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.

Lessons learned

I've just finished reading through my MATH 179 students' final exams. There I'd asked them each to write in response to three different questions; briefly: (1) how has our course challenged your assumptions about mathematics?, (2) How does UNC Asheville, as a liberal arts university, differ from traditional state schools?, and (3) how would you describe your own "writing process" as you completed one of the writing assignments required of you in our class this semester?

The students' responses to the first question were rather erratic and varied wildly in quality, ranging from unelaborated lists of concepts like meddos, abaci, or alternative bases of arithmetic to carefully crafted descriptions of personal change. Sadly, there were far more of the former than of the latter, and as often as not I suspected the students of writing what they thought I wanted to hear rather than what they truly felt. The responses to the second question were predictable, focusing on small class size, the potential for one-on-one interaction with faculty, and a curriculum emphasizing development of the student thinker as a whole.

The students' responses to the third question were the most eye-opening to me, and gave me a good deal of direction as I develop as an instructor of first-year (rather than disciplinary) writing. For instance, it's clear that in the future I'll have to be more explicit in discussing the purpose for various stages of the writing process. Several students pooh-poohed the need for revision, saying, essentially, "I don't like to write in drafts, because I'm pretty happy with how it sounds after the first draft." Others bemoaned being asked to write outlines or rough drafts; one student said, more or less, "they just get in the way of what I know I need to write about...I just want to get to the point." One of the stronger student writers explicitly questioned the validity of the writing process, indicating he felt it was a waste of time and served only to objectify what is ultimately a very subjective and personal activity. Although one could argue that this student is really a budding post-process theorist, I think it's more likely that he's simply not yet learned why we do all of the things we do when we sit down to write a piece.

I noted too that students have a very hard time viewing writing as anything but linear. Perhaps because reading (traditionally, anyway...let's not speak about the way one reads on the web) has always been seen as a linear process, something done from start to finish, from the first word to the last, students have come to think of writing as a similarly linear activity. For example, no fewer than three (out of 18) students complained about how hard it was to write the introduction to a paper. "Once I get the introduction out of the way," they seemed to be saying, "the rest of the paper comes really easily." The upshot? "You can't write anything else until you've written the first paragraph, because that's where you lay out what you're going to say." It hasn't occurred to these students (and, admittedly, I did a pretty poor job this term in helping them to see) that perhaps they're better off saying what they want to say first, even elaborating it a bit, fitting together all of the important pieces of evidence, and maybe even wrapping it up by discussing their own conclusions based on that evidence, before even thinking about writing an introduction. Most of us who write academically know well that the introduction (or abstract) to a paper is often the very last thing we write: it's so much easier to lead the reader into what you wanted to say after you've already said it.

These ideas are ones I'm not used to having to make explicit in teaching writing, as I'm not used to working with novice academic writers. I've got to keep these ideas in mind for the next time I have a chance to teach first-year students general academic writing skills.

Wrapping up

The end is near! (I realize I've been saying this a lot lately, but it's more and more true each day.) We've got just under a week of classes left, with only one more meeting of MATH 480 and three each of Calc II and 280.

My last "regular" meeting of MATH 179 was this morning. We dedicated the day to a free-wheeling discussion of the past semester, focusing on the three central purposes of the class: (1) learn how math is experienced from place to place and across time, (2) learn a bit about a liberal education in general and UNCA in particular, and (3) get some practice in writing academically in a specific discipline, and for specific purposes.

Students pointed out some of the course's strengths: it helped fine-tune their writing skills, gave them a broader view of mathematics, and offered a forum where they could talk about random school-related matters with students in similar situations without fear of looking like fools. They pointed out some rough spots, too: the first text we used (by Ascher) was awful, and the 9:25 time slot wasn't the most conducive to student excitement. A couple of students indicated that I should teach the course again, and I suspect that the next time around it'll go much more smoothly. The students' take-home final exam, due in a week or so, asks them to reflect more intentionally on the three points I mentioned above; I'm sure I'll get from it an even greater sense of the course's strengths and weaknesses.

The MATH 280 students get their last exam tomorrow, too, and like the last it'll be a collaborative one. I was tremendously gratified by the students' reception of the last exam: the learning environment the students created and cultivated as they worked on the exam was a rich and fertile one. I have no doubt that many of the students, particularly those who may struggle more mightily than their quicker peers, learned more from working with one another than they would have from any other form of assessment. (Incidentally, I've been keeping track of collaboration networks on the collaborative exams I've given over the past couple of semesters, and I hope to begin tracking the data more carefully to try to analyze correlations between indices of collaboration like number of collaborators and measures of success like initial exam scores.)

The last 280 exam asks students to provide a few more "traditional" proofs, but, like its counterpart in MATH 179, it also asks the students to reflect on the course and identify specific topics they found interesting and write at length about those. My hope is that the exam will help me to discern which areas students are most interested in, and about which they feel most "iffy." This'll help me plan the course more appropriately in the future.

Speaking of course-planning...I've already got some ideas in mind (involving multiplication tables) for first-day activities for MATH 461 (Abstract Algebra I) this coming fall. I'm excited! I know it's several months off yet, but I'm looking forward to it already.

Before I go, I have to give a shout-out, to my colleague Kelli and to our first-year junior Kamryn, for spearheading the successful founding of a UNC Asheville chapter of the Association for Women in Mathematics (AWM): wonderful first meeting! I look forward to seeing what activities the students can put together in the coming months.

P.S. -- I sent the first draft of my manuscript to the publisher today. Much happiness.

Confessions of a sometimes-mathematician

Sometimes you have to let go, and let people lead their own lives.

As regular readers may know, this term's the first time I've taught a first-year colloquium at UNCA. (How I've avoided teaching one for this long is beyond me.) Therefore, before the last few weeks, I've never had to advise non-math majors. Many of the students in my MATH 179 course are "undeclared," and though many of them have some idea as to what they want to do with their lives, a few of them have almost no direction whatsoever. At the end of the day, I don't think this is necessarily a bad thing, as long as you're open to a little exploration and self-discovery. As much as it might annoy a highly-driven type-A person like me, some people just aren't sure where they're headed, and don't feel an immediate need to figure it out.

