Another day past.

I felt skittish in my Calc I classes today, and awkward. I felt elsewhere, out-of-place, out-of-sync.

MATH 280 made up for it, though. I was definitely at home in Karpen 033 this afternoon.

This afternoon, I started thinking about what I'd like to talk about at a Writing Across the Curriculum conference.

What do I have to say?

What am I qualified to say?

Hey, even if I have to say so myself, I think I'm pretty damned good at teaching math students how to write math...but is that enough? I don't know how hardcore into the scholarship of teaching and learning I'm expected to be in order to "have something to say."

I'd like to talk about my rubric-building exercises: how does one set out to teach math students to teach themselves what to look for most in quantifying quality in math writing? How does one teach them that, given a few ground rules and a little practice, *they* are as qualified as *I* am to render an assessment of a proof's goodness?

Good enough?

Hmmmm...there's a kernel of irony here, isn't there?: maybe I've just got to teach myself that *I* am as qualified as anyone else is to render an assessment on the goodness of my own writing-instruction methods, at least in the context of my own classroom.

Is it that easy, or is that just a bunch of relativistic hooey?

Ah, fugeddaboutit.

I'm going to go rustle up something to eat.

To be continued, for sure.

## Monday, August 27, 2007

### Mirror, mirror, on the wall...

Posted by DocTurtle at 8:14 PM 0 reflections

Labels: Calculus I, Foundations, MATH 191, MATH 280, self-authorship, theory, writing

## Friday, August 24, 2007

### A doughnutty day

Oof.

How many doughnuts does it take to fill a coupla Calc I classes?

Evidently not as many as I'd expected. I got six dozen tummy-busters this morning (yep, that's 72 doughnuts) at Krispy Kreme on my way into campus; I thought it might make a nice end-of-the-first-week-of-class gift for my calculus compadres. With 63 students, I thought maybe I'd have a few doughnuts left over after the carnage had ended. I finished up with something closer than two full dozen to spare, and that's after I'd had my own unhealthily large share.

Mathwise, in Calc I today we rummaged about in a pile of various useful models: polynomials, root functions, rational functions, transcendental functions of various sorts. On Monday we'll work at building new functions from old ones. I was a bit bored in today's class (although it was likely difficult for the students to tell; as usual, I got a number of comments regarding my characteristically high energy level), I don't feel I got the students to *do* enough today. Geez Louise, I've finally gotten to the point in my career where I feel out-of-sorts if I talk for more than 10 minutes in a row in class.

Meanwhile, today marked the first entirely-math-related day for my 280 folks. We started our discussion of mathematical statements, comparing the notions of validity and truth, introducing the notion of a "grammatically correct" ("well-formed") mathematical statement, universal quantifiers, existential quantifiers, and order of quantification. I handed out the first homework assignment and exhorted folks to get a jump on Exercise #2, solutions to which are due to the first HW Committee on Monday. (This exercise asks them to come up with their own mathematical statement involving a universal quantifier and an existential one such that when the order of the quantifiers is reversed, the truth value of the statement is negated.)

In between classes, I was as busy as a caffeine-addled (or doughnut-sugar-soaked) bumblebee. Though I did have an hour or two in which to start re-reading a paper I'm hoping to work on with one of my colleagues here, and to chase down another dead end in the random tree project, meet 'n' greet sessions with students filled much of my time. I've now met with about half of my new students, and as usual have been awed by the diversity of people who come into my classroom. Some are confident mathematics practitioners who come equipped with eagerness and excitement, others are trepidatious, to put it mildly. Some are fully aware of their own particular study habits and learning styles, others haven't a clue what I'm talking about when I ask after the techniques that help them learn best. ("I dunno, I just do the problems, maybe look at some examples," is a common response. I'm always heartened when a student quickly tells me, "oh yeah, it really helps me to draw a picture to help understand," or "when I explain it to my friend, I really catch on.") Variety, variety: one 280 student I met with today credited his math classes with giving him the ability think more clearly and to improve his memory, while one of my calc students unhesitatingly mentioned her math anxiety and indicated that she merely hopes to finish out the class with a better grade than she earned the last time around.

