Sampler

During the past week I've attended, presented at, participated in, or facilitated two conferences and three faculty development workshops, on topics ranging from pure mathematics (group theory and graph theory) to the pedagogy of writing and advising first-year students who are brand new to a liberal arts university. I've written dozens of pages of notes, collected several handouts, worksheets, resource lists, and sets of slides. I've talked with, listened to, or sent e-mails to dozens of colleagues. In between doing all of the above I've been spending most of my time reading up on the history and philosophy of science and its teaching and on the role played by gender in mathematics achievement.

Below I've collected a number of quotes, factoids, observations, and questions dealing with all that I've been doing for the past week or so. Every one of these items deserves further follow-up; maybe I'll get around to addressing some of them over the summer, maybe not. I just want to get them out there for now.

1. Contextualization (and resocialization) of science. First, more from Thomas S. Kuhn's The structure of scientific revolutions (pp. 136-137), on the apparent linearity and cumulative nature of science and the related transmission of scientific knowledge:

Textbooks thus begin by truncating the scientist's sense of his [sic, here and following] discipline's history and then proceed to supply a substitute for what they have eliminated. Characteristically, textbooks of science contain just a bit of history, either in an introductory chapter or, more often, in scattered references to the great heroes of an earlier age. From such references both students and professionals come to feel like participants in a long-standing historical tradition. Yet the textbook-derived tradition in which scientists come to sense their participation is on that, in fact, never existed. For reasons that are both obvious and highly functional, science textbooks (and too many of the older histories of science) refer only to that part of the work of past scientists that can easily be viewed as contributions to the statement solution of the texts' paradigm problems. Partly by selection and partly by distortion, the scientists of earlier ages are implicitly represented as having worked upon the same set of fixed problems and in accordance with the same set of fixed canons that the most recent revolution in scientific theory and method has made seem scientific. No wonder that textbooks and the historical tradition they imply have to be rewritten after each scientific revolution. And no wonder that, as they are rewritten, science once again comes to seem largely cumulative.

My question: does it have to be this way?

My answer: no. But an elaboration of that answer will have to wait for now. I have a good deal more to say in reflecting on the social constructivist point of view of science, first (or at least first explicitly) elaborated in Kuhn's work, as in the following passage (p. 42):

Though there are obviously rules to which all practitioners of a scientific specialty adhere at a given time, those rules may not by themselves specify all that the practice of those specialists has in common. Normal science is a highly determined activity, but it need not be entirely determined by rules. That is why, at the start of this essay, I introduced shared paradigms rather than shared rules, assumptions, and points of view as the source of coherence for normal research traditions.

The nature of the rules to which specialists adhere is fluid and dynamic, susceptible to the exigencies of the day-to-day applications of those rules. Rules, as applied, evolve, and they evolve in accordance with their usefulness as judged by practitioners of the specialized discipline concerned with those rules. More than anything else reading Kuhn makes me aware of the need to be more intentional about including opportunities for my students to explore, discover, interpret, investigate, and describe the concepts we consider in any given class; they must be made, to the greatest extent possible, to feel like they as much authors of scientific discovery as I am. (Particular attention to the role of youth and neophycy in scientific "advancement" is critical as well.)

2. Invention versus discovery. I've often asked my Calc I students to think about the difference (if there is one) between invention and discovery when they take part in the Newton v. Leibniz project. Kuhn will serve as an excellent source for those students interested in learning more about the distinction between the two notions: pp. 51-53 contain a discussion of this distinction, highlighting invention as the adjustment that goes on in a paradigm's conception of science in the wake of a discovery: one may be the first to objectively observe a physical phenomenon, say, but until the significance of that phenomenon is understood and elaborated, and the discovery's relevance is described, one cannot be said to have invented a thing. In a sense invention is the recognition of the importance and relevance of a discovery via its incorporation into the normal scientific tradition that operates within a given scientific paradigm.

What might students have to say about this?