In the past two days I've met with two such students from my MATH 179 class. Both of them worked with me to hammer out tentative class schedules for next semester, but only after a good deal of discussion about possibilities ranging across the curriculum. (Oddly enough, they both might find themselves in Ancient Philosophy as they explore that route.) Neither wants much more from college right now than the experience of being in college, and right now I don't think there's much more they need to get out of it. They'll have to worry about that down the road a piece, but for the time being they'll be safe taking some core classes and getting baseline requirements out of the way. Until they reach a fork in the road, I'll help them along in whatever way I can.

Others can't afford such leisure and latitude. One of my advisees is about to graduate with a pure mathematics major, and I can't help but think she's found no more than the merest passing interest or passion in any of the math courses she's taken here. More than once in the past four years I've encouraged her to take another path if something truly striking struck her ("really, I won't take it personally"), but she's stuck with the math program, passionless as she may appear about it. I hope we've served her well.

Someday, perhaps, she'll find her path. But I've got to let go and let her do that for herself. After all, I can't help her find her way if I'm not even sure from day to day just what it is I want to do with my life. As I've confessed to some of my closest friends (and as I admit here now), I've given serious thought in the past year or so to setting math aside and diving more deeply into rhetoric and composition, areas in which I've been more keenly interested for the past couple of years. Put simply, for the past year or so rhetorical theory has gotten me far hotter than any theorem I've been able to prove.

But I love math, and I love math research, and I can't see myself setting aside the last two decades of work I've done to get me where I am. Moreover, I can do more good where I am now (as a solid math researcher with strong background in rhet/comp) than I could elsewhere (as, for instance, an ex-pat mathematician who took up rhetoric on a full-time basis). Finally, there's nothing stopping me from being a mathematician who geeks out about markers of metacognition at conferences on writing theory.

Who am I now? Who will I be tomorrow? We'll see. Sometimes you have to let go, and let your own life follow whatever course it seems bound to follow.

Reboot

Long day. Full day.

I got to campus by about 7:30 so I could take care of some bureaucratic matters and get ready for the "reboot" of my MATH 179 Ethnomathematics course. My intent today was to transition smoothly from the textbook by Marcia Ascher we've been using to guide the course so far to Stanislas Dehaene's (about the relative merits of these books I blogged yesterday).

That we did: we spent about half an hour talking about least common multiples, primes, and relative primeness. These concepts all come up in understanding Mayan calendar. Though some of the students were shaky with the concepts at first, they all seemed willing to engage them, and moreover the concepts are more concrete than many of those we've worked with so far this semester (like Marshall Islander stick charts and months based on Jupiter's moons). This concreteness helped students get a grip. We even dished a bit about Gauss's formula for the sum of the first n natural numbers and the conjectural infinitude of twin prime pairs.

The smooth slide into Dehaene that I'd hoped for came with an exercise I'd put together to test one of the points Dehaene makes in Chapter 4 of his text (which students are to read for next Tuesday): people who grow up with either English (13 of the 14 students present today) or French (the remaining one of the 14) as their native language have difficulty in remembering any series of more than seven or eight randomly generated digits shown to them 20 seconds previously. All 14 students were successful at remembering five, six, or seven digits, and at eight people started to falter: two students failed to get all eight, and several more failed to get nine. No one remembered ten correctly.

Incidentally, native Cantonese speakers can generally manage 10 without difficulty. If you'd like to know more, read Dehaene's book.

I felt renewed energy in class today. I'm looking forward to seeing what the students think about the reading for next week.

From Ethnomathematics I went straight to my colleague Louise's LANG 120 (our first-year composition) course, where I was guest-lecturing on the subject of writing in mathematics. Louise hoped that I could expose the students to some of the conventions of mathematical writing, partly in order to demonstrate that many of those conventions, and the criteria by means of which the quality of writing can be measured, are not all that different from those of academic writing in general. I think I succeeded in this to some extent. It was a lot of fun! It seemed like a good class, with some very outgoing and eager students.

In Calc II I (boy, that looks weird) tried out a new format for the students' quiz. Rather than offering a collaborative quiz (as the last few have been), and rather than making the quiz a fully solitary activity, I gave the students a brief "consultation period" in the middle of the exercise. The students had several minutes to get a start on solving the problem posed to them, and then I allowed them roughly two minutes to confer with one another in whatever way they wanted to, sharing ideas, checking their answers against each others', etc. This period over, they returned to solo work before submitting their quizzes. My theory was that this format would help students in much the same way "think/pair/share" exercises help them: the initial brainstorming and groundwork is done on a solo basis, but then conference with their colleagues helps to refine their initial thoughts as they take shape.

I can't tell how people felt about this, or if it had a noticeable effect on performance on the quiz. (The students did pretty well, making, for the most part, the errors I'd expect them to make. Nothing out of the ordinary.) Only one student offered feedback on the format on the quiz itself, indicating that she felt it threw her off more than it helped her.

If you're in my Calc II class, how do you think it went? If you're not in my Calc II class, how do you think you'd respond to this activity? Would it help you? Hinder you? What up? Feedback, please!

Travelers' tales

It's been a much better day today.

A highlight: another meeting of the minds in my office, involving three 280 students who'd stopped by after class for a debriefing on their recently-returned homework assignments. I'd mentioned in class just an hour before how beneficial others had found such debriefings, and these three had come hoping to reap the same benefits.

The meeting was wonderful. Sibyl, Nigel, and Quark joined me in a rough circle in front of my desk, and we went through the homework together, one problem at a time. Sometimes it took no more than a minute to iron out the wrinkles they'd worked themselves into, and other times we spent ten or twelve minutes puzzling through a problem's subtleties. They fed my wisdom and of their own, which was often richer. It's good for the students to see that others often have the same struggles they do (more than once they made the same kind of mistakes on the same problems), and it's good for them to hear their peers' explanations (often more lucid than my own). We ended the meeting with my giving them a few impromptu inductive exercises for practice with the technique. (Prove: n nonparallel lines in the plane determine n(n+1)/2 + 1 regions, bounded or unbounded.)

I left our meeting invigorated. I love this aspect of my job.

