eager "YES!" than I've yet gotten from them this semester. It was tremendously encouraging.

I developed these IBL guides a few years back as a means of guiding students through what I've always thought is one of the least exciting topics in Calc II, but I've never taken the time to put together similar sheets for most other topics in the term. Given the students' obvious receptivity to this method this term, I'm going to continue with this set-up for a few more sections and see how things go.

I know I've indicated somewhere in a previous post (I don't have time to find which one precisely) that I've always had an inexplicable resistance to upping the students' centrality in the first-year math courses, and I've only very slowly inched towards the edge of the cliff from which I must take that leap of faith. I'm delighted that, now that I'm standing by the canyon's rim, my students are ready to push me over.

Yes, I realize that I just wrote that my students are ready to push me over a cliff, and moreover that I'm happy about it. I made no mention of the bungee cord tied around my ankles.

I'll leave with the following bon mots from a student of mine: yesterday one of my Calc II students compared my lecturing style to the kid's show Blue's Clues. "The way you pause when you want us to respond is totally like on that show. There'll be silence, and then someone will mumble something in response." Never having seen the show, I had to check it out on YouTube. Sure enough, two minutes into a ten-minute clip in which the title character and her human companion Chris join another cartoon dog ("Magenta") in a scavenger hunt, I was treated to an example of what my student had described, almost verbatim:

"I don't know what shape this is, kids. What shape is this?"

"It's a triangle!"

I LOLed.

Though she was kind enough not to mention it, I also noticed how my classroom manner is not unlike Chris's: we both overact the hell out of almost every line we're given.

"Okay, girls and boys, today we're going to work on integrating powers of sine and cosine!"

Great. My Calc II class is like a children's show.

On that note, I'll bid adieu. My working day's about to begin.

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.

A different class

I've always found it compelling to think that our ancestors from thousands of years ago were no less clever, no less smart, than we are today, and that they merely had a bit less experience, had had a few fewer millennia in which to sort things out by trial and error and intentional experimentation, than have we. Given several dozen more centuries in which to try their hands at various critical and computational maneuvers, certainly they'd have come to many of the same conclusions as we have by now. (You must admit that we've been given a distinct advantage by the astute application of printing technology and modern methods of data storage, data recovery, and data transmission.)

One day at some point during my third year of undergraduate study at the University of Denver I was idly toying with some polygons that I'd circumscribed with a unit circle and I noticed it wasn't hard to recover an inductive formula for the lengths of the polygonal segments that made of a circumscribed 2ⁿ-gon a circumscribed 2ⁿ⁺¹-gon instead. With a little basic trigonometry (it turns out that the Law of Cosines works best) you can arrive at an iterated radical formula for the number π.

I was flabbergasted, thrilled by my discovery, and the next day I told my adviser, excitedly, about what I'd found.

His response was something along the lines of "oh, Euler's formula!" I'd recovered a formula first noticed by the great Swiss mathematician Leonhard Euler (the 300th anniversary of whose birth was recently celebrated in the math community), akin to an even earlier formula, the first successful arbitrary approximation of π, due to the French mathematician and astronomer François Viète.

If you're going to get scooped by someone, Euler, one of the most prolific mathematicians in history, is not a bad one by whom to be scooped. Still, that discovery that your discovery is not a discovery at all, or at least not a new one, can be unsettling. Certainly it's happened to us all, and it happens more frequently when you make it your business to ask tough questions. How often do even the biggest names in math research get one-upped by slightly cleverer colleagues?

Asking tough questions is the job of the mathematician, so it's imperative that young math-minded minds get used to tackling tough questions in a controlled environment, one in which the answers are already known to be known, and in which tough but tractable questions can be set up for what they are: challenges and tests of skill, yes, but not traps meant to lure the student into a sense of hubristic invention.

Put another way, if you know from the get-go that the discovery you're about to make is not a new one you can take your attention from the statement of the theorem on the page in front of you and place it where it really belongs, on the path you're about to trace out that will lead you to the theorem at its end. That same path, you'll know as you walk along it, is the same as or similar to the one taken by hundreds of highly intelligent human beings who came before you...but like they did before, you'll make your way along the path yourself, and the fact that the land at which you'll find yourself at the end has already been mapped out and explored doesn't make that land any less beautiful or wondrous.

Discovery is like that.

While running this morning I thought of a discovery activity I can use in MATH 280 this coming fall when it comes time to rap about equivalence relations, a topic that proofs dauntingly difficult to a large number of students.

I'll gather several dozen small objects of various kinds and bring them to class in a big ol' bag and empty the bag onto the classroom floor.

"Sort 'em out," will be the order of the day.

"How?" I can imagine students asking.

"You tell me." They'll pick through the pile of stuff scattered before them, and after a bit of trial and error patterns will emerge: the Tonka truck matches up with the lemon-shaped lemon juice bottle (for obvious reasons), and by the same logic the wingnut and the nickel get tossed in the same subpile, and the magnolia leaf meets up with the mango. Without realizing it, the students have constructed an equivalence relation, creating classes whose elements exhibit demonstrably reflexive, symmetric, and transitive properties.

"Can you do it another way?" The next iteration takes a bit more thought, and perhaps now inorganic objects are grouped together while once-living things share a different class. Or perhaps size proves to be the most distinguishing characteristic. Somehow a new partition emerges, and another equivalence relation is born.