They're a good lot.

Well, off to bed. Tomorrow brings my first grading session of the new semester, I've got a thick stack of calculus homework to peruse. We'll see how the new grading system works out.

Posted by DocTurtle at 10:17 PM 0 reflections

Labels: Calculus I, Foundations, MATH 191, MATH 280, research, writing

## Thursday, August 23, 2007

### Conversations

Had a little doorway chat with Karl (longtime Math Lab student employee, now graduated) this afternoon about the universality (or lack of it) of mathematics: to what extent is math just waiting around for us to discover it, and to what extent is math itself an artifact of human invention, the residue that's left by the human mind as it makes its imprint on all that it encompasses? The whole conversation started when I was showing him a book of logarithm tables I picked up at a garage sale or flea market somewhere a long time ago, and I wavered indecisively between the words "invented" and "discovered" when searching for the right word to describe the initial human engagement with logarithms.

"Since you said 'invented' first," said Karl, "I can tell which camp you're in." This led to a discussion of whether mathematics can truly be universal, a position neither of us defends. Karl mentioned recent research (see this link for more info) into the language of a certain Amazonian people suggesting limits to traditional Chomskian analysis, and I let him know about Anthony F. Aveni's Uncommon sense: understanding nature's truths across time and culture (University Press of Colorado, Boulder, 2006), an interesting book I worked my way through this summer. Aveni discusses the scientific undertakings of the members of various ancient and modern societies and provides accounts of culture-specific scientific knowledge that might seem patently alien to practitioners of science as defined by the Western European Enlightenment tradition. I'll definitely be looking through that text again when I start to put together my thoughts on the history of math technology course I hope to run.

Rewind several hours: as I walked into campus this morning I thought about our discussion on the topic of "Good Proof/Bad Proof" in 280 yesterday. "Damn," I thought, "that was a nice conversation." I really felt that we got right at the meat of the matter (or whatever vegetarian substitute one would like to put in its stead), and the students themselves were quick to point out, unprompted, what it is that makes a given proof a weak one or a strong one: does it use notation correctly? Consistently? Does it prove the claimed statement in full generality? Does it use correct grammar and punctuation, use complete sentences? Does it "lead the reader" conversationally through the thought processes of the prover? All of these questions get at the issues of clarity, correctness, completeness, and cohesion, my "Four Cs" of assessing the quality of a proof. Above all else, the exercise helped them develop (oh, that meaning-laden term!) "ownership" of the process of mathematical discovery: they have the same right that I do to question the validity of a proof, to test the hypotheses of a theorem. Math's truth does not inhere in a single individual no matter how much experience that individual possesses, and even the greenest of mathematical parvenus, equipped with the right tools and techniques, may approach, with healthy skepticism, a given mathematical statement with the confidence of a professor emeritus. I think that yesterday's exercise helped folks see that, and I hope that it gave them the confidence they'll require to feel free to explore the problems we'll face the rest of the semester.

I'm really glad we took time out for that activity.

Posted by DocTurtle at 4:24 PM 0 reflections

Labels: Aveni, ethnomathematics, Foundations, MATH 280, self-authorship, theory

## Wednesday, August 22, 2007

### Hmmm...

Good day, folks, it's been a good day.

I spent most of my free time today beating my brains senseless (a misleading phrase which seems to suggest that there was some sense in there in the first place) on a couple of twisted research problems I'm working on with my colleague Tip: after a month or so of work we've managed to get a handle on the expected degree, at time *t*, of the *n*th vertex constructed through applying any one of an infinite family of random tree construction algorithms. At this point we're trying to recover the same statistic for a broader class (or other related classes) of algorithms, and to no avail.

Hey, and every now and then I took a break and taught a couple of classes!

