Day of Higher Ed, part 1

My colleague Casie tipped me off that today is The Chronicle of Higher Education's Day of Higher Ed, a day on which those of us who work in higher education are supposed to raise awareness of what we do, and how much of it we're doing. Contrary to the perception of some (mostly conservative, in my experience) suit-types who view higher education with disdain and its employees as fat and fatuous sinecurists, most of us actually work quite hard.

More forcefully, most of us work our asses off.

So, what's in store for me today? (Keep in mind that I'm coming off of a weekend on which I spent roughly 12 hours grading Calc III exams...) Though I can't guarantee I'll get to it all, here's the medium-term docket:

1. Write an end-of-year project report for the National Science Foundation for my 2011 REU (estimated time to complete this: 5-6 hours)

2. Draft a welcome and informational letter to this year's crop of REU students (time to complete: 1-2 hours)

3. Send out rejection letters to the nearly 300 students who didn't get into this year's REU (time to complete: 1 hour)

4. Grade exams for my two students who are taking Abstract Algebra II as a reading course (time to complete: 1 hour)

5. Respond to final drafts of MLA students' latest written projects (time to complete: 3-4 hours)

6. Write three or four rec letters, including an incredibly urgent one for which the requesting student gave me less than a week's notice (time to complete: 2-3 hours)

7. Plan workshop materials for an upcoming writing workshop I'm leading at NC State (time to complete: 3-5 hours)

8. Review and respond to proposed Abstract Algebra curricula floated by other members of the Mathematical Association of America's Committee on Undergraduate Programs in Mathematics (time to complete: 3-5 hours)

9. Generate open problems list for this year's REU (time to complete: 2-3 hours)

10. Begin housing and payroll paperwork for this year's REU students (time to complete: 1-2 hours)

11. Draft a precis for on the latest curricular proposal by the Curriculum Review Task Force's Curricular Sustainability Subgroup (time to complete: 2-3 hours)

12. Complete drafts of research articles currently underway with three different sets of undergraduate coauthors (time to complete: 10-20 hours)

13. Oh yeah...teach!

Anyway, I'd better get to work on that list, me with my oodles of free time. More later, perhaps...

An astronomer, a physicist, and a mathematician...

...are on a train in Scotland. The astronomer looks out of the window, sees a black sheep standing in a field, and remarks, "how odd. Scottish sheep are black."

"No, no, no!" says the physicist. "Only some Scottish sheep are black."

The mathematician rolls his eyes at his companions' muddled thinking and says, "in Scotland, there is at least one sheep, at least one side of which appears to be black from here."

There is danger in inductive thinking, as even Hume acknowledged.

I've been reading up on epistemology for my MLA class's initial foray into the philosophy of mathematics: Bacon, Hume, Descartes, Kant, and Popper...and of course Lakatos! We're going to get in pretty deep. For good measure we'll prove Euler's formula for polyhedra and try to understand what's so unsettling about the Law of Excluded Middle, the Axiom of Choice, and the proof of the Four Color Theorem.

You know you want to join us!

Salt

The world says salt
and we say six,
a number which may or may not
really exist.

I oughta know better...

I'm about 30 pages into Michael Gurian and Kathy Stevens's The minds of boys: Saving our sons from falling behind in school and life (San Francisco: Jossey Bass, 2005), and I have no earthly idea how much more I'll be able to read. It's tendentious, fatuous, and overwritten, relying primarily on anecdotal evidence to prove its points, and when more critical evidence is given it's given second-hand, safely filtered through reference to other texts and not to the studies those texts rely upon.

Bluntly, it's pap.

Why read it? I thought I'd try to get a grip on views contrasting with that of Cordelia Fine (see this post), who holds that the biological bases for gender differences are blown entirely out of proportion, and that acculturation more than anything else is responsible for differentiation of gender, including differentiation in intelligence and academic performance. Gurian's name shows up an awful lot as one of the giants of biological determinism, and he's written stacks of books on gender difference, so I thought I'd check him out.

His website's not particularly promising, listing scant credentials relevant to the books he writes and boasting membership in three professional organizations, one which doesn't exist, one whose website hasn't been updated in seven years, and another which appears somewhat reputable. I'm not sure I'm one to lobby the former objection, having just finished a book in an area I'm not "credentialed" to write, but I would think that one of the foremost "authorities" in gender difference, its ramifications, and its ameliorations, should have some sort of post-graduate degree in psychology, neuroscience, or at least counseling or social work. Gurian's most advanced degree is an M.F.A., and before that he holds a B.A. in philosophy. In fact, his most promising claim to authority is his unwavering insistence that he has authority: his website is a fantasia of self-promotion, and he mentions his own institute on just about every other page of the book I've begun.