3. Revision (in writing) as revolution. The parallels between Kuhn's portrayal of scientific revolution (especially as it compares to political revolution) and revision of one's writing as a process are too great to be ignored: revision occurs concomitantly with the recognition of the inadequacy of what one's written to account fully for one's perception of the subject of the writing. That is, revision is undertaken in response to a perceived discrepancy between the author's intent and the author's ideas as communicated expressly on the page. Compare (p. 91): "In much the same way, scientific revolutions are inaugurated by a growing sense, again often restricted to a narrow subdivision of the scientific community, that an existing paradigm has ceased to function adequately in the exploration of an aspect of nature to which that paradigm itself had previously led the way."

To continue the parallel between these "revolutions" would force us ultimately to recognize what students of writing are often loath to admit (and what teachers of writing already know very well): writing is a social process, as much, if not more so, than is scientific discovery.

I should note that in the following pages in Kuhn's text (pp. 92 ff.) he most clearly articulates the role of social forces in shaping scientific revolutions. When competing paradigms come up against one another, adherents to one or the other must be prepared to argue in favor of their particular paradigm.

4. Portfolios (again). Enough about Kuhn (for now, at least). Let's get to some observations on the International Writing Across the Curriculum Conference, the first two days of which I was able to attend at the end of last week.

In talking with my colleague Nero (currently at the University of Hawai'i, Hilo) I found myself suddenly able to articulate, far more clearly than I've ever been able to before, what exactly a portfolio means of assessment might look like in a mathematics course. One or two communication outcomes would join one or two affective or metacognitive outcomes, and these would join two or three content-centered outcomes as a basis for the course's assessment. (I already generate such outcomes for all of my courses.)

Throughout the semester the students would be given a variety of assignments, successful completion of each of which would demonstrate achievement at one or more of the outcomes on the list described above. These assignments could include more traditional problem sets (though likely not sets of problems pulled from a textbook), written components of projects I already assign (like Newton v. Leibniz, Confectionary Conundrum, etc.), reflection or response papers in which students explore their personal and emotional engagement with mathematics, and so forth.

Students will have a chance to perform unlimited revision on many of these assignments, so that if a student isn't happy with a given iteration of a given assignment, she can revise her work to improve upon it. (As regular readers know, I'm still working on adjusting my revision policies.)

In the last week or so of the semester the students will be asked to select four or five assignments from among those completed during the semester to represent their mastery of as many of the course learning outcomes as possible. They will then write a brief (no more than five or six pages) paper in which they articulate explicitly the role served by each of the assignments they have chosen to include in the portfolio: why include this piece? Mastery of which outcome does it purport to demonstrate?

I'd like to recruit one or more of my colleagues to help me assess completed portfolios the first time around; there ought to be some sort of validation process.

Thoughts?

5. Intentionalize, intentionalize, intentionalize! This coming summer's REU students will receive yet more intentional instruction in writing than any previous year has received. At least two of my College of Charleston colleagues will be coming up to help impart their wisdom on rhetoric and composition. I'll be giving the students more models of professional writing than they've been exposed to in the past. And, most notably, I will obey the exhortation of the four presenters from Virginia Commonwealth University and place more emphasis on the "middle" stage of student research writing.

What do I mean by this? Much like the faculty at VCU (as described by the four presenters mentioned above), I find I've been very intentional about helping my REU students find sources at the outset of their research program, and I've been very intentional about helping them through draft after draft of their week-to-week research reports once those reports have assumed a certain level of coherence. But, like the aforementioned faculty, I've been somewhat remiss in offering the students explicit instruction in the middle stages of the process: how does one evaluate sources? How does one compare them? How does one decide on the relevance of a particular source to one's own researches?

I've decided that I'm going to require the REU students to follow their initial literature searches (which most of them do) with the construction of an annotated bibliography in which they highlight the important contributions of each source, summarize the relevance of each source to their own work, and prioritize the source, ranking it alongside the other sources they've found in terms of its strength of contribution, its clarity, and its relevance to their particular research project.

Will this make more work for the students? You bet it will. But since I'm only going to be asking the students to produce a draft of their report every other (rather than every) week, I feel it's a fair amount of work to ask of them.

