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!

Overdue notices

I'm overdue to post (so saith a few faithful readers, including my mother-in-law, my wife, and two of my most diligent students). Enough, already! So here I am, I'm writing.

I've been here, I've been busy. The last few weeks have been exciting ones, and next week promises its own supply of hecticness (hecticity?).

What's new, pedagogically speaking?

Well, yesterday was the deadline for applicants to this year's REU. This afternoon I counted up the number of distinct applicants from whom I've received at least one document (application, statement of purpose, transcript, or letter of recommendation): 64. Not bad. That's about what we had last year, though I believe the ratio of men to women is higher this year than last. I've yet to break it down geographically, but I think this year's pool is more widely distributed throughout the country than last year's. I'll probably begin looking at the applications in earnest over Spring Break. No sooner: next week's going to be a bear. Tuesday brings a colleague from Davidson College to campus (hello, Twyla! thanks in advance for making it out! and sorry again for the confusion in scheduling!) for the Research Seminar. Then comes the Math Literacy Summit we've been planning for several months, highlighted by the public lecture and keynote address given by Dr. Robert P. Moses. That's Wednesday night and Thursday day. Saturday sees the start-up of Super Saturday once more (I'll see if I can enlist some student helpers...the downside is the onset of Spring Break, taking many students away from campus). By Saturday afternoon I'll be beat.

What else is new?

This past Friday my Graph Theory gurus had their first exam, an in-class ditty that represented my first attempt in almost two years at writing an in-class exam for an upper-division course. It's known far and wide that I'm not a big fan of such a format for seminar courses, and it was hard for me to write an exam that I felt made fair demands of the students' knowledge and was doable within the alloted time. The end result, I feel, was a hair (and no more!) on the long side, and a tad too easy. (I'd be interested in knowing how the students feel about both of those appraisals.) As it is, the students went down to the wire time-wise, several staying for an extra ten minutes to finish up, while the average was high, around 86%. Maybe I didn't ask for enough proofs? Maybe I was a little light in grading? I don't know. The only question that seemed regularly to ensnare the unsuspecting was the problem asking the students to compute the number of homomorphisms from the star with 3 vertices to the star with 4 vertices: there was a broad array of answers to that question.

How's the day-to-day activity in that course been? The students' presentations of solutions are getting tighter, more succinct, more precise. For the most part their diagrams are more descriptive and intuitive than they'd been during the first few weeks, and their proofs are more straightforward and understandable. Most interesting are the differences in presentation styles between the various members of the class. Some are silent until all has been written on the board; their presentations then consist of little more than "voilà! Pas de lacune a remplir!" Some are so verbose they can barely put chalk to board to draw a single tittle without prefacing it with a megillah of mathematical exposition. I wonder to what extent they find others' presentations are affecting the style of their own? (This sounds like a perfect question for a mid-term evaluation!)

We've gotten to the point where the students have a grasp of the basics, I can probably branch off in whatever direction I'd like to in terms of the ground material. The most recent problem sheet (the seventh, available here), deals primarily with components, paths, and chromatic polynomials. I believe I'll make trees the focus of the next sheet, unless someone has a better plan. Lorelei mentioned the other day to me that she'd like to see more applications, so I'll do what I can to work those in (colorability has many applications, and they'll soon be ready to take off in that direction). Markus came to me on Thursday, indicating that his relatively light schedule is granting him plenty of time to take on some independent study in graph theory, so I gave him some reading on graceful labelings, maybe he can join Trixie and Sieglinde in their pursuit for new graceful trees.

Speaking of which, we'll soon have our strongest showing at an MAA Southeast Sectional meeting since my arrival at UNCA: at least five faculty members and four students have indicated interest in going to the meeting, and I'm trying to get three of these students (including Trixie) to present in the student poster session.