About that book...its primary thesis is that our schools are in crisis (his words, not mine): boys are falling behind and failing in disproportionate numbers...and it's because our educational system does not take into account crucial differences between the ways boys and girls learn. Truly this is a crisis, Gurian insists: "Yes, we're sorry to say, there really is a crisis" (p. 20...did I mention the text is nothing if not inflammatory?). The extremist language he uses to introduce numerous "statistics" (none properly cited and all treated uncritically) "proving" boys' educational crisis is particularly chauvinistic: for all his righteous indignation you'd think that it's men and not women who for the last several centuries have been underserved by Westernized educational systems. Once or twice he throws us a bone and insists that he has equity in mind: "Calling attention to the college problem for males is not to decry an individual's particular qualities, nor to lament women's successes in increasing their college attendance, wages, and financial independence from males" (pp. 27-28). Such mots ring hollow, however, and I can't help but come away feeling that a hundred years ago Gurian would have been a phrenologist, palpating female skulls in order to point out their "obvious" mental deficiencies.

Ugh.

I'll read on, but I can't see this getting any better...

UPDATE: I'm on page 39 now, and can't help sharing this bit of nonsense: "Research in the 1990s clarified ways in which our schools fail our girls, especially in areas of math and science, the dynamics of self-esteem in the classrooms, and computer design instruction. Because our culture recognized a girls' crisis, it has addressed those problems and to a great extent has changed things for the better as far as teaching girls is concerned."

Yay, according to Michael Gurian sexism in educational practice is now over! It's a thing of the past, a problem we're no longer wrestling with. I'm so happy to live in a post-sexist, post-racial America, where we're all colorblind, a black man can be president, and the mathematical sciences are roundly dominated by women.

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...

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.

Back to school

UNCA serves a large number of nontraditional-age students who are returning to school after taking time off for other things, and many of these folks are older than I am. Therefore I'm used to not being the oldest one in the room when I'm teaching a class; I've only been the eldest in maybe five or six of the 50-60 class sections I've led at this school.

But I'm not used to being one of the youngest in the room.

My MLA (Masters of Liberal Arts) course, Number sense: The philosophy and psychology of mathematics, met last night for the first time. I've got eight students in the class, four or five of whom are my senior in age and in life experience. It's a great bunch, and I can tell I'm going to learn more from them than they're going to learn from me.

We started off with some freewriting, through which I asked the students to probe into their own mathematical pasts. I hoped to find out what it is that makes these folks tick mathematically and to determine what they perceive to be the most basic and fundamental of mathematical operations. If we can get at the these operations, we'll be in a position to start our study of mathematical cognition where our brains begin, with approximations of enumeration.

I'm delighted to report that there's considerable diversity in the class when it comes to mathematical background. I found it interesting that the two gentlemen in the class reported more facility and familiarity with mathematics than their feminine counterparts (with one exception). Both of them described delight at working with statistics and geometry, and obsession with game-lake mathematical puzzles. The one woman with more mathematical experience is a former UNCA math major with whom I had the pleasure to work when I last taught Abstract Algebra I (in Fall 2008). This is her first semester of study in the MLA program.

The other five folks have considerably less mathematical background, but will provide perspectives from other points of view, reporting interest and expertise in psychology, history, and philosophy. I'm excited to learn from Samantha, who is taking time off from her work as a teacher for special-needs children. She mentioned how frustrated she is with mathematics education, and hopes that our class will give her the skills to help improve the way students are taught math at a young age. More power to her! Her high expectations for the course will definitely keep me on my toes.

After we probed our mathematical pasts for a bit, I presented a few exercises and experiments I hoped would whet their appetites for the material we're about to study (from Dehaene's Number sense). With no promise that this link will be evergreen, you can find Mathematica files for these exercises on the course website under the entry for January 12th.

In the first exercise I challenged students to hold in their memory progressively longer randomly generated strings of digits, demonstrating the means by which we tend to use our linguistic faculties to store such strings in short-term memory. (This use underlies linguistic differences in the ability to memorize and compute with numbers: the brevity of number words in many Asian languages allows native speakers of those tongues to outperform speakers of other languages in basic memorization and numerical manipulation.) The second exercise demonstrates how the arrangement of objects affects our sense of their number: more densely packed objects tend to appear more numerate than those that are sparsely spaced, and more orderly-arranged objects appear more numerate than those that are randomly placed. For the third activity, I asked the students to report any sort of synesthesia they've ever experienced: do they sense that numbers have specific appearance, texture, color, smell, relative spatial position, or gender? I think the students found it odd when I reported I've always had a very well-defined sense of the gender of every digit.

We wrapped up with a short discussion of formal expectations for the course, a topic I'm trying to de-emphasize as much as possible. I'm looking forward to next week's discussion. I'm curious to see if the students will find Dahaene's book as intriguing as I have.

On deck

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

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

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

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

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

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

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

Further bulletins as events warrant, likely soon.