It occurred to me this morning, in sitting in on a faculty development workshop focused on our LSIC courses, that the same sort of exercise should be required of students in our MATH 480 course in order that that course warrant its Information Literacy Intensive designation. Just two years ago I suggested that the department begin requiring students in MATH 480 to produce an expository paper; this suggestion met with almost no resistance. I hope this new suggestion will go over equally well.

6. Other thoughts for the REU. What else will I be asking this year's students to do? Nothing excessive, I believe. It seems to me that I should require the following of the program's participants:

History and context. Every draft (not just the last) of every student paper this summer will be required to have a section describing the history and context of the topic the student is investigating. This section, like the rest of the paper, may be rather sparse and tentative at first, but like the rest of the paper it will become more full and flourishing as the summer goes on. I believe it's important, though, that from the very onset of the program the students become accustomed to contextualizing their work and establishing its place in the field.

Visuals. One of the VCU folks mentioned above presented a metaphorical means of constructing an annotated bibliography and literature review, comparing the process of finding, evaluating, prioritizing, and applying sources to planning a conference, at which participants must be placed at various tables, grouped in various sessions, and so forth, according to interests, purposes, and points of view. The most striking aspect of this presentation to me was the insistence on a visual representation: the presenter required her students to come up with a visual means of portraying their evidence. I am going to start requiring each REU student to include at least one visual representation of her or his work in the bi-weekly presentations they'll be delivering. That visual may be the same from week to week, but if the visual remains unchanged I will ask the student to justify her or his reason for retaining the same visual. This, I hope, will encourage students to reflect upon the way in which they are representing their work through nonverbal means; this reflection could lead to further discovery and, of course, refinement of the visual rhetoric the students use in describing their work.

Elevator talks. Even the strongest undergraduate research students have trouble articulating their work clearly and concisely. I'm going to begin asking every student to open her or his presentation with a no-more-than-one-minute "elevator" version of the presentation. What is the main focus or question of your research? What method or methods are you using to try to study that focus or answer that question? How does your work fit in with others' work on the same topic? I hope this additional intentionality will help students develop the ability to communicate their work in the hurly-burly world of conferences and cocktail parties.

7. QEP. As many of my colleagues in the Southeast part of the country know, QEP stands for "Quality Enhancement Plan," and is the means by which the Southern Association of Colleges and Schools (SACS, the accreditation agency for an enormous number of institutions of higher learning in the Southeast) asks the colleges and universities it oversees to plan and implement institution-wide changes to enhance student learning.

I'll have a lot more to say about this in the coming weeks, months, and, if all goes well, years, but I'll simply say now that I am more committed than ever before to making writing the focus of UNC Asheville's QEP. I will do all that I can to lobby for this position.

8. Inkshedding. Perhaps the most delightful thing I took away from this past Wednesday's workshop on writing instruction of ESL students (ably facilitated by my colleagues Hannah and Tabitha of UNC-Chapel Hill and NC State University, respectively...thank you both so much for coming out!) is a new form of low-stakes writing to which I'd not before been exposed. "Inkshedding" is much like a collaborative form of freewriting. As they would be in a freewrite, participants (in groups of three or four) are asked to write on a given topic for a set amount of time (three minutes, say) or until they have written all they would like to on the topic at hand. When finished, each participant places his writing in the center of the circle and waits for someone else to do the same. The papers are then exchanged, and each person reads what the other has written and then responds in writing on the first writer's paper. Once done responding, the second person places the paper in the center again and takes another. And so on. In theory, the process could continue endlessly, readers writing in response to others' responses to their own responses, and so forth.

Beyond its obvious pedagogical usefulness, I think this would be a fantastic way to construct collaborative poems, or at least generate ideas and images for rich poems or other pieces of fiction. I'm eager to find a few folks who are willing to try this out. If you're game, let me know!