And speaking of Trixie, how's Calc II? They too completed their first exam this past week, and overall the grade distribution was pretty fair, with a course average of about 76% after corrections were made. Oddly enough, though, there was a profound difference between the two sections of the course: the first section's post-correction average was about 68%, the second's around 86%. I kid you not. After corrections 12 out of 16 students in the second section got either an A or a B (8 As, 4 Bs), while only 10 of 30 students in the first section earned that bragging right.

What, as they say, is up?

Could it be class size? Admittedly I find it much easier to engage the second section as a whole and as individuals, owing largely to the fact that it's got about half as many students. Moreover, the students are less intimidated by speaking up in front of one another, and by presenting on the board. They're also much less reluctant to get into groups and work on problems together. Whether any of this has anything to do with the size of the class, I don't know, but I can't help but think that class size plays at least a small role. (Incidentally, I'm happy to report that I'll be teaching two sections of Abstract Algebra I next fall, each considerably smaller than the traditional single sections that have been run in the past. Our program has been so successful in courting majors that we're having to run two sections of the upper-division courses. America gonef! [You'll have to excuse the Yiddishisms, I've been making my way through Leo Rosten's delightful Hooray for Yiddish: a book about English. I'm easily influenced].)

Could it be the time of day? But one might think that the sluggish 9:00 a.m. class would be more ideally situated in that regard than the post-lunch but usually-punchy 12:45 p.m. class. Or maybe I just think it that way because I'm a morning person. Perhaps there's something to it: the morning section's usually slow-to-rile and torpid, the afternoon section's much more get-up-and-go.

Could it be...the luck of the draw? I've got great students in both sections, but they're just more highly concentrated in that smaller second section. Maybe it's just coincidence that the second section's so much more lively.

Whatever the cause, the difference between the two sections is as that between night and day. I love both of them, but I find myself often wishing wishing wishing that the first section would wake up and stop dragging its heels!

What else?

Faculty talks have ended in the Senior Seminar, I capped them off with a presentation this past Wednesday, on open problems in graph theory. I'm proud of the fact that we had three speakers from off campus, and that we'll soon have at least two more visitors coming to speak in the Research Seminar. I truly believe that our department should attempt to cultivate a more active research environment, and I think we're well on our way towards achieving that goal.

Student talks begin in the second week after Spring Break, two-by-two they'll fill up the last six weeks of class. I'm looking forward to those talks, the topics look to interesting.

What else?

The Writing Intensive committee (well, technically it's a subcommittee, but who's keeping track?) has sprung back into life, continuing our analysis of WI applications and beginning our conversation on the assessment of the success of already-WIed courses. This is a sticky wicket of a tricky schtick: How are we to judge whether a WI course has met the goals it laid out for its students? What materials must we demand of the course instructors in order to perform a proper assessment? How many of a course's learning goals relating to writing must be met in order to call the course a success? And if a course is less than entirely successful, what consequences do we as a committee mete out? It's unrealistic to aim for the ideal right out of the gate (assuming the ideal can be articulated from the outset anyway): all but the perfect instructor is going to stumble here and there, and no course is flawless in design and execution. Therefore it's pointless to pull someone's WI away should perfection not be attained. We don't want to smite those who fail in providing this or that element of their class's purported learning experience. Instead, we wish to encourage the instructor to look carefully at her course's goals, to look at the students' products in attempting to meet those goals, and say, "this was done well. But this, when asked of the students, proved unrealistic. Better I ask that they reach for the moon with their hands at their sides!"

How many of us are so reflective? I'd like to say that I am, but who am I to say?

In the first of what will be several meetings of the writing assessment project this past week (didn't I say I've been busy?) I told my colleagues that last semester's 280 course taught me to be truly conscious of the role played by writing in my own particular discipline...indeed, I think I learned more in that class than my students did. My approach to writing as a tool for learning has changed because of that class.