A second highlight: I realized this afternoon, while rereading Stanislas Dehaene in preparation for MATH 179 tomorrow, why it is I've had such a hard time getting into the mathematical aspects of my Ethnomathematics course this semester: the book, by Marcia Ascher's Mathematics elsewhere: An exploration of ideas across cultures, is awful, at least for my course. It's a bad fit. It's boring, condescending, pedantic, preachy, and is aimed at entirely the wrong level. (It's too dry to intrigue most math majors but too mathematically sophisticated to appeal to non-mathematicians.)

By comparison, Dehaene's book, though not strictly speaking about ethnomathematics (it deals more with the psychology and neuroscience of math and math learning), is intriguing, engaging, and written in a manner that's neither condescending nor confusing. I think the students will find it much more interesting than Ascher's text. We'll start drawing from it tomorrow and next week.

It's been a good day; I'm looking forward to another one tomorrow.

A closing note: my thanks to all of you who offered me support after my day of frustration yesterday. I'm sure regular readers will recognize that blogging is a cathartic exercise for me (need evidence? Check out the "anxiety" and "bitching" tags on the right!), and I often feel tremendously better after writing.

Vox populi

The students have spoken. It's clear from the feedback I got at the end of class today that I've gotten a bit off-topic in MATH 179 for the past couple of weeks as we've undertaken a more in-depth analysis of the university's structure and the impact that budget cuts will have on that structure.

The students definitely want to get some more math into the course, and we'll begin doing that on Thursday when we look at the Mayan calendar more closely. I'm also thinking of bringing in some reading from Stanislas Dehaene's The number sense: How the mind creates mathematics. I found this book fascinating reading a few years back, and I think it'll start some interesting conversations. Ditto Anthony Aveni's Uncommon sense: Understanding nature's truths across time and culture.

My bad for letting us yaw of course...let's get back on track!

Everything you always wanted to know about UNCA (but were afraid to ask): Vol. 2

We had another round of "Everything you always wanted to know about UNCA (but were afraid to ask)" in MATH 179 today. It came at the end of the class (after further discussion of Jupiter's moons and an overview of the Integrative Liberal Studies program at UNC Asheville), so we didn't have enough time to address every question asked, but we got a few in. We'll finish the rest on Tuesday.

I'm impressed by the students' candor and curiosity, and I'm equally impressed by the students' knowledge of campus functioning. Some of these kids are definitely up with what's going down.

Questions?

• What is the cause for the recent budget cuts?
• What are the advantages/disadvantages of declaring a major?
• If 179 is one of the writing intensives, what are the other 2?
• What all will the new health and wellness center have to offer?
• Will any majors be dropped with the budget cuts?
• Where do I find info on clubs?
• What are the rules for the disc golf course? And how many holes is it?
• How much of my tuition is going toward building the community/campus gym?
• Where do you apply for a learning disability? (I write on the side WITHOUT lines, I'm such a rebel!)
• How many hours are required to graduate?
• When will the construction for Governors Village be finished?
• If I took a Spanish at my other school and it counted as both Spanish requirements if you passed and I passed then I scored below that level on the placement test what happens?

We got around to the first four of these today. I have to admit to a little relief that some of the others have yet to come up: I'm going to have to look up the answers to a few of them!

Of moons and months and mathematical manipulation

This morning in Ethnomathematics we finally had a chance to spend some time trying to reconcile the "lunar" and "solar" calendars one might use were one stationed on Jupiter and decided to use the largest of the Galilean satellites, Ganymede, to reckon your "months."

A Ganymedean month (the time it takes to make one transit around Jupiter) is roughly 7.15 (Earth) days; meanwhile, Jupiter's tropical year (the time it takes the planet to make one transit around the sun) is 4331.57 (Earth days). Just as on Earth, where a tropical year doesn't contain an even number of lunar months, resulting in a "drift" between lunar-based and solar-based calendars, there are an uneven number of Ganymedean months in a Jovian year. What to do?

After a bit of numerical piddling and fiddling, we figured out two reasonably accurate ways of making the numbers jibe. Both rely on the fact that each Jovian year we have a "remainder" of 5.82 days that don't quite make up a full Ganymedean month. If we let 11 Jovian years pass, we've saved up 64.02 spare days...this figure is very close to 9 full Ganymedean months, which give us 64.35 days. Therefore, if we add 9 months every 11 years (which can be done in some systematic fashion, much as is done with the Jewish lunar calendar), we've add only 0.33 extra days. This overage is tantalizingly close to 1/3...so why not simply take away one day every three 11-year cycles? This day too can be chosen systematically, removed from the middle of the 17th year of the 33-year cycle it corresponds to, for instance.

Neat!

I'll leave it to my readers (I'm sure your curiosity is now piqued) to puzzle through the details of the other solution we arrived at, which had much the same flavor and involved slightly more frequent adjustments.

Fun stuff...now if we can only figure out how to make these systems fit nicely with the 365.24-day Earth tropical year, which some of our Jovian emigrés still insist on using...

Whaddaya mean, "liberal arts"?

Last week I asked my MATH 179 students to reflect on what "liberal arts" meant to them. They were to write an ungraded page or two about what they thought it is that distinguishes a liberal arts institution from one of a different sort. Their responses to this exercise were uniformly astute. They've got a good grasp of what a liberal arts education entails already, so my goal for the next of the semester need not be so much to introduce them to this educational philosophy so much as it will be to help them investigate its subtleties and implications.

Many focused on the relatively intimate learning environment at liberal arts colleges, bolstered by their smallness: "Because of the small student-teacher ratio, students are closer to their professors and receive personal attention" and "I like the fact that it [UNCA] is a small campus, the students and teachers are very nice, and there are classes here that you probably would not find anywhere else," in contradistinction to "UNCW [UNC Wilmington, where] most of the lecture classes were very large. I had several hundred people in my Biology class and also in my Algebra class."

