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.

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.

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.

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.

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.

Never too many cooks

Howdy, folks!

Yesterday's installment of Super Saturday set a "Math Discoveries" record for most student volunteers, with six stalwart students showing up, breaking the mark of five, met earlier this semester and originally set in Fall 2006 when five students came out to help organize and oversee a series of mathematical games. Many thanks go to nearly-omnipresent Beatrice and Belladonna, the irrepressible Tallulah and her roommate Betty Sue, who isn't even in my class but thought it might be fun to come along, Sieglinde, admirably representing my morning Calc I class, and first-time shower Trixie, whose hand-painted polygons were a hit with everyone (I likened them to stained glass, and one of the little kids thought they bore a favorable resemblance to wood chips). Thanks also go to the eleven Calcsters who, though unable to attend, each contributed a hundred or more poster board polygons to the effort, they were very much appreciated.

As with any successful Super Saturday, I'm not sure who had more fun, the college kids or the young 'uns. Both big and little fingers had a hard time at first in managing the assembly of cubes, dodecahedra, and icosahedra. Once we got the hang of it, though, it was smooth sailing, and it was hard to stop. Several of the little kids left for home with their own polyhedra and handfuls of unwed polygons, some just took a few to use as templates to trace their own. Tallulah and Betty Sue vowed to spend some time this weekend making more polyhedra with which to adorn their room (maybe they'll start a craze?). Several of us old-timers hung out well after class was over, finishing off particularly large polyhedra and chatting about next semester's schedules.

A moment of crowing: I've won two of these six over as Math majors, and I've still got hopes for a third!

Another member of my morning section approached me last weekend regarding a Math major, and I was heartened to hear while speaking with him after class on Friday that he was convinced in part because of this blog. Orville, you have no idea how happy that makes me! Welcome aboard! Any questions you've got about the major, please ask. I like to think that my approachability and others' is part of what makes our department and our major such a popular one, and such a strong choice. I really do believe that ours is not only one of the best programs on the campus, but also the friendliest.

Going backward in time...on Friday afternoon I spent an hour with a couple of my colleagues in talking over Chapters 3 and 4 of Bob Moses's Radical equations. Both remain somewhat cynical, one regarding the entire venture, another regarding the relevance of the Civil Rights Movement in all of this math talk. For instance, one doubted the strength of the "ball bouncing" analogy invoked by Moses: if you want to get their (the kids') attention, says Moses, go to the corner and start bouncing a ball. At first they might not take notice, but gradually they'll come, and they'll ask questions to find out what it's all about, and before long you'll have a game going. "I can't really see myself bouncing the ball," my colleague admitted.

"I don't think Moses's point is that every person reading this book has to be a ball-bouncer," my other colleague pointed out. "Everyone has a part to play in this, and many people will be acting behind the scenes in some organizational capacity, you don't have to stand on the streetcorner with a ball."

"I don't think Moses anticipates that every person reading this book is going to rise up and become a part of the movement," I added. "If for every ten people who read the book only one hops aboard, then that's fine."

Something is better than nothing, someone better than than no one.

I feel that our discussion was a good one, and it's helping me to understand the weaknesses of Moses's approach, as well as its strengths.

This past week I approached the director of the Teaching Fellows program at UNCA, asking her if she thinks she'd be able to interweave a reading of Moses with her program, much as she did with Jonathan Kozol's The shame of the nation last year. I've yet to have a real-time conversation with her on the matter, but from our one e-mail exchange I think she might be up to the collaboration. I'd love to get some of my students on-board with the reading circles. How 'bout it, readers, are you up for it? (Don't try to hide: too many of you have outted yourselves as regular readers for me to think you're not out there!)

I've had some more thoughts about what a more learner-centered Calc I class might look like...having been reminded this past semester just how loathsome "word problems" are to math-minded freshpeople, perhaps it would be best if they spend a semester never seeing a problem that isn't a "word problem." That is, from Day One through Day Sixty (or however many days there are), every example considered would be embedded in the context of some application, no matter how simple or straightforward. No computation would be without at least some interpretation requiring a modicum of extractive analysis. Sure, it would be a bear at first, but the students would grow stronger and stronger as the semester wore on, and by the time they'd get to topics in which "word problems" are traditionally replete (related rates?), it would be nothing new to them, and they'd breeze on through.