9. Gender matters. I'm currently reading a book that I picked up (in the simply marvelous bookstore Caveat Emptor) in Bloomington, the site of the IWAC conference last week, Mathematics and gender, edited by Elizabeth Fennema and Gilah C. Leder (1990, New York: Teachers College Press). This collection purports to analyze the different ways in which gender influences math performance, success in math coursework, and affective responses to mathematics and its study. Unsurprisingly, men and women differ with regard to their experience with math, and factors such as confidence, perception of utility, sex-role congruency (the "math is for men" stereotype), fear of success, and attribution of performance to one or another cause (effort, ability, or outside forces such as sheer luck) all strongly, and differently by gender, affect an individual's mathematical understanding and performance.

I've yet to read much in this book that's given me reason to adjust the way in which I teach math, aside, perhaps, from Lindsay A. Tartre's study (Chapter 3, "Spatial skills, gender, and mathematics") suggesting that in women there is a far stronger correlation between spatial skills and mathematical performance. Might I do well to place particular emphasis on visual representations of problems when working one-on-one with a female student? I already attempt to adapt my explanations to whatever mode it is in which I know a given student most clearly understands mathematical ideas.

It's something to think about. I may have more to say about this book as I get into the later chapters, which deal with the role of the teacher and the classroom dynamic in assisting or impeding students' mathematical understanding.

That's enough for now. I realize that this is one of the longest posts I've written in a long time. Believe me, I've tried to keep it short! I hope to be able to elaborate on one or more of the above issues in later posts, especially as I begin to implement some of the proposed changes to my REU and to my regular courses.

To be, as ever, continued!

Snippets

From Leon Botstein's Foreward to Writing to learn mathematics and science (eds. Paul Connolly and Teresa Vilardi, 1989, New York and London: Teachers College Press, p. xiv): "The use of ordinary language in the teaching of science and mathematics enables the teacher to connect what otherwise might seem an arcane and distinct set of languages, thought processes, insights, facts, and understandings to experience, in the everyday sense that Dewey realized could constitute the basis for motivating learning, memory, and long-term comprehension."

From Paul Connolly's "Writing and the ecology of learning" (op. cit.), p. 7:

[In the all-too-typical science classroom] the important feature of education becomes saying the right words, not learning how to use one's own words. In such circumstances the "language" of science remains for many students a set of foreign words, dead as Latin, to be memorized from a book. It is not the constructive speech of a vital culture. Students then regard "definition" as a chain of words that bonds one to "truth" and "reality"; if one link is forgotten, the whole chain of understanding is broken. They have...no notion that meaning is recomposed in each new personal performance on the public instrument of language.

From Sheila Tobias's "Writing to learn in science and mathematics" (op. cit.), p. 49 (cf. my Charleston colleagues' and my recognition of the importance of visual rhetoric in the legibility of mathematical writing):

Even a cursory examination of math and science textbooks reveals that these books are not meant to be read as we understand the act of reading....As I write elsewhere, clarity in books in other subjects is achieved through repetition, using different words to restate a single idea, slowing the pace, using a spiral kind of organization that keeps coming back to the same idea at different levels, using topic and summary sentences to nail down what the paragraph contains, and always foreshadowing the point to be made later on. In math and science texts, we find, instead, pages of information with virtually no repetition, no varying of pace, few topic and concluding sentences, as few words as possible, written with the expectation that the reader will not proceed to the next sentence or point without having thoroughly mastered the one at hand.

From Thomas Kuhn's The structure of scientific revolutions (1962, Chicago: The University of Chicago Press), p.11: "Men [sic] whose research is base on shared paradigms are committed to the same rules and standards for scientific practice. That commitment and the apparent consensus it produces are prerequisites for normal science, i.e., for the genesis and continuation of a particular research tradition."

And, p. 20, on the evolution of scientific genres: "No long [after the establishment of a paradigm] will [a scientist's] researches usually be embodied in books addressed, like Franklin's Experiments...on Electricity or Darwin's Origin of Species, to anyone who might be interested in the subject matter of the field. Instead they will usually appear as brief articles addressed only to professional colleagues, the men [sic] whose knowledge of a shared paradigm can be assumed and who prove to be the only ones able to read the papers addressed to them."