Some commented on the liberal arts curriculum, focusing on the curriculum's breadth as well as disciplinary depth ("Usually liberal arts colleges have more core classes than other colleges. I think this is because the college wants us to have a broader view of horizons rather than focus on the one thing that we like") and on the ways in which the curricular offerings are tied together ("Liberal arts colleges tend to look at education holistically, meaning the curriculum is usually intertwined and well-constructed relating certain areas of study") . They acknowledged that tackling this curriculum is not always easy, but should ultimately be rewarding: "Due to liberal arts requirements one must go outside of their comfort zone and take classes in topics that they are not the best at or comfortable with. Although can be cumbersome as a student at times it helps integrate all areas of study to further the student's knowledge in general."

What are the rewards? Employability, one would hope: "When I graduate I think all those requirements [clusters, writing intensives, etc.] will make me a more rounded person and make me look better to an employer" and "...a degree from a liberal arts school is better to find a job that is not specific, because it shows you have skills in many areas." There's also something to be said for the recognition of the "human" in human scholarly endeavors: "The liberal arts environment at the school has had a major impact on the way I am taught and how I learn here at the University of North Carolina Asheville. [...For example, the Newton v. Leibniz project in Calc I] seemed so absurd but it really helped me realize that the topics we were being taught and the rules that we were learning were not just things that appeared out of thin air. Every subject, every theory, and every idea has a back story, a history, a time, a place, and a brilliant mind who thought it up."

Several students expressed (sometimes extreme) satisfaction at attending a liberals university (a few have spent at least a term in a non-liberal arts environment). Some even expressed concern for their peers elsewhere: "I see a liberal arts education to be very important because I have noticed a terrifying trend that not many of my peers that attend non-liberal arts universities are very secluded and "protected" from the world and the culture in which they live."

So far they know what they're talking about.

Ahem...

Despite the late start (again!), yesterday was a full day, pedagogically speaking.

Things got underway in earnest with presentations by my 179 students, on the campus offices and organizations they were to investigate and report upon. Though I've not yet looked at them closely, the brochures they came up with to "advertise" these offices' services look quite good. They're creative and colorful, and some of them almost look professionally done. The presentations showed signs of nervousness and trepidation, which is natural with students at this point in their careers. As I always remind them, getting up and speaking in front of other is hard, no matter how often or how many times you do it. I admired both their willingness to get up and speak and the respect they showed one another as audience members.

I am not, on the other hand, admiring their highly imperfect attendance. Though there's a core of students (15 or so of the 21 now registered for the course) who come unfailingly, the remaining students come only to one out of every three or four class meetings. I've never seen such nonchalance in my courses before.

It's easy to come up with reasons for this. Even I have to admit that the course's subject may not be one which is inherently engaging for students not already interested in math, so many students may find it hard to get excited about the course for its own sake. Since mine is the only 179 course running this semester, several students are in it simply because they have to be, and no one wants to do something on compulsion alone. Finally, 179 courses in general (not just mine) are sometimes seen as "throwaway" courses that students expect to do well in with minimal effort. This is in part because many instructors fail to motivate the class by making its purposes clear.

I hope that in this I'm succeeding. I'm trying to strike a balance between writing instruction (to meet the course's WI goal), overview of the campus and its surrounding community, and actual course "content." It's a precarious balance, and sometimes I feel as though I'm making it up as I go along, and this worries me. However, several activities have seemed to work well ("Everything you always wanted to know about UNCA..." was well-received, and the students seemed to have fun making their meddos). I hope it proves meaningful in the end.

Speaking of public speaking...I ended my workday at Mars Hill College, a small school about 20 minutes north of here. The director of their nascent writing program had invited me to come and lead a workshop for the faculty who are spearheading efforts to get their writing-intensive course program off the ground. There was good disciplinary variety in the participant pool, with social work, journalism, history, literature and composition, religion, and philosophy well-represented. Given that Mars Hill can't possible have more than a hundred or so faculty members, I'm guessing that I met with something like 15% of the school's faculty yesterday. However, I must admit that I was a little chagrined, as were several of the attendants, that the sciences didn't send any of their folks over. (There's talk of me coming back to work with them specifically.)

There was energy in the room: given that Mars Hill's QEP is focused on writing and information literacy, there's strong institutional buy-in for their new WAC program, which features phased-in writing-intensive courses and student writing-fellows. The campus is outwardly stoked about the QEP: banners hang from lampposts exhorting "Write! QEP 2010-2013." It's clearly not just a "top-down" mandate, though; the people I met with yesterday were very much on board with the idea of writing across the curriculum. Their energy was authentic and intrinsic, not simply something generated by an administrative fiat.

Owing to this energy, the workshop was a fruitful one, with active (sometimes even fervent, but always friendly) discussion and idea-sharing. We talked about identifying roles for writing, coming up with learning goals writing could help to meet, structuring writing, using low-stakes writing activities, and responding to writing (while guiding students to do the same, through peer review). We packed a lot into two hours! Judging from the exit slips I received from the participants at the end (thanks for this idea, Libby! I'm going to do this from now on), most people were most interested in learning more about low-stakes writing and its uses in the classroom. The most interesting were the technology-driven (and very game-like) "constrained" forms like tweets and texts, as well as my newly-minted "Intrigue, Confusion, and Confidence."

I picked up some ideas, too. One of the conference participants offered a low-stakes writing activity he's used to help his shyer students get their say in in-class discussions. Rather than requiring them to speak up in front of a couple dozen of their peers, he asks students who are hesitant about contributing talking points to email their ideas ahead of time so that he can collect these ideas and share them with the class on the overhead. Those students' discussion points are on display, anonymously, when the class comes in, and in this way even the less outgoing students are able to make their voices heard. I asked if this method of starting the class conversation has helped the shy students come out of their shells (as, for instance, they might be called on to defend the points they've made via email), and he indicated that this is the case.

At the end of the workshop I had a chance to dish about the reception on the part of faculty of UNCA's own WI program. I said that I think it's generally good, and that though there was resistance when ILS was first phased in several years ago, most of the younger faculty have bought into it authentically and this has led to a campus culture in which writing-intensive courses are broadly accepted. I honestly think that this is the case: there are grumblers and groaners, but for the most part the mission of the intensives is unquestioned, and teaching WI courses is a valued activity.

Thoughts?