Something to think about.

The difficulty, clearly, would come in fitting computations classically done for their own sakes out with realistic applications. I ain't despairing. It can be done, it'll just be difficult.

By the way, for those keeping score at home: I don't think I'm arguing for a return to a "reform" curriculum for calculus. I'm not, for instance, suggesting that the formal definition of a derivative be put off for weeks beyond its natural point of introduction. I'm suggesting that a more "traditional" curriculum be retooled to accommodate meaningful and motivating examples.

Before I close this post, I should mention that tomorrow we try Newton v. Leibniz. I'm happy in that I think both of the primary parties in the trial, in both classes, have done a good job in preparing. The worst-case scenario would have involved Newton and his team damning the torpedoes and cruising ahead with full sail while Leibniz et al. drifted around in front of them on a chunk of creosote-soaked flotsam.

We'll see how it turns out. For the time being, I've got some Mathematica code to write to help me out with my research, and I promised a few Calc I kiddies that I'd try to slap together a practice version of the third mid-semester exam, to be handed out this coming Thursday.

Random thoughts, volume 2

I had a few interesting conversations today, with colleagues and with students. I also found myself unfairly piqued during my second section of Calc I, and I feel an apology is in order to my students.

Let's start there: Quiz 4 came today, asking for a brief rundown on the two fundamental interpretations of the derivative. As I would soon forcefully point out to my students (post-quiz), it's not as important to me that they memorize the formula for the derivative, nor that they master every one of the rules for differentiation we will soon study, as it is that they understand what derivatives mean, and how it is that we can see them in nature, and put them together to help us understand natural phenomena. As I put it to them, Mathematica can do all the derivatives for us, and much more quickly than we ever could. What Mathematica can't do is study a natural process, recognize that there is an interrelationship between two or more dynamic quantities, chart those interactions over a long enough course of time to posit a model that describes the way they depend on one another, and use that model to put together the derivative that gets fed into Mathematica at the end of the line. Mathematica, in this sense, represents the mathematics of the past, when it wasn't yet the case that there was a handy formula that one could apply to find the derivative of a given function. That was then. This is now, and the future is yet to come. The mathematician of the future needs to know more than a mechanical rules for finding derivatives (as important as they are to be able to apply well); she needs to know how to use derivatives, how to interpret them.

Of course, when I said all this to them, it came out ne'er so fluently as it did just now, above. Figures, huh?

Performance on the quiz was...meh. It wasn't horrific (I've given harder quizzes), but it was by no means stellar. A couple of my best students cornered me after the second section and asked if they were going to be okay from this point on. Tallulah: "because I didn't do well on that quiz." "I doubt anyone did," I told her. "The folks in the first section pretty much biffed it. It was a hard quiz, largely because you're not used to being asked questions about concepts rather than computations." She's a fantastic student, she'll recover splendidly.

I realized even as it was happening that I was (unfairly) letting my frustration with my students' conceptual misunderstandings get the best of me for a few minutes during that second section's class. I threw markers and punched the board like I always do, but I did so with more vigor than is typical, partly to dissipate my frustration. "How dare they not get this? Damn it, what, they think they can coast in this class if they spit up a formula or two?!"

My righteous indignation subsided as the class went on, and I realized by the end that if they'd not focused on the concepts over the calculations, it was as much my fault, and my colleagues' faults, as it was the students', for not asking them to refocus their attention elsewhere in the first place. I've got to try harder at that myself. For some reason, I have to admit, Calc I has proven the most resistant of all courses to redesign along the lines of discovery and application-based learning. Only now am I beginning to understand what a truly problem-based Calc I class might look like, and I admit that this semester I'm falling far short of that mark.

I promise a less angry, less frustrated tone tomorrow, folks: you really are a great bunch of students, and I enjoy working with you very much. Let's make tomorrow's class a good one, huh? I'll bring some donuts tomorrow morning, and we'll start off with a couple of conceptual exercises to get our creative juices flowing. Sound like a plan?

Good.