Finally, p. 21: "Although it has become customary, and is surely proper, to deplore the widening gulf that separates the professional scientist from his [sic] colleagues in other fields, too little attention is paid to the essential relationship between that gulf and the mechanisms intrinsic to scientific advance."

Gems

Though not all of them are directly pedagogical, I've done a few things today about which a few words might be here said. They're definitely uncleaned jewels, diamonds in the rough. Extracting these dusty gems from the cracked earth in which they still lie, polishing them, cutting them, and showing them off will give me something to do for the next few months.

Let's see...

...I hurriedly hit "publish" after blogging just an hour or so ago, elliptically, about dinner this evening with a candidate for the University Writing Center Director's position. Obviously it would be inappropriate for me to say more than I already have about the candidate or even our conversations, but suffice it to say the conversations I had with the candidate and with my own colleagues on the search committee taught me much about the university's functions.

This afternoon I spent a few hours in researching the history of the writing across the curriculum movement and the writing-to-learn movement, and in the process put together a few ideas for first-day low-stakes writing activities I might try out in my Calc I course this coming fall. I hope to do a few focused freewrites, the first of which will be designed to let the students write a little bit of a mathematical autobiography, while the subsequent ones will ask them to bring forth from the cobwebbed corners of their brains whatever it is they remember from the last math course they took, whatever and wherever and whenever it was. These latter freewrites can then funnel into a class-wide discussion.

I also spent a couple of hours reading the several remaining LANG 120 essays I've been assigned as a first-round judge in this year's First Year Writing Contest. Not long after telling Maggie that I'd been disappointed that none of them had "popped" for me (there had been several fairly good ones, but none that were clear front-runners), I read three papers, back-to-back, which were strong in every way: compositionally, grammatically, citationally. Their authors had rich vocabularies, a knack for imagery and inventive, challenging sentence structure, and a strong compositional thread along which the reader could pull her/himself from beginning to end. I was happy to find these!

This morning's run gave me the perfect chance to do a bit of self-analysis and self-assessment. I found myself asking, at this point at which the close of the academic year offers the closest thing I'll ever get to the end of one chapter and the beginning of a new one, what it is I like about myself, and what it is I don't like so much: the former, ideally, I can play up and the latter, play down and try to work away...or at least admit and accept with greater awareness.

I don't know what this says about the way in which my brain makes sense of the world around me, but I found that when I wrote them down at the end of my run most of the likes and don't likes fell into dialectical pairs, nestled together like fraternal twins in some sort of psychic womb. I'm not sure, as open as I am, that I'm up for sharing every one of these pairs, but here's one which deals more directly with my role as an educator:

"I like that I'm a good problem-solver. I don't like that I often slip too quickly into problem-solving mode."

I've dealt recently with this issue in my personal life, in incidents in which I've started trying to puzzle out often inchoate and inappropriate solutions to problems about which whatever friend with whom I'm speaking really just wants to bitch. The mathematical, and, some might point out, stereotypically masculine, part of me wants to simply get at the problem and root it from the ground. This isn't always the appropriate course of action: sometimes all my interlocutor wants is for me to shut up and commiserate or empathize. (Another one of my "like/don't like" pairs, closely related to the first: "I like that I'm a good listener. I don't like that I seem to think that gives me license to respond, when often all my partner in conversation wants is someone to listen.")

Pedagogically, I see the effects of premature problem-solving first-hand in many of the interactions I and my colleagues share with our students on a daily basis. (I thought of the following at dinner this evening while we were all discussing the ideal meeting between a writing consultant and a writing center client.)

Imagine that a calculus student has a problem with a run-of-the-mill textbook problem, and he comes to his teacher for help in solving it. He's bravely stepped into his teacher's office, interrupted his teacher's work, and sat himself down heavily in the chair across from the teacher's desk.

At this point the weakest teacher (often eager simply to get back to her work) will simply work the problem out for the student. (Novice teachers are often prone to this.)

The stronger teacher will take the time to ask the student what work he's done so far in trying to solve the problem on his own and try to build on what's already there. (This is where I am most of the time.)