I felt much more comfortable in Section 1 of my Calc I class this morning. (I made damned sure that the classroom door was open before traipsing up to my office so that I wouldn't be greeted by a hulking phalanx of students waiting outside the locked door.) After going over a few bureaucratic points I'd intentionally let slide on Monday, we spent about twenty minutes figuring out what sort of grading system we wanted for the HW: as planned, I split everyone into groups of four and asked them to consider, if they would, the weight the wanted to assign to each of the sets of HW problems graded in a certain fashion. (Details are on the syllabus; roughly, a fraction of the problems, selected randomly as I've done in the past, will be graded "carefully," lots o' juicy feedback and whatnot, while the remainder will be graded "quickly," based solely on whether or not the correct answer was obtained. All told, 20% of the overall grade will come from HW graded in one way or another.) After reaching a consensus within each group, we reconvened and briefly discussed the pros and cons of each grading system. Each group then reported the figures arrived at, and we agreed to go with the straight average of the resulting weights. In Section 1, we ended up with something like 65/7% for the "careful" method and 75/7% for the "quick" method, and everyone seemed pretty satisfied. The same exercise, run by the second section, yielded a more arcane division that we decided to round off to 5% "careful," 15% "quick." Though there wasn't out-and-out rebellion, I sensed a little bit more resistance to this decision. I asked anyone who was unhappy with the result to let me know confidentially, in case they didn't want to be put on the spot in front of 30 of their peers.

Once all of those parliamentary matters were out of the way, it was time for...MATH! Oh boy! We actually ran short of time in the second section, having spent a few more minutes still on discussing what it is that students and teachers can do to work together in bringing about a classroom atmosphere conducive to effective learning. (This discussion yielded a wealth of good ideas, and everyone seemed to agree that the best environment involves an active classroom in which a variety of learning styles were addressed and in which both teacher and student come prepared to play their respective parts. Duh.) Even in the first section we only got to scratch the surface of our discussion on functions and models. We'll pick it up there tomorrow. These "environmental exercises" are fairly new to my teaching repertoire, and I'm really glad I included them in this semester's proceedings, I think they'll make for a healthy place to study.

Meanwhile, 280 went *very* well. Folks came prepared (some folks came *very* prepared) to discuss the writing samples on chemistry I'd given them to think about (this one here), and we had an active conversation on the relative strengths and weaknesses of these samples. Although everyone agreed that the second fragment was the strongest, there was a healthy debate as to whether the first or the third came in second. Ultimately I think we were all willing to admit that the though the third fragment was the most informative, concise, and precise, it was simply inappropriate in terms of its format (given that the assignment was to construct a *formal paper*).

From this we went into the mathematical version of the exercise (here), on which there was greater unanimity: the third fragment was the clear winner, followed by the first, with the second coming in last: the second one simply isn't a proof, and while the first provides very little context and lacks clarity (and a bit of correctness), at least the basic idea of the proof is sound.

I have to tell ya, I had a blast in all three of my classes today, and I hope they all continue along the lines we've established so far. I'm totally digging all of my students, and so far they're all really engaged.

It's all good. Tomorrow's calc sections will dig more deeply into some interesting mathematical models. We'll see what they have to say about Jane Austen's *Northanger Abbey*. And of course my 280 folks get the day off! Yay!

Ladies, gentlemen: have a nice night. I'll catch you on the dayside.

Posted by DocTurtle at 8:34 PM 0 reflections

Labels: Calculus I, course prep, Foundations, MATH 191, MATH 280, research

## Tuesday, August 21, 2007

### Day Two: A survivor's tale

Made it!

Yesterday wasn't so hard as it was looooooong. And dealing with the tail-end of this dratted cold didn't help matters. I realized as I got into campus yesterday morning (at about 7:05) that though I'd done a huge amount of preparation well in advance (two or three weeks ago), because of the cold I'd not really done much prep work for my classes during the last half-week, during which time I'd ordinarily have gone over my notes again, gotten my Day One crap in order, made sure all of the technical details were tended to, and so forth.

