JoaLDG: no coin flipping in the exam hall!

Leave your calculators and quarters at home: yesterday was the linear algebra final exam. So today it was graded. It took a little under five hours, from nine-thirty to a quarter past two, for maybe a hundred and fifty papers. I had one problem to grade. For five hours. A little numbing, but not so fatiguing as you might think from it being doing math for so long at a stretch: it consumes much more energy to shift to thinking about a new problem than to continue looking at the same kinds of solutions to the same problem. Grading in parallel is easier as well as more consistent.

Actually this one problem was in two parts, a and b, and the two parts didn't have anything to do with one another, so really it is two problems. They were couched as “true or false” questions, but when they add “Explain.” to the end that actually means: “prove or disprove.” Since this isn't the math majors' linear algebra class, it will perhaps be not too surprising to you to read that generally speaking the option taken was “not prove.” I was not unduly burdened by doing “really two problems.” It turned out that the composer of part b badly mistook the difficulty of his problem: the average over the entire class for that problem was about one-and-a-half percent. There were earned a total of twelve points in all those hundred-fifty papers, eleven points from seven hundred and fifty. (After a vast, vast number of zeroes were awarded, after I began to feel that I was back in kindergarten learning fine motor skills by forming the numeral 0 over and over again, well, I started counting.) And of those four people sharing twelve points only one earned full credit, so congratulations to MJA.

This is, I confess, a fair disgrace. (I should mention that part a was rather better done – I wasn't counting but I'd guess the average was somewhere around forty percent, plus or minus ten. So I wasn't just a zero-scribbling bit of broccoli for those hours. Maybe you find that a fair disgrace too, but at least it's a spread.) The other one is really just a waste of everyone's time. I can't blame the problem-poser for this disaster; the problem was merely a great blind-spot for almost everyone.

Something is shameful if it's counter to some law – what law do I mean here, calling this a disgraceful situation? Really just what I said: no spread of grades.

Grading problem sets I am acutely conscious of their existence as a pedagogical tool. Ultimately everything I do is predicated on the need to indicate to the students the deficiencies and successes of their technique. So for example if the problem is to compute some certain numbers or vectors and the bottom line is incorrect and the reason why is exactly that there was an error in carrying out one step of an algorithm a little while earlier then substantial partial credit is earned. This is only natural: I am not grading the answer but rather the mastery of the material and the techniques of the class. This mandates more nuance than up-or-down.

There is, of course, a different point of view, that overly discretised up-or-down view, which we'll call the “either the building is going to stay up or it's going to fall down” vantage. I think we can all agree this is the standard to which we'd like to hold our civil engineers, but it seems a little draconian for first-year math students, and moreover, as I said, unpedagogical. I must even object: what do I care for the right answer? If I wanted the right answer then I'd do the problem myself. And if some miss the nuance in my nuanced feedback, all the worse for them, but I don't mean to pander nor panic.

And yet there is a well-known and popular case of this discrete approach. The Putnam contest for undergraduate math students applies a similar rubric: “not a solution” gets in the range of zero to two points (from ten); “solution” gets in the range of eight to ten. The problems are, to be sure, not easy, but the easiest problem on the Putnam paper isn't so hard that you couldn't manage it in three hours (if you were an energetic young undergraduate math student), and yet the median score, the score which half the contestants don't exceed, is perennially zero or one point, out of one hundred and twenty. (One year the median score was three, and some wag remarked that this was a reflection of how “ridiculously easy” the contest was that year.)

Well! If I graded linear algebra problem sets in this style, I can only think I would quickly discover how expendable I am – just after all but a handful of students from forty drop out in fear of failing the class. And if this seems draconian even in a prize exam, well, in the first place, it is the rule of the game, and everyone comes to play the game. But moreover it has after a fashion a certain logic: every problem in our linear algebra textbook may be all-too-easy for me to solve three different ways – there's a reason I'm qualified to grade this stuff – but if you give an incorrect solution to a problem no one knows how to solve, how can we really say how close you are? Every false theorem is one mistake away from being proven. It's not a hopeless problem to say how near or far a proof is – there's a reason they pay those research mathematicians so well – but the iron prison of the idol called Rigour lets no one free who hasn't really a complete and correct solution.