The strongest teacher will ask the student what problem it is that the student's trying to solve in the first place. "Is there a problem at all? What, really, are you being asked to do? Can you articulate it for me, maybe in your own words?"

For a confident student with reasonably strong mathematical skills, a student who's generally on target most of the time and who only occasionally really just needs a little nudge in the right direction, the second teacher's action will have served the intended purpose admirably: after an initial survey of the student's own work, it might take little more than one more step worked out jointly for the student to see where to go next on his own.

Yet the second teacher has instantly leaped into problem-solving mode, presupposing there is a problem in the first place, presupposing that the student knows what in the hell he's being asked to do. Does he? Should the student be a weaker one, a less confident one than his peers, perhaps he's not even sure what's being demanded of him. In jumping at once into problem-solving mode the teacher has missed an opportunity to help the student to develop a perhaps-yet-more-important skill than problem-solving: problem-posing. One cannot hope to become an effective learner if one is adept only at answering questions; one needs also to know how to ask them in the first place.

I'm sorry if this train of thought appears to be riding on rails laid out by a track layer on LSD. I'm really just writing this out in order to better understand it myself; I'm actively (even as I make this next keystroke) engaging in writing-to-learn. (Kids, let this be a lesson to you!)

I'm afraid that's about all I've got right now. It's been a busy Sunday, and it's late. I've got more meetings with our University Writing Center candidate tomorrow, and an early meeting with one of my undergraduate research students, so I've got to hit the hay. Perhaps I'll soon say more about other relevant "like/don't like" pairs. We'll see.

Should one ever chance...

...to be asked to serve as the "outsider" on a search committee, or as a member of a search committee not in one's area, accept! I've learned a tremendous amount about the workings of my own university (regarding funding, internal operations, interdisciplinary initiatives, distance learning projects, etc.) from my experience serving on the search committee for our new University Writing Center Director.

Rocks and gravel

I've just finished reading Jonathan Kozol's Ordinary resurrections, begun just a few days back, and I feel, as I usually do on reaching the end of one of his books, an overwhelming sense of tiredness bound together with a bright and unbreakable thread of hope and strength.

It's a good book on which to end this academic year, surely the most tiring yet of my career (although, as I've noted on this blog before, not the busiest). Reading it at the close of the year gives me at once a sense of closure (which I've sorely needed) and a feeling of renewed energy. The book ends on the same note, as Kozol describes a gift of art Pineapple, one of the children of whom he speaks about most often in the book, gave to him: "an imitation stained-glass window that she made from tissue paper, brightly colored with green paint and with a wash of light-blue ink...When I asked her recently if it was supposed to be a rising sun or a setting sun, she seemed at first to not remember what I meant...'You decide,' she told me uselessly...at the risk of being sentimental about somebody whose sunny disposition brings a lot of joy into a world that has too many cloudy afternoons, I like to think it's rising" (p. 339).

Is it?

Kozol often admits in this book and others he's written since that he's getting older and feeling weaker. Although not yet as frail as he is physically (he's over seventy now, if I'm not wrong, and was sixty-four when he wrote the book I've just finished), I feel, on certain days at least, that I can empathize: each passing year steals away a little bit more of my relevance, my credibility, and my coolness.

"You may have them eating out of your hand now," my department's chair sometimes warns me, a hint of glee in his voice, "since you're not all that much older than they are. But just wait until you're my age." I'm warned that I won't be seen so much as the cool older brother but rather the stern-but-caring father. Just this past semester I encountered a student out of whom my most earnest attempts at cool cajolery could coax nothing. Magda, scarred, I suspect, by recent unpleasant experiences with mathematics, was timid in her dealing with me, reluctant to take part in any activities in class, and unresponsive to personal offerings of assistance I sent to her by e-mail. "I have to be honest that I've gotten a sense of 'defeat' from you for much of this semester," I wrote her. "I know you'd mentioned earlier this semester that you'd considered math as a major, and I hope that that's not an idea you've abandoned entirely."

I urged her to reply, but I never heard back from her.