As it was, I staggered into Rhoades Hall and stopped at the calculus classroom on my way to my office: I wanted to make sure the computers were on and functional. Of course, three of them ended up being inoperative, and the projector for the instructor's terminal at the front was wonky. (The projector problem I managed to fix before class got underway, but in the course of the in-class exercise I'd scheduled, my students discovered that two additional computers were not hooked up to the internet, further limiting the number of workable stations. Grrrrr...) I left, shutting the door behind me, blindly trusting the ITS people to come through and open it up a bit before 8:00 so the students would have access to the classroom in order to get settled in. (Notice the words "blindly" and "trusting"?) Up to my office (by about 7:25 or so) to assemble my thoughts and get my gear in order.

Of course, I'd only remembered while walking in that I'd yet to write up a sign-up schedule on which students might pencil in "meet 'n' greet" appointments with me during the first two weeks or so, so I had that to arrange. Thankfully I've still got several sheets of poster board left over from Super Saturday activities; I was able to fashion a schedule out of that. I had just enough time (by now it was 7:40) to go over my first section's students' faces a few more times before heading downstairs, where I supposed I would find them cozily resting their tushies in their desks' seats, eyes forward, brains engaged and ready to learn.

Instead I found them lining the hall on either side, in some places two deep along the walls, stretching twenty feet in either direction from the classroom door. (I found out this morning that someone over at ITS has been dropping the ball on opening labs in the morning...it's not a big deal, I can open the classroom myself, I just need to know to do it! Grrrrrr again...) No worries, mate.

After a quick by-your-leave with the lock and a twiddling of the knob, we were in, and I spent the first ten minutes with the names. I had a relatively rough time with that first section, I think my accuracy was something on the order of 50% or 60%. Bleh. I let my poor showing get to me, I think, and I was out-of-stride for the rest of the class. Looking back, I don't think it was a wasted hour, but I certainly wasn't very comfortable at the time. I had too much to say, and not enough time to say it, and I let the first group activity run a little too long. The students seemed like sharp ones, though, and I look forward to our next meeting on Wednesday. (It's weird having Tuesday as my "off day," and not Thursday!)

The second section of Calc I? Let's just say we had it goin' on. It went well. I was able to adjust by correcting most of my missteps from the morning class. And perhaps this was a function of the time of day, but the students were definitely more engaged. We had a few laughs while I was playing the Name Game (I think I only missed one or two, and though I needed the assistance of initials for three or four students, I achieved something around 95% accuracy in a class of 32), and we segued nicely from our brief discussion of the syllabus into the first group exercise, modified from its morning form: I moved the responsibility for the concept review from my shoulders to theirs, and they carried it well. After a brief brainstorm, we filled a board with calculus concepts, zeroing in on the notion of "function," the heart of the Google pedometer exercise I'd planned for them (see my exercise Pedantic Pedometers for more information). They pulled that one off without a hitch; Henrietta (remember, folks: fake names!) gave a really clever way of interpreting the Google Maps path as a function, and we were off and running.

After class, Henrietta and a few others (Tallulah, Beatrice, and Belladonna) joined me in my office for the semester's first meet 'n' greet session. We talked for about a half-hour, had a good ol' time. All four of them are fired up about the class, and their enthusiasm really helped me look past my cold and get in the groove. All four were also very open about their own particular learning styles, and our discussion really helped me articulate my thoughts on how our course can help to accommodate those different styles. Thank you all around, kind students, for helping me get a grip on the start of the semester!

MATH 280's turn came next. After a pretty easy round of Name Game, I started off with the same opening exercise I used last semester: from the ground up (i.e., starting with suggestive data and working from there), build the statement of a "theorem," introduce appropriate notation and terminology, adding definitions where needed, and prove the darned thing. Stating the theorem isn't so hard: "the sum of an even number and an odd number is an odd number"; coming up with a proof means first coming up with definitions for "even" and "odd," which Theodoric handled well, with a little input from Quincy and some other folks. These definitions in order, a proof is only a careful logical flourish or two away. We finished in time to spend a few minutes going over the syllabus. I regret that I didn't have a chance to mention the writing assessment grant; that's the first item on the agenda for Wednesday's meeting.

