Georgia on my mind

I just got back from a lovely overnight trip to Kennesaw State University, where I had a chance to give a talk on the mathematics of the Incan khipu (often spelled quipu) and hang out with Zima, one of my best friends from grad school, who's on the faculty at KSU. Zima'd asked me to spend an hour with her department's faculty, talking with them about writing in the disciplines and writing-to-learn activities, which I was more than happy to do.

One of her colleagues gave me some neat teaching ideas, including the following writing exercise: take a valid mathematical statement (printed out), one with which your students are not familiar, chop it up into its individual words, and scramble it. Give it to your students and challenge them to recreate a valid mathematical statement from the scrambled words, using every word exactly once. This exercise helps students to make sense of the grammar and semantics of mathematical prose, whose density often obscures its meaning.

Later in the afternoon (after a lovely lunch playing catch-up with Zima), I delivered a presentation titled "The traditional mathematics of Peru: khipu and khipumakers" as part of KSU's Year of Peru activities. The audience was made up of faculty and students from across the KSU campus, including a good number of math-anxious folks who were more interested in the "Peru" part of the talk. Overall, I think the presentation went well, even the bit where I had all of the people in the audience making their own khipu cords. Khipu (about which you can learn much more here) offer the most salient example of Incan mathematics, as well as a touchstone of cultural determinacy: khipu demonstrate assertively that math is a cultural artifact, a product of human society. (Moreover, they're beautiful, as a peek at the gallery at the above link will show.)

I'm back home now, and am looking forward to tomorrow's attack on a new Calc III problem set, and a couple of meetings on the curriculum review (well, not really looking forward to the latter, but they'll come nonetheless...). Meanwhile, I'll savor the last sweet sips of today.

Yup, it was a good day.

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.

Breadcrumbs

I feel like Hansel and Gretel (being both at once would be an apt ontology for this post), following a trail of breadcrumbs as I wind my way through a forest.

In each of the three meetings we've had so far, the students in my MLA course have raised some interesting (and as yet unanswerable) questions. Many of these concern the classic "nature vs. nurture" matter that infects every conversation involving human abilities and achievements. For instance, is it nature or nurture that leads to mathematical (and otherwise) savantism? That is, do "human calculators" owe their skills to some advantageous neural network structure in their brains...or do they develop those skills through hard work and constant application of ordinary neurological machinery?

Opinions were definitely divided on this matter during our last meeting: some accepted Dehaene's explanation that practice makes perfect, and that those who have plenty of time to practice are liable to more closely approach perfection; others weren't convinced. "Maybe none of us are born geniuses," one student said, "but some of us are born with better propensity to achieve genius than others are. Just like most of us will never be professional basketball players, as we lack the physique it would require."

This analogy might remind us that after all the brain is as much a piece of our anatomy as are our arms and legs, and our interpersonal differences in overall anatomy carry over to interpersonal differences in our brains as well. No doubt some of those differences predispose us well to certain kinds of genius inaccessible to others? It's up to each individual to nurture latent talent in the most efficacious way from that point on.

The risk we run in arguing like this is making dangerous generalizations along the following lines: "brain difference X translates into ability difference Y" and so forth. As I quipped mysteriously on Facebook the other day, brain does not equal mind anymore than map equals territory, and if we pretend that we can extrapolate everything we need to know about a person from the structure of their braincase, we anachronize and become phrenologists, feeling for bumps, risking claims about racial or sexual superiority. Remember that it was held for a long, long time that women were less bright than men because their brains were less massive...and that when this point of view was assailed from all sides, its adherents did all they could to rescue it, falling back on more and more convoluted sophistry to save their theory: "well, it's not brain volume per se, but volume of gray matter...or at least the ratio of gray to white matter...well, maybe the degree to which it's all convoluted..." (See Stephen Jay Gould's marvelous The mismeasure of man for a blow-by-blow debunking of such arguments.)