A liberal conversation the liberal arts

We had some great conversations in my Ethnomathematics course today, a day we devoted to trying to obtain a better and broader view of the university we all work at. After doing a peer review of each others' brochures (advertising campus offices and organizations of important to new arrivals to campus), I asked the students what it was that drew them to UNC Asheville in the first place.

The list they came up with was a comprehensive one. According to them, the draw of the school has to do with the fact that UNCA

• is cheap,
• is in the mountains,
• is in North Carolina,
• is in Asheville,
• is close to family,
• offers a "real" [I think he meant "more comprehensive"] education, as opposed to a conservatory,
• has a small campus,
• has a green campus,
• has promising programs of interest [teaching licensure and Chemistry were mentioned specifically],
• isn't pricey,
• has nice faculty,
• has an accepting community,
• is part of an active larger community,
• is situated in a "hippie town,"
• has no football team,
• offers diversity,
• offers a large number of extra-curricular activities,
• offers small classes,
• allows students flexibility in their studies, and
• offers a broad variety of courses one wouldn't find elsewhere.

I pressed the students a bit on this last point: what courses do you think distinguish UNCA from other institutions?

"This one," one student offered. "This doesn't seem like the kind of class you'd be able to take just anywhere." Students also cited a number of other courses, many of which have an interdisciplinary flavor to them: a jazz course on the popularity of The Beatles, physics courses on music and sound, Asian studies courses, and teaching licensure courses relating to every discipline across campus.

Toward the end of our conversation, we began to drift toward the topic on which I asked the students to write a brief reflection for next week: what is it about UNCA that makes it a liberal arts institution? Judging from their responses to this question, I think the students have a good idea already, and I'm looking forward to further discussion:

"I went to UT [University of Tennessee] before this, and there all of the teaching was straight out of the textbook. You had to read the text and then you'd get lectured to about it. Here there's so much more variety in the way things are taught. And the classes all fit together and build on one another. They affect one another."

"There are connections between all of the different components of our education."

"We have a chance to learn in small classes like this one, instead of large lectures with several hundred people."

"The range of courses we can take is huge. And they sometimes consider the same ideas from different points of view."

Unwittingly the students are hitting some of the reasons I came to UNCA and enjoy my career here so much: I have the chance to actually get to know my students and their hopes and dreams. I have the chance to teach rich courses with a similarly rich assortment of pedagogical techniques. Moreover, I do this all with strong students who would never be able to afford the cognate experience at a private school like Bucknell, Bard, or Davidson. It's a good life.

I'm looking forward to reading the students' reflections...and to their presentations on the campus offices to which they've been assigned. More to come!

Everything you always wanted to know about UNCA (but were afraid to ask)

Today's MATH 179 class was the most fun yet...at least for me. I felt both major activities went pretty well, and I finally get the sense that the students in that class are opening up to and relaxing around one another, and that they're acknowledging and understanding the sort of work that I'm expecting of them.

Our reading today treated three examples of calendars that reconcile lunar cycles with solar cycles (including the very hard-to-understand Jewish calendar). As an entry into discussing the calendars, I used a low-stakes writing activity I've been developing for a bit now: "Intrigue, Confusion, and Confidence." I ask each student to write three sentences: in the first the student indicates something in the reading which intrigued or excited her; in the second she indicates something she was confused about; in the third she indicates something she feels confident she could explain well to someone else.

After writing for five minutes, I have the students pair off and compare their lists. I ask them to fill one another in if one is confused about something the other knows confidently. I ask them to come up with at least one point they'd like to bring up in discussion.

Five more minutes later, we come together as a class and compare the things we've all come up with which are intriguing, confusing, or inspiring of confidence. Students are heartened to see how similar their lists often are: they all have trouble on the same things. (Like the Jewish calendar!)

The second half of class was devoted to an activity I called "Everything you always wanted to know about UNCA (but were afraid to ask)." I passed around identical index cards and asked each student to write down any questions they had about our school to which they wanted to find answers. "I know how it can be embarrassing asking your friends or professors. Some questions seem like silly ones, and you don't want people to think 'he doesn't know that!?' " After I collected the cards, I read them aloud, one by one, pausing so that anyone in the room who had a response could pose an answer.

The ensuing discussion was fantastic. Two or three of the students proved to be particularly knowledgeable about our campus, and I feel they'll make excellent mentors as the semester goes by. It was an empowering conversation, I felt:

1. the students realized lightning bolts would not smite them for asking questions;

2. the students realized that their peers had many of the same questions they did (thereby making those questions by definition "not dumb"); and

3. the students realized that they personally often had the answers to questions their peers might pose, making them the authorities.

I'll definitely repeat this exercise several times this semester, and I'll probably use it in other courses, about course content, too.

So far

So far, though it took a little while to build up steam, this semester's been great.

After three straight days of canceled classes and (and a couple more late starts thrown in the next couple of weeks), we got enough ground to get some traction, and I'm starting to feel like I'm in the groove.

Both my Calc II and 280 classes are large...I'd even call the 280 class huge: 33 in Calc II and 36 in 280. Yes: three, six. I'm always anxious about classes this large (I seem to attract them), but the students in both have set my mind at ease: they're all very engaged, outgoing, and willing to both ask and answer questions, in both classes. I've got fantastic students in both, and they're making the classrooms' atmosphere lively, spirited, safe, and fun.

I think I've done a good job in encouraging a healthy learning environment so far this term, downplaying grades, up-playing collaboration, throwing in a good number of writing-to-learn opportunities...the students are receiving this well. I sense a greater-than average willingness to learn on these students' parts. It's going to work out well.

I was particularly excited about today's 280 class. This morning we had our "LaTeX seminar," in preparation for which I asked everyone to install a compiler and a text editor on their computers and bring them, if possible, to class. Roughly three quarters of the students had laptops with them, and most of these (after a few fits and starts and glitches involving flavors of Texmaker and odd configuration settings) were able to get LaTeX up and running within five or six minutes. And they liked it. Comments like "This is so cool!" were fairly frequent. It's the best reception LaTeX has ever gotten in a 280 course.