By the end of 280 yesterday, it was only 2:35, but it felt like 8:00. I spent the rest of the afternoon on research and on getting all of my scattered course notes together. By 6:00 p.m. I felt like I was where I should have been by 7:00 a.m. I was so tired, it was all I could do to keep at the enigmatic graph theory data Mathematica was spitting out at me, and by 7:00 I gave up and played Nelinurk until Maggie came by at 8:15 to take me home, exhausted.

Today's been more manageable, chocked full of meet 'n' greet appointments and assorted meetings with colleagues, but nothing overwhelming. I actually feel human. I'm more healthy, I'm more well-rested, less nervous, and looking forward very much to the second round, tomorrow. Lemme at 'em!

Posted by DocTurtle at 10:02 AM 1 reflections

Labels: Calculus I, course prep, Foundations, MATH 191, MATH 280, sick

## Sunday, August 19, 2007

### Check

Syllabi written, ready, posted? Check.

First week's assignments, activities, *et cetera*, same? Check.

Names and faces committed (more or less) to memory? Check.

First day's classes mapped out mentally? Check.

All systems go? Check.

So why the nerves?

I've always got nerves the night before the first day of classes.

I've been teaching at the college level for eleven years now, you'd think I'd be used to this by now.

Tomorrow will be, as it always is, exciting. I'll be meeting roughly 75 new people, and reacquainting myself with a dozen or so familiars, with whom I'll be wrestling with complex mathematics for the next three and a half months or more. I'll be hurling myself into a classroom where the latitude I grant my students in their interactions with me permits them to challenge me, call me out, put me on the spot, at any one of the fifty minutes that'll make up our class period.*Three times* tomorrow, I'll do this.

God, I love it.

I'm sure mistakes will be made, as much by me as by my students. I'll forget to mention something (it always happens), I'll misstate my meaning, I'll misunderstand someone's question. Mistakes will be made, yes, but in equal measure, so too will be made discoveries. If all goes (at least somewhat) according to plan, I'll arrive on the far side of this coming semester a bit wiser than I am now.

I can't wait.

But I can, and I will.

See you there!

Posted by DocTurtle at 7:07 PM 0 reflections

Labels: anxiety

## Friday, August 17, 2007

### Turnabout is fair play

Lo, the semester is nearly upon us!

I've spent the morning working with several of my colleagues from across the university in preparing for the pilot Writing Assessment study we're all working on together, so writing in mathematics is of all topics foremost in my mind right now.

I realized as I was driving home that since one of the first tasks I'm going to ask of my Calc I students this coming semester is to write a brief reflection (perhaps an impromptu paragraph) on what math means to them and how it makes them feel, it would only be fair if I were to provide my own extemporaneous thoughts on what math means to me.

Thus, today's post will simply be a 10-minute freewrite on the topic of mathematics, and what it's meant to me in my life. Starting...now...

I've always known that I wanted to do something related to mathematics. Statistics, maybe (I at one time wanted to be a statistician for the Atlanta Braves...that shows the power and influence of TBS on someone growing up in the middle of nowhere Montana!), but mathematics, sure. For a little while, maybe, I dallied with palaeontology (what little boy didn't love dinosaurs?) and cartography (a little bit of math there, for sure), and astronomy (coming of age at a time when Carl Sagan's Cosmos held sway)...but math won out.

It was in my blood, I think: what else could have made me count out the variouis piece of sorts of dog food that I found in Candy's bowl? (Yes, I did this...I'm happy to report that I was quite young when I did it.) What else could have compelled me to pepper my speech, at the age of 10, with proclamations like "according to my calculations"?

Once I accepted that math had its hold on me, once I became comfortable with the fact that my destiny was somehow pre-ordained, I became a lot more comfortable with who I was.

College...yeah, college gave me a lot of choices, but none that I recall ever consciously taking: I went to the University of Denver, close enough to home that I was still in the same world as my parents, far enough away that they couldn't jaunt over and see me every weekend. Why DU? I think the reasoning went something like this: "DU has a strong engineering program [it does, and did]. Therefore, since math and engineering are intimately related [this I knew back then, but I doubt I could have elaborated it on any further than that], it must be that DU has a strong math program. Socrates is mortal and all that crap."