Writing is playing less of a role in my Calc II class this semester than it did in either of last semester's classes, and while I've not shone a spotlight on writing in Graph Theory, it's ever present. (The work I've done in 280 over the past year is most evident in the structure of the students' proofs on the blackboard: I'm thrilled whenever I see clear statements of hypotheses, an explicit indication of proof technique, summarizing sentences that indicate when and why a proof has been concluded, and so forth. In all only two or three of that class's students didn't have 280 with me, and almost daily I see elements of my own idiosyncratic style that have rubbed off on them.) I'm going to take a few minutes during this coming week to refocus the students' attention on writing and encourage them to keep an eye on the criteria for solid mathematical writing as they put together their solutions to the problems selected for written submission.

What else?

Um...hmm...giving a recruitment spiel on the upcoming REU and speaking ongoing graph theory research to a wonderfully receptive and warmly inviting crowd at Morehead State University in Kentucky (y'all were great, thank you so much for having me!), writing about a dozen or so REU rec letters for current and old students, joining a couple of colleagues in a presentation to the university's Foundation Board, agreeing to help organize this coming May's Writing Intensive workshop, and shaking off a nasty cold that took me out of commission for a few days...see? There's a reason I've not been around!

If you'll now excuse me, it's Maggie's birthday (which one, I will not say, though I doubt she'd mind if I did, she's not embarrassed by her birthdays), and we've got to go celebrate it in the manner of her choosing.

As usual, all comments, questions, queries, suggestions, insinuations, epiphanies, innuendi, graffiti, scritti politti, revelations, retorts, ripostes, and recriminations are welcome on the comments page.

Until next time, live well, and try to learn something new today.

Never too many cooks

Howdy, folks!

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

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

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

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

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

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

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

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

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

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

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

Something to think about.

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

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

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

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

Super Saturday IV

I'm not sure why I got up at 6:30 this morning...at least I got a head start on the grading. I'm halfway through the Calc HW at this point, but 280'll call this afternoon.

At this moment I'm sitting in Zageir 227, 45 minutes remaining before our fourth Super Saturday session begins. I've got nothing to do for a bit, so I thought I'd take a moment to check in here.

How'd the week wind up?

In 280 we talked about injectivity and surjectivity, concepts that are substantially more puzzling in the abstract than they are in the familiar context of real-valued functions. We had a bear of a time working through a proof of the equivalence of the two most frequent characterizations of injectivity. There was something about the subtlety of the contradiction that was tripping up even the best and the brightest in the class. The confusion likely had to do with the convoluted structure of the statement we were trying to prove. Roughly speaking, the statement looks like (A ⇒ B) ⇔ (C ⇒ D), for various statements A, B, C, and D, so at one point we're dealing with a hypothesis that looks like (A ⇒ B) ∧ C. Say huh?

In Calc I we rapped about the way a function's graph is determined by its derivatives. On Monday I'm going to have them do a few exercises as a class to drive home the meaning of properties like concavity and inflection, and how they relate to increasing and decreasing.

Oh, by the way: they made substantial recovery on their exam revisions, raising the class average to roughly 77%, up from the woeful 68% last week. Several students offered touching apologies or narratives on their submitted revisions. Magdalena, for instance, mentioned how she saw herself in the inner monologue I included in the post following last week's grading frenzy. "Oh my god, he's inside my head!" she said. She vows never to become so reliant on the solutions manual again.

They're good kids.

Newton v. Leibniz takes place a week from Monday; this coming Monday I'll get the rough drafts of their "arguments," many of which I will share with opposing teams so that each legal team may adequately prepare a defense having had access to the other's evidence and data. So far most of the teams seem to be functioning pretty well together, with a couple of exceptions. There's been a little tension stemming merely from personality differences present in one or two of the groups. In a perfect world, perhaps...

...you know, when I was a student, I didn't so much like working in groups. I didn't have to do it all that often, but when I did, I got it done with as soon as possible and moved on. I can't say now how I must have come across to my teammates. Bossy, maybe? Headstrong? I hope not. I know that I'm the sort of person who wants to get it done "right," even if that means that I have to take the entire project onto my own shoulders and do every last bit of the work myself. Better that than that the project turn out shoddily because one of my teammates had the audacity to not be as smart as I was.

I suppose I might have been bossy, without meaning to be.