About sexual superiority (and getting back to our trail of breadcrumbs): one of my students turned me onto Cordelia Fine's Delusions of gender: How our minds, society, and neurosexism create difference (New York: W.W. Norton & Company, Inc, 2010), a thorough discussion of modern attempts to pin gender differences in intellect on neurochemistry. Phrenology's not dead, it just looks a lot different than it did 150 years ago: now we don't look for protrusions in the skull, just higher-than-normal levels of fetal testosterone, and we don't come right out and suggest that women aren't as smart as men, we just say they have a greater propensity to empathize than they do to systematize. Fine spends much of her time pointing out flaws in modern phrenologists' methodologies, though not as many she might; I've noted a few flaws she could have mentioned but didn't.

It's a stimulating read, and I plan on sharing a few chapters with the MLA students. It's also leading me to other sources. Some of these are general in scope, like Jan Morris's Conundrum (New York: Harcourt Brace Jovanovich, Inc., 1974), a first-person account of the author's transition from male to female and the worlds she lost and gained in the process. Others are more specific, like Nash and Grossi's analysis ("Picking Barbie's brain: Inherent sex differences in scientific ability?" Journal of Interdisciplinary Feminist Thought 2(1), Article 5) of Simon Baron-Cohen's methodologies...which analysis might doom some of the infant studies which Dehaene cites, as well (O, circularity!).

I also plan on offering up a few excerpts from Richard E. Cytowic's The man who tasted shapes (Cambridge: MIT Press, 2003), an odd novel/memoir/neuropsychology text dealing with the phenomenon of synesthesia. Though not directly related to our course, I think certain passages bear tangentially on our discussions of brain function and will lead to interesting discussions.

More fun to come! Exciting. I hope the students are getting as much out of the class as I am. I may have to offer this course again soon in the Honors Program. Something to think about for next year...

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.

Data mining

I spent a few hours this morning running the numbers on the students currently enrolled in our Honors Program, hoping to get some objective data on students' participation in the program as I move toward my new position in the fall. I made some heartening findings.

Namely, each of the school's three major disciplinary divisions (humanities, natural sciences, and social sciences) is pretty equally well-represented in the Honors Program. School-wide, 9.25% of all declared majors take part in the Honors Program, and the participation rates of the individual divisions range from 7.68% to 9.91%, quite tightly centered on the overall mean. Counting the courses these students take in Honors gives further evidence to this balanced participation: overall, a student in Honors who has declared a major has completed 3.65 Honors courses on average, and the means for the various divisions range from 3.58 courses per student to 3.81 courses per student.

There are certain departments that are particularly well-represented in Honors, including a few that are quite large (and that are therefore somewhat immune to sample-size bias). For instance, five of our seven departments with at least 100 majors can boast that more than 10% of their students take part in Honors, including one department with 122 majors, of whom 17 (13.93%) are enrolled in the Honors Program. At the other extreme, there are four departments, each home to anywhere from 26 to 31 majors, with no students in Honors. (One of these is a relatively new department, one which graduated its first majors just a couple of years ago.)

Interesting.

I don't believe these data to be "actionable" in any way...and besides, the results don't indicate dramatic action. I'll stay the course for now...though it might not be a bad idea to talk with the folks in those four departments to make sure their students are aware of the opportunity...?

We'll see.

Defragging

For the past few weeks I've been hammering away at several different poems, but I've got little more than a couple of callousy handfuls of bent-nail fragments to show for it. Odd images (starry-breasted crows, fog-covered tenement-like brick blocks, concrete tunnels like birth canals), and random thoughts.

Keeping with this fashion, here are a few random thoughts on academics and academia:

1. I wonder at the extent to which we are all isolated in our disciplines, and to which we do most of our work in rooms with four walls and very often no windows.

2. I wonder at the sterility of the Platonist, universalist, formalist conception of mathematics, itself an isolating philosophy, allowing as it does a detachment from the world and from others as we engage in our mathematical work.

3. I wonder at the mechanisms we feel we must make and maintain in order to "deliver" our curricula. The more I learn about the inner workings of the Honors Program, the more I wonder if there are simpler ways to put it all together.

4. I wonder at our assessment practices, at every level, from the individual student to the institution as a whole. To what extent are they arbitrary, effective, replicable? To what extent are they doing what we need them to be doing?