It turns out that I was...well....yes, and no. DU's math program was a relatively small one, but stocked with folks I've come to appreciate as above average teachers and researchers. I was lucky, in some ways, that things turned out as well as they did, I think.

It was a good place to be, at any rate. Being a small program, the majority of the classes above a certain level had to be taken as randomly-strewn "topics" courses...or as independent studies. But the smallness, thus limiting my choices of courses, also meant that I got much more one-on-one instruction with some fine teachers.

Then there was grad school...oh, too much to say. I realize as I'm writing this that I'm straying away from my original intention: what does math mean to me? Well, that's how freewriting goes, I guess...what does math mean to me?

It's a language.

It's a way of life.

It's fun.

It's exciting.

It can be terrifying.

It can leave me lying awake at night, pondering imponderables...like the epistomological questions it raises...and the metaphysical ones...to what extent am I "faking it"? I ask myself sometimes...how can one "fake" mathematics? What does that mean? Does that mean I'm merely making up the rules as I go along, and as long as I'm careful enough to write the rules so that they accord with the rules everyone else has written so far and as long as they accord with each other, it's all good, it all makes sense in the end?

In that sense, I guess...

I don't know.

I feel warmed by mathematics. It's familiar to me, I don't remember a time without it, and I pity people who can't appreciate it. Or don't appreciate it.

Pity's a strong word, maybe that's not what I mean, but in any case, my ten minutes are up.

Posted by DocTurtle at 2:42 PM 2 reflections

Labels: assessment, Calculus I, MATH 191, theory, writing

## Wednesday, August 15, 2007

### Notes to self, part 2

We're just a few days away from beginning the new semester (Monday, August 20th: do you have your calendars marked?). In between periods of vegetative depressurization from the newly-ended REU and continuing research in probabilstic graph theory, I've been spending a bit of time during the past couple of weeks putting together various activities for MATH 280 and MATH 191. Much of my planning is outlined nearly illegibly in the margins of my copy of John C. Bean's *Engaging ideas: the professor's guide to integrating writing, critical thinking, and active learning in the classroom* (Jossey-Bass, San Francisco, 2001), but I really need to compile it all in one place. Ergo...

During the coming semester I will be putting greater emphasis on "decentering" activities that promote cognitive dissonance, unorthodox points of view, and healthy skepticism. In presenting students with counterintuitive mathematical ideas, I can imbue them with a sense of surprise, wonder, and curiosity. In asking them to develop the ability to see a problem from all perspectives (literally and metaphorically), I ask them to become stronger problem-solvers. In encouraging them to question unproven assumptions, I not only charge them to be more careful in their calculations; I also open up to them unexplored fields of inquiry. Where would geometry (and by extension, much of modern mathematics, not to mention the physical sciences) be had a number of brilliant minds not questioned the validity of Euclid's Parallel Postulate?

Now, how to do all of this?

I hope to introduce the students to the idea that mathematical discovery, along with its recording and transmission, is a dialectical process that involves the researcher in conversation with herself and with others. The act of discovery is almost never a burst of light illuminating the void, but rather is born from the steady nurture of an ever-growing spark. Discovery begins with the posing of a question, the wrestling with a problem. One's first thoughts on a problem are generally chaotic and messy, and harken back to solutions to analogous problems and inchoate modifications of earlier ideas. After long hours of talking with others and lying awake at night staring at the ceiling, and after countless pages of notes have been scribbled, studied, and redacted, more complete ideas take shape. Bean (p. 20) speaks on the nature of the written manifestation of this process: "the elegance and structure of thesis-governed writing -- as a finished product -- evolves from a lengthy and messy process of drafting and redrafting."