Meanwhile Ethnomathematics is steaming along, picking out a course through the Marshall Islands. Literally, actually. We've spend the last week talking about the mathematical aspects of the mattang, the stick charts used by traditional Marshall Islander navigators in order to plot their path from one atoll to another in the sprawling and sparse archipelago. The students have even been working on building their own mattang out of various materials, including everything from pipe cleaners to pretzel sticks.

Late last week I tried to get the students to cast aside their "Western" assumptions about the make-up of maps by asking them to create maps from our classroom to various important offices and organizations on campus, maps which could not make reference to human-made objects, could not use any sort of reference to fixed units of distance (feet, miles, paces, etc.), and would be followable by someone who had not had a hand in creating the map in the first place. I realized after the fact that I should have included additional stipulations: no text, and mandatory use of "nontraditional" materials: the vast majority of the texts included copious textual commentary and were drawn on paper. (I admit that I'd hoped to see more "tactile" map-like objects like the mattang.)

Nevertheless, the students are doing well in what I think is a highly nontraditional course. I'm eager to see what they come up with for the brochures I'm asking them to make, one for each of several important campus services (like the Math Lab and the Student Health Center).

That's all I'll say for now. I'm sorry this is a bit of a banal post, but I've had nothing heinous happen so far this term. I could say something about the book (coming along, in bits and pieces) or the state's budget situation (in a word, bleak...think "how in the hell can we keep providing the quality of education we pretend to provide?"), or even about my upcoming visits to various writing programs around the state, but I'll leave those for other posts to come soon.

Day One...finally

Well, yesterday was our first day of classes. I knew we'd get there eventually, but Snowpocalypse 2011 gave us three days of canceled classes before we finally got things underway with a two-hour delay.

My "first class" therefore turned out to be the first meeting of my first-ever first-year seminar, on ethnomathematics. Although we were a bit rushed (trying to take care of two days' worth of business in a single day), I think things went well. At 14 students, the class is the smallest I've taught in years (not counting, perhaps, one or two sessions of the senior seminar); this is going to offer me a welcome relief. It'll also make my student-centered teaching techniques more effective.

After introductions and bureaucratic whatnot, we got started with a "Think/Three/Share" on campus services, offices, et cetera. (One of the purposes of the course is to introduce students to the university and to academic life, and to help students understand the purpose of a liberal arts institution.) Unprompted, the students came up with a substantial list of university offerings, including everything from the Health and Wellness Center to the many offices lining the student union. In a few weeks I'll be asking them to go around to each of these points of interest, interviewing employees there, and designing brochures to advertise those offices' services to prospective students.

After this exercise, we got the mathematical ball rolling with an activity I thought might help the students to identify certain mathematical "conceits" we hold. I showed a clip from the movie Contact (1997), in which Jodie Foster (as Carl Sagan's Eleanor Arroway) and friends are first contacted by an alien intelligence. The alien transmits a series of radio pulses their way, consisting of 2 bursts, then a pause; 3 bursts, then a pause; 5 bursts, then a pause...the first few dozen primes are pulsed out, and from this nonrandom behavior the scientists infer that whatever's sending the message must be a smart cookie.

Of course, there are several assumptions here that are arrogant and "humanocentric." Ranging from the most specific to the most broad:

1. The pulses are sent assuming knowledge about (and concern for) divisibility, the conceptual underpinning of prime numbers),
2. the pulses are meant to be interpreted in base-10 arithmetic, and
3. the pulses are broadcast with the assumption that we use a discrete number system in the first place.

The students didn't catch the first of these, but they caught the second two. I was impressed! We didn't have a whole lot of time to follow up on this exercise before we had to go. I only had a small bit of time to explain their first writing assignment, an "ethnomathematical autobiography," which, besides having the most Greek I've ever fit into a single assignment title, will challenge them to speculate on the ways in which one's upbringing might affect one's perception of mathematics, in the context of their own lives. We'll do some writing exercises on Tuesday (peer review and metacognitive writing), and Thursday will see us begin talking about math in other cultures.

My second class was the first meeting of my Calc II kiddoes. This class is packed! The room's designed to max out around 30 students; I've got 34 in the class now, and one or two more who'd like to get in. It was almost standing-room-only: once the desks ran out we had just enough chairs (and a single stool) to accommodate every last tush in the room. Moreover, the room itself is rather small and cramped. Fortunately we meet in a different room on Mondays, Wednesdays, and Fridays; I hope this other room is larger.

However, this begs a pedagogical point: what effect does it have on learning, moving the class from room to room on different days of the week? It can't help but have a noticeable impact, I would think. I've got to be open about this: I'm becoming increasingly ticked about the lack of respect the administration has shown the Mathematics Department here, in the form of allocation of space and resources. Every single semester we teach with too few rooms and too few seats. Despite serving more students (as measured by face-time equivalents) than any other department except perhaps Literature and Language, we always get the short end of the stick when it comes to classroom space.

It's verging on the absurd now. I teach in four different buildings this semester. Four. There's one day of the week (Wednesday) on which I'll teach in three different buildings in one day. I understand if occasionally one might need to be sent out into some far-flung corner of campus for one class or another, but considering most departments teach courses which meet almost exclusively in that department's faculty's home building, I can't help but feel we're being slighted. (I just did a quick survey of our course offerings; at a glance most departments teach either one or no sections outside of their home buildings.)

At the risk of sounding crass, that shit ain't right. Can you imagine the holy hell that would be raised if the chemists were asked to teach their small upper-level courses in the basement of New Hall, where all of the foreign language people dwell?

Anyway....

Aside from being cramped, the first day of Calc II came off rather well. No major kinks. We spent much of the class time building a rough concept map of Calc I, which I hope to include in my book.

Speaking of which: I finished off about 26,000 words over break, and am now about 80% done with a first draft. Draft versions of Chapters 3, 4, 5, and 6 are nearly done, and 7's coming along as well. Chapters 1 and 2 might end up getting merged into one, and I'll likely get a start on those/that this weekend.

For now, I'm off...in twenty minutes I've got the first meeting of this semester's 280 course. 34 students! The last time I taught it (Fall 2009), I had 15. Crikey.

Wish me luck.

In the interest of full disclosure

It's been a busy break.