5. I wonder at the effects of our educational system, both intended and unintended. How often does a student's passion for perfect grades overpower her passion for learning?

6. I wonder at things as they stand for things, and am reminded of William Carlos William's red wheelbarrow:

so much depends
upon

a red wheel
barrow

glazed with rain
water

beside the white
chickens.

Guinea pigs

I have to admit some trepidation on my part going into the second meeting of my MLA course this past Thursday. Despite my strong record of teaching in mathematics, I have relatively little experience in leading wholly discussion-based courses, and this inexperience coupled with my sense that a few folks in the class are leery of mathematics in the first place made me worry that the conversation we'd have together would fall flat. Though it took a little while for the conversation to get going, my fears soon proved unfounded.

We began in small groups, where I dealt myself into a conversation with the two older gentlemen in the class, both of whom expressed some measure of skepticism about the reading. Uriah seemed appalled by Dehaene's seeming to alternate between making claims of revolutionary understanding of the brain's functioning and retreating to more palatable and defensible observations. Quinn seemed to accept some conclusions but was put off by the relative (to mathematics) lack of "rigor" and less rigid notion of "proof" in psychology literature. I found these reactions heartening, as reasoned skepticism is generally salutary, something to be expected: these two men are among the more mathematically experienced in the class (surpassed only by Bonnie, a former UNCA math major whom I taught in Abstract Algebra back in 2008!), and the insistence on rigor is a more traditionally masculine trait.

When we returned to a full-class conversation, many more ideas came out, primed by my request for each student to identify those aspects of the reading they found most intriguing, most confusing, and most well-received. It would be difficult for me to summarize all of what was said, so I'll focus on a topic we spent much of our time, dealing with the following image:



I crafted this image last spring for my Ethnomathematics course, in order to serve as a Rorschach test of sorts, testing respondents' notion of numerosity. For the longest time I've found it fascinating that as a species we tend to distinguish objects based upon contiguity and connectedness, "topological" aspects, not on color, shape, or other more "geometric" aspects. That is, most people will respond, if asked "How many objects do you see here?" that there are four, for there are four noncontiguous bodies present.

This, however, is the unskeptical answer, unaffected by the sort of "questioning bias" that doomed the Piagetian experiments Dehaene outlines in Chapter 2 of his book. Specifically, when asked to decide which of two rows of small objects is greater in number (the lesser quantity being arranged in a longer row so as to mislead), young children will often respond incorrectly simply because they suspect trickery on the part of the questioner. In our situation, the skeptical respondent, suspecting trickery, might respond (as did Quinn in my MLA class) "one," seeing a single paw, or even "two," differentiating the two objects on the basis of color and not contiguity.

For quite a long time we discussed the evolutionary advantage of enumeration based on contiguity, and various related questions came up: what would have to be true of a species whose members enumerate objects on the basis of other aspects? What could be said of their mathematics? Can a mathematics of "continuous quantity" model our "discrete" mathematics fully and effectively?

If nothing else, everyone in the class wanted to learn the likeliest response to the "how many" question above...if the question could be posed in a less leading fashion. "I've got a sample of over 60 students I can test tomorrow," I said, referring to my Calc III classes. "Why don't I get some more data?" The students loved this idea, and we debated how the Calc III students should be prompted for the purposes of this informal survey. It came down to between "What do you see?" and "Describe what you see." People seemed more satisfied with the second, as it seems to beg for a more elaborate response, increasing the likelihood of a description featuring some kind of numeric content.

Yesterday I began both sections of Calc III with the experiment, providing no context beforehand, so as to minimize bias in the students' responses. (I did inform them of my intent afterward, and gave them all the option of retrieving their responses if they'd prefer that they not be read by others. I hope that no one on our IRB is reading this...) I've not yet looked over the responses, but I'm already looking forward to the analysis we'll do this coming Thursday night. I'll be sure to post some sort of summary results here once we've had a chance to sort it out.

Anyway...I'm counting last Thursday as a success. I think this course is going to run itself. I'm not so worried anymore.