And I wasn't always my sweet current self, either. Though sarcasm is my native language, over the years I've learned to tone it down with others whom I don't know so well. Sarcasm, spite, vitriol. My mother loves telling the story of how I frustratedly called one of my kindergarten colleagues "stupid" because she couldn't grasp the nuances of the game I was trying to play with her. I was insufferable as a young 'un, but I've gotten more mellow with age.

Why can't we all just get along?

More later...I'll let you know how my first session of the departmental Moses Learning Circle went yesterday afternoon (short version: I'm not so sure my colleagues are sold on the claim that mathematical literacy is a civil rights issue, or the claim that improved numeracy is a societal panacea), but for now I've got to get ready for the SS kiddies who'll be here in a few minutes (Tallulah and Deidre are already here, ready to lend a hand).

Avanti!

Pre-summer reading and subsequent ruminations

My summer reading has begun with a bang.

I've been working my way through a book my colleague Tip lent to me, Radical equations: civil rights from Mississippi to the Algebra Project, by Robert P. Moses and Charles E. Cobb, Jr. (Boston: Beacon Press, 2001). Tip met up with Bob Moses, a noted civil rights leader, Harvard-trained mathematician, and founder of a grassroots math program (the aforementioned Algebra Project) targeting middle-grades mathematics students on the fringe, at the Mathematics and Social Justice Conference Tip attended in Brooklyn a month or so ago. Clearly Tip's thought deeply about the ideas in this book, especially as regards his nascent community-building math program and the NSF grant proposal he and I are soon to set about writing, and he was kind enough to lend his (signed!) copy of the book to me.

I'm about three fourths of the way through the book right now, and though I've found much of the text itself dry and unengaging (particularly when it gets bogged down in the toledoth of Moses's project's converted teachers; at some points it reads like a biblical genealogy: "and Norma Jean begat Shannon, who begat Tom, the Explainer-to-Children. Tom taught twenty years, and then begat Sylvia. From the Delta did Sylvia come, and her students were good in the eyes of the Sunflower County Board of Education..."), one or two of its ideas have really struck me. I've been won over in one respect in particular. To best describe my conversion, I ought to mention a conversation I had with a colleague at a school I recently (in January) visited.

We were talking of undergraduate math majors; I had mentioned that at UNCA we have something on the order of 80-90 majors at any given time, and my colleague was thoroughly impressed. He said that at his school (a school nearly twice as large as UNCA), there were perhaps half as many undergraduate majors. Ever wanting to be helpful, I made some comment along the lines of "I've got some ideas that might help you to bring those numbers up."

My friend's response was something like "I don't think we're looking to do that."

I was taken aback. Not only had I not scored a point on this man's scoreboard (he and I share a long history, and I've never wanted to let him down), but my view that more young people should be helped to embrace mathematics had been met with hostility. This man had no interest in bringing math to the masses; he preferred to let it stay the closely-guarded territory of the few and the proud.

Moses's book has helped me to make out this not-so-well-hidden trap many professional mathematicians fall into: too often we think of math as a religion, a cult of worship into which only a select few are to be inducted, and in which only an even smaller few are allowed to become high priests. We take pride in our weeding out of the undesirables, defined as anyone who doesn't have a natural knack for math, who comes pre-programmed with a love of abstraction and analysis.

I've seen our culling at work at several levels, more clearly at some institutions than at others, and I'm ashamed to say that I've even taken part in it, though unwittingly. Math professors have a tendency to emphasis the arcanity of their field, its abstruseness, its disconnect with reality. Many of us take pride in the difficulty of our field, and make every attempt to show off our own intellects by making math seem imponderable, impenetrable, dense. I saw this most clearly at Illinois, where research mathematicians would show open disdain for all but the brightest undergraduates in their classes ("most of them are dullards, I've got a few who might prove capable"), lending a hand to the top 5% while leaving the others to drown; "they're just not cut out for it."