This incident can't help but make me think of a noontime meeting several of my colleagues in the department and I shared with a pair of textbook company representatives. While my colleagues and I pored through elaborate boxed lunches bought from an off-campus catering company, the two textbook reps gushed for several minutes over the features of their company's latest Stewart-clone calculus textbook.

They tried to make their case, I'll give them that. I'd indicated that I honestly don't use whatever calculus textbook I've been assigned as much more than a source of examples and exercises (generally the exposition in such textbooks is godawful, and the organization is unmotivated, at best), supplementing the textbook substantially with descriptions, worksheets, activities, and projects of my own devising, including a number of nontraditional writing assignments. "We're very proud that [their text] contains [some unsubstantiated and moderately large number of] pre-written group projects." I nodded, unimpressed. (So does every Stewart-clone; generally these projects are as unmotivated as the integument of the text itself.)

About fifteen minutes into their demonstration, and after about ten minutes of fiddling with recalcitrant teleconferencing software, the reps got the lead textbook author himself on the line. He sounded tired, as though he'd made these dogs and ponies to dance two or three times already earlier that morning (or at least that week). He showed us a number of the features of the on-line text, most excited about the numerous Mathematica-driven animations he and his colleagues had developed for the textbook. (Such animations are not in themselves bad things; however I fear that without empowering the students to learn how to create their own animations, the animations alone do little more than provide an alternative visualization tool.)

After trotting the poor schlub of an author around the rink for a few more laps, the reps resumed their own presentation, and reached the part of their sell about which I was reminded above. "The beauty of the on-line scoring capabilities," one of them said proudly, "is that once a student has submitted her homework and the homework is graded, the computer can generate a personalized letter indicating to the student what sorts of problems she got wrong, telling her where she needs to focus her study. This can be done automatically, for a class of two hundred students. They'll think that you cared enough to write a personalized note to every one of them about their work."

I don't want to give the impression that this past year was all about weariness and defeat. To the contrary, I feel I've had some tremendous successes in the classroom.

While the textbook exercise fell on its face this past semester in Topology, it went over fantastically with the MATH 280 folks in the fall, and I can't think of many moments in my career at which I was prouder than I was when Ulrich and Uriah spoke on the project at the Elon conference (including in their presentation the phrase "writing-to-learn," and defining the phrase correctly!).

While I feel I lost the "faith" of some of my favorite students, like Magda, above, and like Tish, who became openly disillusioned about topology by the semester's end, I feel like that loss was only temporary, and that through open communication and understanding I've won that faith back.

While I feel my attempt at crafting a more meaningful system of feedback met with mixed success, the success I've earned is great enough to encourage me to recraft the system and try it again next term, rather than simply to abandon it.

While I've had my share of dealings with troubled and troubling students this past year, I've also had my share of dealings with marvelous ones. Jacobina's transformation from a hesitant and nearly math-phobic student at the outset of Calc I to one of the most outspoken, confident, and competent scholars in the stronger of my two sections of Calc II was nothing short of astounding. Words cannot describe the pride I have of her. Equally exciting is the undergraduate research Tonio and Siegfried have done, and the potential Ino and Iris have to do similar research this summer.

These small (and not-so-small) victories and many more like them are the rocks and gravel out of which, as the old blues standard goes, a solid road is built. Where's the road headed?

I've got back-to-back conferences coming up in less than a week (the International Writing Across the Curriculum Conference in Indiana and the AMS Eastern Sectional Meeting in New Jersey), then a half-week of Integrative Liberal Studies workshops in which I'm playing some role or another. A week after those come to and end the REU students start arriving. I have high hopes for this year's program; they're clearly a bright bunch with a lot of strengths. I've got a book proposal in at a good publisher, on which I'm still waiting to hear a decision. I've got great ideas for my courses next fall. It's going to be a good year.

The road is a long one, but the sun's shining yet.

I think, indeed, it is rising.

Time to put the "I" back into "integral"?