From the second section of Calc I, it was off across the quad, to the second of the semester's Writing Intensive meetings (the first was this past Monday) for me. As I anticipated, I'm enjoying working on this committee, conferencing with a group of peers who feel as strongly and as passionately as I do about writing-to-learn and writing-across-the-curriculum and writing in general. I'm starting to get a good sense for the way writing is integrated academically, campus-wide, rather than simply in my own courses and in those of my math colleagues. The bar is quite high; the quality of writing instruction university-wide is solid. Nevertheless, there is room for improvement, and I found myself in a heated exchange of hallelujahs with Lexington, the WI committee's acting chair, as we walked back to our shared building after the committee meeting.

We agreed that the university has made tremendous strides forward in terms of embracing writing-across-the-curriculum, undergraduate research initiatives, outcome-based curricula, discovery learning opportunities, and so forth...but that there's also a lot of work to be done before perfection is reached. "If we're going to advertise that we're using discovery learning," Lexington said, "we've got to start doing just that, and to do that we're going to have to get serious about giving people the resources they need to do that." We agreed that we need to try to drive class sizes down (I mentioned my conversation with my own Chair last week regarding getting my Calc I classes capped at a lower level), we need to offer kids the opportunity to engage in alternative classrooms early and often, we need to make a focussed, directed, campus-wide effort to provide these opportunities to students from the get-go.

After a brief stop at my office and a moment in the Math Lab to unstick the stuckness of a few of the Calc I kids in computing the derivative required of them in the team project, I was off across the quad again to 280. Davina caught me before class with a few concerns about her service on one of this week's homework committees. For one, she wasn't sure about what to write on a person's submission if he said something like, "I'm stupid, I can't figure this out," or something along those lines. "That's a hard one," I agreed. More substantially, she wasn't sure she was giving the right kind of feedback, and she felt like she was being hypercritical, telling people to reword this, change that, and so on. I suggested that she might try to balance positive and negative feedback, and to offer comments like, "I'm having trouble understanding this, could you make this more clear?" or "This is a really good insight, it really helped me to see this point more clearly!" I later reiterated some of these ideas to the whole class, and wrote on the board: "Recall that the purpose of the committee work is not to homogenize, but rather to help people to clarify their own individual ideas."

The subsequent committee reports were good ones. I was particularly impressed with Davina's discussion during the presentation she and DeWayne gave on the homework problem they'd been assigned. I admired the way she was willing and able to come out and say that her serving on the committee definitely helped her to better understand the concepts involved in the problem they'd reviewed: "seeing how other people did it made me see how I could make my own writing more clear and more concise." I'm glad she came out and said that, and I hope her sentiment is shared by the others. I'm certainly going to ask the students about their committee experiences explicitly when I pass out midsemester evals in a week or so.

Came then (after another half hour of set theory) the trek back to Robinson Hall, where I'd spend a few more hours before heading home. I finished grading the second section's Quiz 4s, on which they did marginally better than the first but still not wonderfully. I also got a chance to work with a number of the teams as they struggled through their projects (they're all doing quite well, from what I can tell), and I met up, one-on-one, with several of the 280 students, helping them to polish various drafts of homework problems. They're definitely developing an appreciation for more and more subtle nuances, meanings of stereotypical mathematical phrases ("thus...," "for every...," and so forth), and clarity, clarity, clarity in writing. (A funny, and very heartening note, if I may: at the close of today's committee reports, I reminded the students to keep an eye on the rubric I'd handed out last week as they worked on their math writing, and I asked them if they could remember the "four Cs." In nearly complete unison, they intoned: "correctness, completeness, clarity, and composition." I didn't have to say a goldarned thing. I was a happy man.)

While I was finishing off those Quiz 4s, Cuthbert, Chemistry colleague of Lexington and a big, big man in undergraduate research, came by to ask me if I wouldn't mind providing him with the titles of the projects the REU kiddies worked on this past summer: he was putting together material to take to a regional conference on UG research, indicators of the sorts of things that can be made to happen in a liberal arts science curriculum. I was happy to oblige, and not an hour later I'd send him a list of the topics. We then got to talking about the inclusion of research components in UG courses, and he asked if I'd done much of that. "Depending on how you defined research," I told him, "I do that in nearly every class I teach." I told him a bit about the course that spawned this blog (MATH 365, Linear Algebra I, taught last Fall semester), and he asked if he could have information about the projects those students worked on. I obliged him there, too, sending him the prompts for a few of the research projects (Monopoly, Traffic Patterns, and Come on, Feel the Noise).