Below are a number of the exercises I plan on implementing in some fashion during the coming semester:

- Response writing to Polya; possible guiding questions: "Is Polya's proposed process relevant in a modern problem-solving course such as MATH 280? Take a position on this question, and defend your point of view." "Have you ever applied Polya's process, knowingly or unknowingly, to solve a problem posed to you? Explain carefully." "Do you feel that intuitionism is a defensible mathematical philosophy?" "Use Goldbach's Conjecture to illustrate the difference between constructivist mathematics and nonconstructivist mathematics."
- Decentering exercises focusing on puzzling phenomena such as various sizes of infinity, space-filling curves, fractal dimensions,
*et cetera*. - For the 191 folks, to get them to take a position in a short thesis-governed paper: "Suppose you need to differentiate a function of the form f(x)/g(x). Do you prefer to apply the Quotient Rule, or would you rather rewrite the function as a f(x)(g(x))^(-1) before applying the Chain and Product Rules? Explain the reasoning for your preference."
- For the 280 folks, to accustom them to "mathematizing" messy problems and developing intuition (skills I feel are overlooked in even the more discovery-learning oriented proofs courses): "Consider the game of Nelinurk (see tonypa.pri.ee/start.html). Play the game for a half-hour or so to get used to the rules and the flow of the game. Once you feel comfortable playing, see if you can describe the game mathematically and develop a strategy for optimal play. Explain your strategy as clearly and as completely as you can. (It may help to develop your own terminology and notation as you write.)"
- More "intuition-building" activities for 280 students: estimation exercises? Incomplete proofs? ("How big?...", "How many?...", "Give the outline for a proof of...") As I said above, I feel that the nurturing of mathematical intuition that's done in most proofs courses is woefully outweighed by the emphasis placed on learning how to do formal proofs. (And no, I don't think it needs to be put off until a "problems course" like our 381; good intuition makes for clearer, more succinct proof-writing, and clearly written proofs feed a healthy intuition like Wheaties feed Mary Lou Retton.)
- Taking a page from my own playbook, five or six years ago at Vanderbilt: have 'em write a few "poems inspired by mathematics." It can't hurt, and it might be just what the more humanities-minded students need to get their creative juices flowing.
- Have a "show and tell" day on which I ask everyone (myself included) to bring in all of the notes, scribbles, emendations, and so forth that went into the final draft of a given project. (I might simply ask them to save all of their homework drafts?)
- Class-opening and class-ending one-minute essays: "where do we need to go today?", and "where did we end up in our travels?"
- Mock trials: in 191, the obvious,
*Newton*v.*Leibniz*. In 280, perhaps*Brouwer*v.*Hilbert*? Each side is taken up by roughly half of the class, certain individuals chosen to act*as*the given personages, with others as their supporting staff (*i.e.*, legal counsel). For the 191 debates, I could even have the two sections square off in a "finals" round, one section taking the side of Newton, the other that of Leibniz. - Analogy games (cf. Bean, p. 111): ask the 280 folks to complete the following and elaborate upon it: "Writing proofs is like ________ ." For the 191 classes: "If differentiation were an Olympic event, it would be most like ________ ."
- Precise proof-summarizing and theorem stating: require students to write a proof summary or a theorem statement using a
*precisely defined*number of words, giving maximal credit only for using exactly that many words. Example: "explain the Axiom of Choice in precisely 25 words." - To give the students a taste of "original research," I can hand the 280 folks the data I obtained this past summer on consecutive inverses modulo
*p*and ask them to describe the patterns they see. I can do the same for the Calc I kids, giving them the graphs of the sequences of expected degrees for various of the random tree construction algorithms, asking them to supply likely models (and concomitant analysis) for the shapes they see.

Hmmm...that's all for now. More to come, I'm sure.

Posted by DocTurtle at 11:08 AM 0 reflections

Labels: Bean, Calculus I, course prep, Foundations, MATH 191, MATH 280, self-authorship, theory

## Tuesday, August 07, 2007

### Whoa...

Yeah, it's been a while.

I had no idea that I'd be so busy during the REU that I wouldn't even get around to updating this damned blog even once.

To my regular readers (I know I have a few of you dedicated souls out there, including my wonderful colleague up at Bates College), I sincerely apologize for the lack of activity on this here website. Things got a little crazy around here this summer...good crazy, not bad crazy, but crazy nonetheless.