Winter Break started, effectively, just under three weeks ago. In that time I've been almost completely submerged in writing theory and pedagogy, hard at work on my book. I've written about 17,000 words in the past two and a half weeks, so that I'm now roughly 60% done with a first draft of the damned thing.

I say "damned thing," but of course you know I'm loving every minute of it. Some paragraphs are slow going and take me an hour or more to kill, and others just fly from my fingers. It's all fun, though, and I'm excited about how the book's developing. I'm working away at several chapters simultaneously (I've started every chapter but the second and the seventh), and they're starting to grow and to grow together. By the end of next week I hope to have completed first drafts of Chapters 3, 4, 5, and 6. (That'll leave 1, 2, and 7 to go.)

While I'm at it, I'd like to give a massive shout-out to my writing colleague Libby of East Carolina University, who gave me fantastic feedback on my introduction, and excellent ideas for later chapters. She's the first of my colleagues in composition and rhetoric to read any substantive portion of the book, and her reading, and her response to it, has encouraged me greatly. Peace!

Unsurprisingly, in working so closely on the book I've begun to think about ways I can model to my students good writing and good writing process to an even greater extent than I already do. I'd like to offer them a window on my own writing process in whatever way I can. I'd like them to see rough drafts, wrong turns, dead-end ideas, reviewers' feedback, and revision, revision, revision.

Obviously they're not going to give a day-old donut about every single version I write of one or another dry math paper. But it can't hurt to show them some of my first-draft research notes that are little more than mathematical freewriting (see below), or the copious comments and suggested emendations my editor offers me on draft chapters of my textbook. If it accomplishes nothing else, at least showing these things to my students will show them that I too am human, that I too make (often very stupid) mistakes, and that I too am continually growing as a writer and a thinker.

Without further ado...here's a sample. Below are the first five out of nine pages of notes I scribbled out the other day as I was working on a paper dealing with the combinatorics of complex polynomials. In these notes, if you look closely (I don't recommend it) you'll see everything I indicated above: wrong turns, dead ends, and idiotic mistakes. (My Facebook friends may even be able to notice on one of these pages the (-1)(-1) = -1 mistake that was the focus of my Facebook status for several hours earlier this week.)

After all, I'm only human.












Coming soon: my thoughts on putting together my first-ever first-year seminar, on Ethnomathematics! This course begins in a little over a week, and I plan on writing my syllabus this weekend. Stay tuned...

Three great gifts

Week One has ended!

And we've not lost a one.

This afternoon's session on fractal dimension, led ably by my colleague Nostradamus (my partner in crime this summer), marked the end of the week's action for the REU. While they're still a bit reluctant to speak up in front of one another, they're definitely growing more at ease with working together, as evidenced by their professions to collaboration behind the scenes (from Nils and Ole) and the ease with which they cooperate in class (two pairs worked together to complete this morning's LaTeX exercise).

Speaking of LaTeX (and other mathematical technologies), all eight students now have installed on their computers some sort of LaTeX editor and compiler, and all eight have installed some version of Mathematica.

The students are beginning to show the first signs of focus as they near their initial selection of topics: Billie indicated specific interest in the "use it or lose it" tree construction, as did Daria. Nigel likes the look of the same algorithm, though like Daria he'd like to hear more about Cayley graphs before deciding on what to do. Several students asked more about graceful labelings and generalizations of chromatic polynomials, too.

All in all it's been a good first week. I've certainly learned from it that there's no single snapshot of a successful first week's work: while I've made no small point of this group's relative reticence, in their own way they've been no less successful in their mathematical efforts than last year's bunch, say, a band of brothers and sisters to whom I was often tempted during lectures to say, "shaddup already!"

If any of this year's REU students are reading this, please know that we're only remarking on your quietness because we find it a striking counterpoint to the previous years' groups. There's absolutely nothing wrong with your reservedness: it says nothing about your intelligence, your work ethic, or your eventual success as mathematicians. It's just very different from what we're accustomed to.

As yet I've said nothing in this post about the three gifts to which I've alluded in the post's title. It's time to remedy that.

All three of these gifts promise to expand my both my own understanding of the mathematical world and my ability to convey that understanding to others.

The first gift comes to me from Daria. When I fetched her from the airport on Sunday morning she and I got into a conversation about ethnomathematics, which readers of this blog might know is one of my less minor interests, especially given my rather unorthodox (among the research mathematical community) view that mathematics is not universal but is indeed a cultural artifact, a socially-constructed system that varies from one people to another. Somehow it came up early in one of our first conversations that Daria had recently taken a course in ethnomathematics, and in fact would soon have with her the textbooks she'd used for the course. I asked her if I could borrow them when they arrived, and yesterday she brought them to me. I have no doubt they'll prove a fascinating foundation for my own study of ethnomathematics, and a good basis for the course on the subject that I hope soon to develop for UNC Asheville students.

Both books, Ethnomathematics: a multicultural view of mathematical ideas (CRC Press: Boca Raton, 1998), and Mathematics elsewhere: an exploration of ideas across cultures (Princeton University Press: Princeton, 2002), are by Marcia Ascher of Ithaca College. In a heedless display of randomosity I began reading the second-written one first just a half-hour ago. It promises to be an interesting read. Having read little more than the introduction at this point, I already suspect I'll find a kindred epistemological spirit in Ascher.

For instance, "we now know that there is no single, universal path -- following set stages -- that cultures or mathematical ideas follow" (p. 2). Take that, proponents of mathematical universalism. As I'm fond of saying (and have said elsewhere in this blog), mathematical language is hardly more universal than the English language, and the mathematics of an alien race would likely be as indescribable and indiscernible to us as their courtship rituals.

Or take this line: "most practitioners of modern mathematics value their ideas because they believe them to be context-free; others value their ideas as inseparable from the cultural milieu that gives them meaning" (p. 4). Indeed, it's a blight on modern mathematics that so many modern mathematicians might laud math's seeming baselessness and independence from any fixed ground. This view could hardly be farther from the truth, as math is a highly predicated belief system, the truths it embodies obtaining only when certain cultural norms about truth and knowability are applied. How is it that a mathematician unwilling to state her or his hypotheses, elements necessary for the application of any reasonable theorem, would be laughed from the lecture hall, while it can be commonly supposed among mathematicians that the very science of mathematics does not rest on similar epistemological hypotheses?