We hit a few more points in our short conversation, enough to give me a sense of déjà vu, feeling as though I was reliving the conversation I'd had earlier with Lexington. We talked about the increasingly interdisciplinary nature of scientific study; the need for broader, deeper, more meaningful implementation of discovery learning in our courses; the need for more robust interdisciplinary course offerings than an occasional and disingenuous cross-listing: we need team-taught classes, classes that provide a true interface between one field and another, like one Cuthbert mentioned that took place at his previous institution, in which beginning chemistry students learned their ways around a lab while generating real, unprettified data that could be fed to statistics students who would put it through realistic rigorous analysis, the results of which analysis could be funneled back to the students who ran the lab experiments in the first place in order to help refine their techniques. We agreed that a course that combined the concepts of physical chemistry with those of linear algebra would be a tremendous boon to both the Chem and Math departments; I have no doubt that we'll be talking about this again soon.

Thence, from my office, to dinner at Sorrento's (offering the best Italian in town in a cozy out-of-the-way den halfway to Oteen that sadly never seems very busy), and then home. Once home, I finally had a chance to call Bedelia, my colleague late of Harvard, now of Lesley University, and catch up with her. As it was bound to be, our conversation turned to...teaching! Imagine that! We talked for a bit about a program she's like to get off the ground, a sort of summer prep course to get local (in her case, Bostonian) college-bound kids up to speed on the math skills they might need to succeed in a challenging entry-level math course, while providing them with some useful advice on making the transition to college more smoothly. I compared the program she was describing with UNCA's own SOAR program, and she seemed to agree with the comparison, only she emphasized the local nature of the plan she envisioned. Then we talked shop for a bit, I about my own six courses (if one counts the senior seminar and my three independent study students), she about her Quantitative Reasoning and Pre-Calc classes.

We're both having a good time.

Hey, I've rambled on long enough, it's time to stop. I'm going to make a smoothie, maybe watch an episode of Mr Bean or some other such nonsense, and call it a day. Thanks for reading, I hope you'll free to comment, I always appreciate your feedback!

Happy day after Pi Day!

Hey, All!

Yesterday was the 301st Anniversary of the naming of π (celebrated), and the Math Club event I helped to put together went off splendidly. Over 50 people (mostly my Number Theory class combined with Quidnunc's Linear Algebra class, and a few assorted hangers-on) gathered to watch 6 folks compete in the pie-eating contest and 2 in the π-reciting event. Bocephus finished off about 90%-by-volume of his pie in 3 minutes and 14 seconds, giving him the victory in the first activity, and Ulrich recited 64 places after the decimal to garner the win in the second.

Many photos to come soon.

Meanwhile, my classes are chugging along nicely (I don't think anyone was too distraught over classes being cancelled on Friday). In Calculus we're almost done with shortcut rules, in 280 we're set to talk about relations and functions, and in Number Theory we're headed back to the text to talk about more on congruence arithmetic for a little while before tackling a couple of primality testing algorithms. The first of my Senior Seminar students' presentations comes next week, too, as Beulah will speak about hyperbolic geometry and how it inspired M.C. Escher. She's shown me her slides, and she did a great job in putting them together. If she can work out the timing, I think it'll be a fantastic talk.

Now, I've gotta hit the road to Georgia, hoping to make it to Statesboro in time for this afternoon's Project NExT-Southeast events. Tomorrow morning brings our panel on PBL/IBL. I'm looking forward to that, and I hope we get more than the 9 pre-registered participants.

The proverbial Morning After

Well, it's all over and done with.

I think.

There are a few minor details to wrap up with one or two folks, but nothing major. The third exams, the last journal entries, the presentations and concomitant posters are all graded and ready to hand back to those eager enough to come and retrieve them. (C'mon, y'all! Come and get 'em!)

Moreover, I've finished grading.

Ouch.