A similar exercise may well work to demonstrate order relations. Confronted with a disorderly mess of objects, can the students impose some kind of order on them? What properties does this "order" satisfy? What properties does it not satisfy? Does the order need to be a total one?

Surely the students, without formal knowledge of the definition of the phrase equivalence relation will be able to build several such relations of their own, and having done so will be far likelier to recognize such relations when they encounter them in more mathematical contexts. Moreover, they'll have a greater appreciation for the technical definition of equivalence relations when it's given to them.

That's the power of discovery: you're much likelier to remember and understand something you discovered yourself than something someone else discovered for you and merely told you about.

Why in the hell don't we teach like this more often?

I know an answer to that question already (and my colleagues and students should feel free to supply many more in the comments section): because it's difficult to do so. Setting the stage for incipient discovery is far more difficult than describing what discovery looks like.

I admit that, though I hope that my classes set students up for discovery more often than those of less ambitious instructors, I make use of discovery-based pedagogical methods more rarely than I should. I'm trying, my friends, I'm trying to address that. I hope to devote a good deal of time this summer both to my own discovery (during the hours I spend with my REU students and the other students with whom I'll be doing original research) and to developing means by which I can facilitate others' discoveries on their own.

What discoveries, new and old, await us? I'm tremendously excited to set out on this summer's journey.

I am not Euler, and you are not me. Yet we're all human, we're all clever and intelligent, we're all naturally inquisitive, and we're all equally capable of discovery should we put ourselves in positions from which discovery is easily possible. In this regard no one of us is in a different class.

Write or wrong

Hey, hey, hey! It's a brand new year, folks!

So far this year I've done little related to my teaching, I've spent most of my time reading (gasp!) for pleasure. I've capped off seven books in the last two weeks...golly, it's nice to have free time. This morning, for instance, I finished Wangari Maathai's Unbowed: a memoir, a recounting of her life in Kenya and the founding of the Green Belt Movement, the organization primarily responsible for her receipt of the 2004 Nobel Peace Prize. Before that it was a collection of short stories by Guy de Maupassant, Georges Bernanos's The diary of a country priest, John Griffin's classic Black like me, and a pair of books by Jonathan Kozol and Kurt Vonnegut, reviewer elsewhere in this blog. It's been great to be free to read again, something I'm sadly unable to do much of during the school year.

I've also been mulling over what I'd like to accomplish through my teaching during the coming year.

Last year could be characterized by consciousness: I believe that more than anything else I learned to become fully conscious of my pedagogical efforts, and cognizant of the effect my deliberate actions would have on my students. I made conscious efforts to structure my assignments developmentally, to engage students in meaningful, conscious discovery. I believe that my conscious focus paid off, I feel as though my 280 class benefitted enormously, for instance, and the effort that went into Newton v. Leibniz was repaid tenfold by the students' growth through the project. (By the way, I heard back from Prof. Bornstein, she was delighted to hear from me, and wrote me a wonderful letter on her own ideas on teaching. She looks forward, as do I, to continued correspondence. I need to write back to her...)

So what is it that will characterize my teaching in the coming year?

Discovery, perhaps? That will certainly be a central theme of my upcoming graph theory course, in which I'll be challenging students to rebuild the discipline from scratch.

Or maybe authority? Might I focus my energy on encouraging my students to take the reins in their own studies, to ask the questions that need to be asked, to take responsibility for their own futures?

The line between these broad territories is an unclear one. I look forward to seeing how my classes take shape in the coming weeks.

At present I'm ready for Day One (now a week and a half away, on Monday, January 14th), freshly printed syllabi, worksheets, problem sets, and project outlines covering my desk. All I've got to do now is get some Skittles for the candy machine, in order to be ready for the third installment of Calc II's Confectionary Conundrum, an exercise whose execution I've now got down to an artform.

Good news came yesterday in the form of an e-mail from Texas: both my individual presentation and the panel presentation I'm putting together with a couple of my UNCA colleagues (one from the Writing Center and a second from Sociology) were accepted by the organizers for May's 9th International Writing Across the Curriculum Conference at UT-Austin! This is exciting. It'll be the first time I'll have had a chance to speak at a non-math-related conference, about a subject that's quickly becoming my second specialty (writing in the mathematics curriculum). In my individual presentation I'll be talking about the use of the "homework committees" and other structured peer-review exercises to encourage student self- and peer-assessment and self-authorship. Our panel will discuss the ways in which discipline-specific writing is taught, nurtured, and evaluated in the liberal arts setting. My portion of that program will invite non-mathematicians into the world of mathematical writing, indicating the similarities between math writing and writing in other disciplines. By highlighting the grammatical structures, syntactical rules, stylistic conventions, and assessment criteria that characterize mathematical writing and by comparing these aspects with corresponding aspects of writing elsewhere, I hope to dispel the notion that math writing must be an alien enterprise to non-mathematicians.

This is going to be an exciting conference.

Meanwhile I'm only a couple of days away from departing for San Diego, site of my fifth Joint Mathematical Meetings. A lot going on there (judging an UG poster session, presenting in the expander graphs and Ramanujan graphs special session, glad-handing every mother-lovin' person I can find to drum up support for my REU), but I'm already looking beyond it to May, bringing not only the Writing conference but also my next graph theory conference, to which I hope to drag a few students. (One of my freshpeople is making great progress on graceful labelings over this break! I told her to expect me to try to get her to go to this conference in May. More on that as events warrant...)