Just this past Saturday I took our last-remaining REU student to the airport to wing her way back home in Boston (where she'd pass just a scant 10 days before heading over the sea to do a semester in Budapest, lucky devil), marking the semi-official end of our REU's first run.

Lemme tell ya, I think it went wonderfully, and I think for the most part that the students who took part in it would agree. What amazed (and warmed) me most were the strong bonds the students clearly developed for one another. No later than a week into the program it was clear that they would form a cohesive group whose members would support one another in work and play. They were a unit, and did nearly everything together. Their support for one another was evident in all that they did. And they did a lot: I think we're looking at at least five papers of the sort you might find in either an UG research or a mainstream math research journal with students' names on them. Not to mention a few other papers to be written by faculty inspired by the ideas put forth during the program. I'm tellin' ya, this place was hoppin' this summer. Not only did the program give the students the chance to work on some real math, I feel it also injected new life into the department as well.

I've nothing more specific to say about the program right now, but I'm sure that vignettes and anecdotes will trickle out of me in the coming months as I look back on the program and assess its strengths and weaknesses, especially once we start gearing up to run next summer's installment.

So what else is going on, pedagogically speaking? I've just finished writing and posting the syllabi for my Fall 2007 classes (here is the Calc I syllabus, and here is the Foundations syllabus), underway in a little less than two weeks (is it really so soon?!).

After meeting with Fiona back in May, I felt I was better able to revamp the Foundations syllabus from its Spring incarnation; I'm retaining much of the present structure of the course but am modifying the way in which students share their solutions to selected homework problems. Instead of simply soliciting for student volunteers to present these selected problems, I will be assigning committees of three students each to collect and provide summaries of all students' solutions to selected problems. Following their analysis, the committee will lead a short discussion on the solutions they encountered, indicating which they felt to be the strongest and the reasons for their choice. The details are included in the syllabus, if you have any desire to read more about it. (This sort of peer feedback exercise is a modified version of something I picked up from a session I attended at the Joint Meetings this past January.)

Fiona also helped me tinker with the homework. Many of the assignments will continue largely unchanged, but I've tweaked a few problems, deleted a couple that proved out-of-place or inappropriate, and added a few here and there. Thanks, Fiona, you rock!

As for Calc I, the biggest structural change comes in the way in which homework will be graded: since the "Homework Lottery" didn't seem to go over so well with the Calc I kiddies as it has with my last four sections of Calc II, I'm adding an additional layer of structure to encourage completion of all of the homework. (Last semester's students' primary weakness was in completing the homework.) To wit, I'll still use a lottery to select a handful of exercises to grade carefully, and for all other exercises I will merely indicate whether they are "right" or "wrong." I'm leaving it up to the individual sections' members to decide how much weight to lend to which sort of problem. In this fashion I'll be incorporating the "choose your own grading system" idea I toyed with a couple of months ago. (Incidentally, I'm not allowing a more drastic student-led grade assignment because I want to be able to maintain a high degree of consistency between the two sections; the last thing I want is some struggling student from Section 1 coming to me five weeks in with a petulant complaint that his weak homework grade wouldn't even matter in Section 3 because they chose to make HW count for 10% instead of the agreed-upon figure of 30% in his section.)

In terms of both in- and out-of-class activities, I've put together several new ones for each class. For instance, Day 1 of Calc I sees us playing with Google's pedometer feature as we puzzle through the meaning of the paths we trace on maps, and what the heck the altitude feature has to do with calculus. Day 2 of Foundations will be spent on the first of several activities focusing on the importance of clarity and correctness in written mathematical exposition: students will be asked to compare the distinctions between good and bad writing in a more "traditional" writing-intensive field with the same distinctions in the mathematical sciences.

And so on.

Much more to come in the coming weeks, on the REU, on my classes, on my students, on my random thoughts. For now, I must away and get some more work done on yet another NSF grant I'm putting together with a few of my colleagues from around the U.

Thanks for reading, take care, please come again!

Posted by DocTurtle at 2:05 PM 1 reflections

Labels: Calculus I, course prep, Foundations, MATH 191, MATH 280, research, REU, undergraduate research