In the end, I'm really disappointed that I've got to hand out grades: how difficult is it to boil down all of the interactions, inquiries, examples, applications, projects, presentations, portfolios, and other assorted whatnots we've collaborated on in the past few months, and end up with a residuum summed up by a single letter?

I tell you what: put simply, it's a bitch.

But it's done. And ultimately many people in the class did very well. There's a pretty large number of As and A-minuses, a fair smattering of Bs and Cs of various sorts, and only a puny handful of anything lower.

Beyond the grades, there are the lessons learned. I hope that in the case of our class we've been able to transcend cliche and put some truth into that truism. I can't speak for the students in the class (I'd love it if they'd take the time to speak for themselves in the comments to this post!), but I know I've learned a lot.

1. I am never, ever going to do this with a class this size again. Ever. I'm figurin' the upper limit for this method is something in the ballpark of 15 students. At that point I could have an eminently manageable 5 teams of 3 folks each. With that small a number of teams, I could make the rounds in the classroom during group exercises and be sure of hitting everyone at least once. I could schedule team meetings more regularly to ensure frequent updates, and the teams would be small enough to allow for easier scheduling of research meetings. We started out with thirty-three students and ended with thirty, and as hair-pullingly frustrating as the size of the course sometimes made the daily proceedings, I'm quite frankly awed that more people didn't drop midway. I have nothing but admiration for the patience and dedication of those that stuck with it.

2. This method of learning is not for everyone. Those that fared best were those who were more used to courses run along these lines, and those whose learning styles are at odds with those assumed by a more "traditional" classroom. For instance, those who identified as "visual" learners were likelier to find our class useful. Others, more used to the run-of-the-mill lecture format, felt a bit out-of-place and longed for those infrequent days when I'd stand at the front of the room and yammer. As the semester wore on, I developed a balance between the applications-based guided discovery exercises I'd envisioned for the course and a more lecture-led semitraditional format, all based upon the worksheets I turned out, one or two per week.

3. This method of teaching is not for everyone. As folks who've had me for other courses can attest, my teaching style is probably best characterized by the word "enthusiastic." A number of other words have been used to describe my teaching (few of which, fortunately, are unprintable), but this is the word which predominates in my teaching evaluations at the end of every semester (runners-up include "approachable" and "accessible"). And honestly, without the charisma and energy that I put into my classes, I have NO IDEA how I would have made it through this semester. WARNING: if you plan on teaching in this manner, make sure that you've got lots of free time, and boundless energy. Even with all of the preparation I did in advance, I was still blown away by just how much I had to do to keep up with the work. A lot of this labor was on account of the size of the course, but much can be attributed to the method alone.

4. Some innovation is appreciated. Team quizzes, for example, went over enormously well. I'm keepin' those: you can be sure that every course I teach from this point forward will include some variant of that activity. This blog's been a popular feature, too. For a while there, before everyone was occupied with exams and presentations and other geegaws, I was getting at least one or two comments per post (and as many as 14 at one point), which ain't bad considering all of the other faaaaaaaaar more interesting blogs there are out there that my students could be reading. (By the way, a shout out to The Comics Curmudgeon, one of the baddest blogs on the internet.) Change of Basis, too, will live on, in modified form, as I move into the planning stages for next semester's classes. Look forward to my continued chronicling of my teaching adventures. There are several of you from Linear who are continuing on with me, either in 280 or in Number Theory...keep reading, folks, and keep posting!

I've learned more lessons than this, but those are the biggies.

I've gotta go for now, but hey, 365ers! I really would like to hear what you feel you've gotten from this class, so please feel free to leave a comment or two on this post: let me know what you've learned, what lessons you'll take with you.

This'll likely be the last post I make on 365 for quite a while, but I'll be back soon with updates regarding next semester's courses. And I'm slated to give a talk on the writing component of our class in January at the big annual Joint Meetings of the American Mathematical Society and Mathematics Association of America. I'll be sure to let you know how that goes. (And yes, Fiona, I'll let you know as soon as I hear about the Information Literacy Intensive status for the class...but I'll be seeing you in 280 in the Spring, so I know you won't be going anywhere!)