I'll be sure to blog about these books as I make my way through them this summer.

A second gift, one of recognition and promise for future collaboration, comes to me from a heretofore unknown colleague in South Carolina. Lately my work on the intersections between poetry and mathematics has been getting the attention of more and more poets. Io, a poet and teacher from South Carolia, came across a copy of my paper on using poetry to teach mathematics (the one to appear in WAC Journal), and told me of her interest in the subject. She confessed that abstract algebra had been one of her favorite classes in college, and that she had great interest in understanding more fully the similarities between math and poetry.

Already, in just a short exchange of e-mails, I can tell I've found another likely friend and colleague. I hope to continue my correspondence with this woman as I further develop my own understanding of the ways poetry and math interact.

Side note: next year marks the 50th anniversary of the founding of Oulipo. Perhaps some sort of public and poetical and perimathematical celebration is in order? That's something to think about.

The third gift comes to me from an old student, Sedgwick, who graduated about a year ago with a degree in environmental studies. Sedgwick was one of the star students in the second section of my Spring 2008 Calc II course, a close-knit class that was a lot of fun to work with. He's still, a year after graduation, a regular reader of my blog (shout-out, Sedge!), and after reading a relatively recent post (this one, I believe) on the effectiveness of various components of the Integrative Liberal Studies program at UNC Asheville, and an even more recent post on Don Tapscott's Grown up digital, he wanted to offer a former student's perspective on the ILS Program, and did so extensively in an e-mail he wrote to me about a week ago.

His e-mail is, as is all of his work, thorough, well-thought out, and well-organized. This guy's always been a top-notch thinker. He makes many excellent points about various components of the ILS system. I asked Sedgwick's permission to excerpt his e-mail to me and to form a response to it in the form of an open letter consisting of a blog post here. Having been granted his leave to do that, do that I shall, in a post I hope to write this weekend.

For now, though, the dinner bell is readied to ring, and after a long, long week or work with a new crop of talented young researchers, I'd like nothing better than a few hours off. (If only I could get this damned channel assignment problem out of my head!)

To be continued...

Breakin'

It's Spring Break.

Which doesn't, sadly, mean I have nothing to do. It only means that what I've got to do (and there is a great deal of it) needn't be done on a rigid schedule.

I've got several job-related tasks to take care of in the next week, ranging from the quotidian (prepping for class once school resumes a week from tomorrow) to the leviathan (going through a stack of roughly 75 applications for this coming summer's REU). I've got a couple of meetings tomorrow, one with a student (my independent study in order theory), another with a colleague (Writing Intensive stuff). After that, I'm looking at a nearly completely unstructured week.

I'm in need of some unstructured time, after the busyness of this past week. Dr. Robert P. Moses, noted civil rights leader and founder of the Algebra Project, came for his visit this past Wednesday, and between the public lecture on Wednesday evening (a talk about the degree to which the Constitution ensures a quality education, at which I was delighted to see several of my students!) and the ensuing Math Literacy Summit held on Thursday, there was no shortage of excitement and things to do in our department.

The session I chaired at the summit (a talk on numeracy as it relates to health issues, given by a psychologist at the Duke University Medical Center) led me to the book I'm now reading, Stanislas Dehaene's The number sense: how the mind creates mathematics (Oxford University Press, 1997). This is proving a truly fascinating read!

Dehaene is a psychologist specializing in the neurobiology of mathematical acquisition, his book is a record of many of the facts that have been discovered concerning the way in which people learn mathematics, they way they organize its ideas in our minds, the way math is retrieved from memory. At its most basic level, our sense of mathematics is very little advanced beyond that of many animals, who share with us a precise sense only of the numbers 1, 2, and 3; beyond this is a roughly-reckoned haze of numeric quantities. Dehaene compares our mental conception of number as an "accumulator" with approximate graduations allowing us to give rough estimates of large quantities, but which fails to give precise values for these same quantities.

A few snippets:

• Even as soon as a few days after birth, babies are able to discern between the numbers 2 and 3. (See p. 50.)
  
• We (adults included!) are susceptible to "the magnitude effect": it's harder for us to discern the difference between 90 objects and 100 than it is the difference between 10 objects and 20. Various factors (symmetry, density, etc.) militate and mitigate this effect. (See pp. 71 ff.)
  
• Studies show that when asked to compare numbers, such as 5 and 7, and state which is the larger, instead of behaving reflexively and answering based upon our knowledge that the symbol "7" represents a larger quantity than the symbol "5," we instead convert each of these abstract digits into collections of the requisite number of objects before performing the comparison on these collections. (See pp. 75 ff.)
• We have a tendency to "compress" numbers as they grow, storing them in our minds as though on a logarithmic scale. One corollary of this behavior is that when asked to provide a random sample of numbers in a certain range, people will tend to elect an overrepresentation of smaller values, as though these were more widely spaced than their larger compatriots. (See pp. 77 ff.)
• Since adults compute sums and products (for example) by retrieving the resultant quantity from a memorized table, those whose native languages have exceedingly short names for the ten numerals (like Chinese and Japanese) are able to more efficiently memorize the desired sums and products, and so perform much more quickly and with fewer errors than their counterparts speaking other tongues. (See pp. 130 ff.)
  

These are just a few of the fascinating facts I'm learning about the development and refinement of mathematical thought processes in and by the human mind. Ultimately, one of Dehaene's primary points is summed up nicely on pp. 118-119: "Although our knowledge of this issue is still far from complete, one thing is certain: Mental arithmetic poses serious problems for the human brain. Nothing ever prepared it for the task of memorizing dozens of intermingled multiplication facts, or of flawlessly executing the ten or fifteen steps of a two-digit subtraction. An innate sense of approximate numerical quantities may well be embedded in our genes; but when faced with exact symbolic calculation, we lack proper resources."

To be continued, I'm sure.

For now, I'm off to enjoy some more of this wonderfully unstructured time, probably by knocking off a few more pages of Dehaene's book. Highly recommended!

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.