From Jonathan Kozol's Ordinary resurrections (2000, New York: Crown, pp. 54-55):

I used to edit out these questions that concern my private life when I was writing about children. I think, in part, I did this to avoid the risk of "complicating" things too much by intermingling the details of my life in Massachusetts with the more important details of the stories and impressions that the children chose to share with me. I think that this was probably connected also with the old idea that I, like many other writers, used to have of trying to remove ourselves from any situation we described in order to convince the reader, or ourselves, that we had not become entangled in these situations in a way that might affect our objectivity or have some power to affect the way children chose to speak to us.

I think most of us recognize, however, that we do become entangled and that we're never really neutral in the way that we conduct a conversation with a child.

Couldn't have said it better myself, only I'd simply omit the last three words of the above quote. It's time we face up to the social forces that shape our understanding of the world around us, whether we're talking about our understanding of spirituality and religion (as Kozol is above) or our understanding of science and mathematics (as I often do).

A modest proposal (one having nothing to do with cannibalism)

Hmm.

In the past week or so I've had several really good conversations about this past semester's revisions policy with students in both Topology and Calc II. A few things are clear from these conversations:

1. A large number of students benefited from the policy of allowing unlimited revisions (on exams in Calc II and on homework in Topology). Moreover, they benefited in exactly the ways I would hope that they'd have benefited: they profess to being much more eager to look over their past mistakes, to understand what they did wrong, and to attempt (sometimes over and over again) to correct their errors.

2. A much smaller number of students played the system like a two-dollar fiddle, not taking adequate measures to prepare for the exams (or the homework), putting off studying various topics until after the exam's been given, graded, and returned (or until after time's been found to corner one of the class's brighter students and ask said student for a transcription of her or his homework).

3. Something needs to be done to counteract the inevitable effect of procrastination, so that I'm not socked with a hundred revised exams and homework sets to grade at the same time final exams come due. (I apologize if the mass of grading I had to do during the past week as more and more revisions came in made me a bit more snippy of late.)

One of my Topology students made the following suggestion (thanks, Karl!): include with each successive revision an "expiration date," which, once past, precludes further revision. For example, I might allow a student a week to complete each iteration of a revised assignment: say I hand back a graded assignment on Monday, May 10th; the next revision, should a student wish to undertake said revision, would be due on Monday, May 17th. If that date passes without revision, the student's out of luck. If the student does perform some revision and receives the paper back on Wednesday, May 19th, then he'll have until Wednesday, May 26th to effect further revision, and so on. It would be the student's responsibility to be present in class to pick up graded revisions.

This would definitely help address the third observation above. Moreover, I think it would encourage more students to take advantage of the revisions: instead of putting them off until the inevitably busy end of the semester, students would be pressed to perform revisions right away and receive the benefits thereof.

The second observation above remains unaddressed. What to do about those who play the system? I will say that to some extent the chickens came home to roost when the final exam was graded: it was clear to me from performance on the final exam which students cruised on through with little concern for actual understanding of the concepts treated in the course. However, perhaps it might be appropriate to provide some sort of system of diminishing returns, at least for the lower-level courses (like Precalc, Calc I or Calc II) in which I might allow unlimited revisions: the first round of revisions will earn 1/2 credit back, the next 1/3, and so forth, until after two or three rounds there's really no purpose, from a "credit" standpoint, to revise. I'd like to think the best students would continue to revise anyway, striving for a perfect theoretical mark and a more and more perfect understanding.

Anyway...that's where I stand right now.

FYI: I just kicked off my summer leisure-reading season by starting Jonathan Kozol's Ordinary resurrections: children in the years of hope, and although I'm only about thirty pages in (and enjoying it immensely), it's making me more and more aware of the need for "ethnographic" methods in the "hard" sciences. I suspect this coming summer will see me write a number of posts on this blog in which I discuss the humanistic side of mathematics, in which I experiment with what Laurel Richardson would call "creative analytic practices." Stay tuned!

Axe

The state legislature, like its sister bodies in other states, is currently considering further massive cuts to education.

I hate to sound cliché, but isn't it odd that we live in a society willing to spend billions on bombing sovereign nations into the Stone Age but unwilling to scrape together enough money to offer its most intelligent young members a halfway decent education?

To be continued...