Au revoir, then. To everyone in my class: thank you. Thank you for your hard work, your time, your cooperation, your willingness to try something new, your everything. Take care, and have a wonderful Winter Break!

...Sunday night!

Well, it's over and done with.

The Harvard talk seemed to go over very well. It was an interested audience to which I spoke, a small group of preceptors and graduate students in the Harvard Department of Mathematics. They asked good questions, and they most definitely kept me on my toes.

After we got the ball rolling with the Markov Dance, I described the basic philosophy of the course, and then got into the nitty-gritty details of the way the course is put together. Much of the time we had a hearty dialogue going, in which we engaged in a discussion of the course and its design. They were really interested in finding out more about the team quizzes, the nature of the worksheets we work through, the source of our applications, the dynamics of the group work we've encountered, and how the size of the class has affected the way it's been run. I gave honest answers, often aided by the 11 pages of comments (from which I quoted heavily) you all gave me in your last journal entries. (Thank you all, thank you, thank you, thank you!)

I had a much-needed rest on Saturday, hanging out with Bedelia and her honey, Eugenia, and their beautiful daughter, Isadora. We hung out in their Somerville apartment, ate crepes, and joined them in a walk to the Cambridge Public Library. Good times!

I'm tired. Very tired.

I hope that all went well with all of you this week, and that you've had a chance to look over each others' preliminary reports before revamping them along lines penciled in lightly by your colleagues. We'll all be back together again tomorrow afternoon, when we'll consider Fibonacci-like applications that arose in my own research this past spring, and are arising again as a consequence of the conversations I had with my colleagues in Tennessee this past week.

To be continued!...

Quickie

It's been a little while since I've posted, this week's been slamming.

I do have a lot to say, a lot I've been thinking about as my Hahvahd trip nears and I have to say something about IBL in the context of our 365 course. I want to write a bit more later in reflection on my goals for the course, and how well we're meeting them right now (in particular as regards the learning goals I'd set out in the syllabus).

I've still got some fun course materials to type up, though, so I'll be brief at present.

For the time being, I hope that the inventors won't mind me sharing with my readership the following linear algebra drinking game, made up this past weekend:

Equipment: TI-81-or-later calculators, one per person. Drink (non-alcoholic, of course!).

Object of game: players compete by constructing 6x6 matrices on their calculators, and then computing the determinants. The first to obtain a matrix with determinant lying between 10 and 20 takes a swig. Repeat as desired.

Change of Basis reminds you to drink responsibly.

High rollers

I took the time yesterday to look up some resources on problem-based learning. While much of the literature still concerns itself with medicine (the field in which PBL first arose and became widespread), there were a few sources that gave information on PBL in nonmedical settings, and there were a number of medical sources which provided useful hints on making PBL work in a more generic setting. (The Problem-Based Learning Initiative at Southern Illinois University has proven helpful for some of their suggestions.)

First and foremost among their hints (surprise, surprise) is that in the PBL setting, students must be held responsible for their own learning. This ratchets up the stakes: there are definitely greater risks involved when you're asked to take charge of your own learning. The stakes are high for me, too, obviously: I've got to make sure that the students have access to the skills they need to face the challenges I'll be presenting to them as the semester progresses, if we can ever hope to achieve that elusive "Flow" state.

With stakes this high, things are bound to be interesting this coming semester. I'm going to start off the syllabus for the course with a line or two from David Mamet regarding the vibrancy of theatrical drama produced by actors who've raised their stakes as high as they can go.

I finished the day yesterday by swinging by the room where I'll be teaching 365 next semester. It's pretty spacious, and the desks'll be easy to move. There's not a good deal of blackboard space, but I'll manage, as I hope to be less dependent on boardspace than I've been in previous courses. There's plenty of light, too, and that should be nice.

Having finished up with the design of learning activities appropriate to the stated course goals, I'm going to spend some time this afternoon putting together the "overall scheme of learning activities" based upon the seven primary foundational concepts I want the students to master (vectors, matrices, vector spaces, linear independence, linear transformations, determinants, and eigenstructures). Castle-top diagrams? Nah...that's just not my style. Maybe more of a flowchart. Who knows? I figure I'll end up with some more "wall art" of some form, though.