For too long mathematicians (and scientists of other stripes) have tried to fill their ranks from the lists of the best and brightest of the American studentry (to borrow a wonderful turn of phrase from William Strunk), leaving the dregs to find work elsewhere when they prove themselves incapable of meeting the high standards set for practitioners of math research. This leaves the untouchables, consisting primarily, in this nation, of poorly (read: publicly) educated blacks, latinos, and poor whites, out in the cold.

Moses makes clear that there's something wrong here.

What's called for, in his mind, is a radically different paradigm: rather than drawing from the few whom opportunity and natural talent have buoyed to the top, we ought instead be working to ensure that everyone is given the chance to rise to the surface. With a system wherein all are given the tools needed to excel mathematically, everyone benefits: traditionally underrepresented groups obtain the opportunities they need to succeed academically, and academicians expand broadly the talent pool from which they will one day choose their colleagues and successors. This "bring 'em on, all of 'em," attitude is the core of the Algebra Project, a program committed to making sure that every 6th, 7th, and 8th grader is given the background needed to successfully navigate a college-prep math sequence in high school. As Moses sees it, not everyone will go to college, and while there, not everyone who goes will study mathematics, but everyone should be ready to do so.

It just makes sense.

Moses's work has definitely helped me to come around to this point of view. It's never been so clear to me that I as a mathematician have a good deal of work to do to ensure that I'm doing what I can to grant everyone the chance to experience mathematics, and to succeed at it. Work needs done at all levels, K-12, undergraduate, and higher. And the work done at each stage needs to be interwoven with work done at other stages: vertical integration is called for. I hope Tip and I will be able to capture that spirit effectively in the proposal we put together this summer.

I've also begun the book the Project NExT reading circle has chosen as its first focus of discussion, Ken Bain's What the best teachers do (Cambridge: Harvard University Press, 2004). I'm (we're) one chapter in, and so far I'm unimpressed. It strikes me as a poorly-assembled pile of truisms and platitudes, absent the concreteness and careful analysis of a seasoned student of the scholarship of teaching and learning (SoTL). Not that every SoTL text has to build up a rock-solid wall of data and facts, or deluge the reader with a statistical breakdown of every study on teaching efficacy performed since 1970...but this book just seems "lightweight" to me. So far it's a great "rah-rah" feel-good page-turner, but its place might be on the nightstand of a newbie college prof just out of grad school who needs a little cheerleading and from-the-sidelines inspiration. I wasn't a huge fan of Maryellen Weimer's book, but I found it far more useful and engaging than Bain's, at least to date.

I ought also say a word or two about preplanning I've begun for Fall's classes. Francine's agreed to help me go through the notes and homework problems we used in 280 this past semester, retooling them, getting them good. I've already written a couple new in-class exercises dealing with writing mathematics.

The one I'm happiest about was inspired by a conversation I had with my colleague Lulabelle from the Sociology Department (I'm on a team of folks helping her out with a pilot assessment program for writing across the curriculum). She indicated that as a part of the work that'd need doing for this grant we're collaborating on, I'd have to to be able to train my colleagues how to "read" mathematics. I got to thinking about how I would best do this, and realized that it's likely easiest simply to highlight the linguistic analogues mathematics shares with "natural" human languages: syntax, grammar, orthography. Not only would an exercise indicating these analogues help my colleagues; it would help my 280 students, too.

The exercise consists of a take-home portion and an in-class portion. Each part comprises three written passages of varying levels of quality; the take-home passages are in "English" and discuss the chemical element boron. Students (and my colleagues) should have no trouble in ranking these passages from worst to best, and in explaining their reasoning for the ranking. The in-class passages are in "math," each giving a "proof" of the fact that the sum of two odd numbers is even. Having given them the chance to warm up in "English," I now ask the students to rank the proofs from worst to best, and to justify their rankings. This is a more difficult task, but once it's done my students (and colleagues) should be able to see more clearly that good writing in math is only a half-step away from good writing in any other discipline.

Okay, I've prattled on long enough. I'll end this for now. As usual, feel free to check in with your comments, always appreciated!