So when I grade prove-or-disprove problems on a final exam, what am I doing? Really my goal cannot be pedagogical. The students are permitted to look at their papers in the sequel but my understanding is that this is not typically done, and in truth I don't expect it to be otherwise. I sympathise with those who, coming to the end of a difficult course which they have not yet totally arranged into comfortable, familiar parlor room furniture in their head, would rather take a long, quiet, contemplative silence by the koi pond. So I have no belief that I am still teaching anyone anything.

Rather I am evaluating: who has it and who hasn't it? We need our grades to give a spread, to differentiate the students, and form them into some groups, the men and the boys and those in between, linear algebraically speaking. This does inform one's granting of partial credit. For example, it is very far from my mind that a simple response of “true” or “false” should beckon my pen hand to form anything other than a zero. (No doubt you have already guessed this, since it really beggars belief that only four people should have their coins come up tails, even if your name is Rosencrantz.) If there is no evidence that a proof could be in the offing, there is no reason to say that anything good has happened.

This problem of credit for the right single word has been a serious challenge to me: how could it be that any credit at all is earned by an utterly confused answer merely because one key word happened to be correct? Yet at the same time I want to encourage students to make guesses, develop their intuition, and so forth, even if they can't finish it off. After all, a good conjecture is the first step to a theorem. So on problem sets a one-word answer should worth a little bit of partial credit – typically one or two points from five, depending on the depth of the problem in question and the credibility of the notion that the student just didn't want to bother justifying their answer. (That unmathematical practice I strenuously discourage, even to economics students.) But to avoid giving the slightest encouraging word (or numeral) to errant nonsense I have developed what I have been privately (now publicly) calling the egregious weirdness doctrine. (I like to give names to these sorts of things, as you may have gathered from my discussion of the ultimate linear algebraic sin, but typically I'd rather terminologically paint myself in a more temporally-based judicial role.) This doctrine states that if I am considering a range of possible grades for a submission then if there is egregious weirdness present I will err to the low side. To clarify, some silly examples of egregious weirdness; pretend I've added parenthetically “(good grief!)” after every last one of them: claiming a matrix with a row of zeroes is invertible; writing down two vectors that are parallel after applying Gram-Schmidt; citing a theorem's converse despite having disproven it in, well, every single problem for the last two weeks; and so forth. They're the kinds of mistakes that are manifest and impossible for me to believe derive from anything but the coupling of linear algebraic ennui and the sort of deep confusion I called “the fog of the Nothing,” which “roam[s] between the sky and the space which is beyond, fantastic, undreambound,” in the first JoaLDG entry.

If the egregious weirdness doctrine explains how I can give credit for one-word answers on problem sets without seeming to punish people for just “writing a little bit more” (shouldn't it be a principle that writing a few more lines oughtn't worth an answer less credit?), perhaps its dual explains how I can't do the same on the final exam: only if there's something to praise can I say it worths something on the high end.

I just wish we hadn't ended up with a Putnamesque lack of hosannas on that problem.

* * * * *

Though it's the end of linear algebra for the school year, this is not, I should assure the gentle reader, the end of Journal of a Lower-Division Grader, nor the beginning of a hiatus. At my current rate, and at their tendency to bifurcate themselves into two and three parts, I have stories enough percolating on note cards to keep you and me over the summer.

JoaLDG: Artin on matrices

Preface. This entry is a first part of two planned about the mathematician Emil Artin. It's a little heavy on the math, (no surprise!), so let's heed in advance some advice we're about to quote and remember to pass gently over the oppressive parts without letting ourselves be burdened by their gravity. Listen to the music and not the song -- or was it the other way around? -- never mind, they're both light and airy.


Emil Artin (1892-1968) was one of our most formidable expositors of mathematics for mathematicians. To give just a most obvious and striking example of this talent, there is a reason why all introductory texts on Galois theory sound the same, and that reason is that they all borrow very, very heavily from Artin's book on the same. Artin on this subject was original: it was he who reformulated the work of Evariste Galois (1811-1832) from a theory of the symmetries of roots of polynomials into a theory of the symmetries of field extensions. Considering how this view now completely dominates it is a little surprising to learn that it was only so recently developed – Artin's book Galois theory was published in 1942, from his lecture notes, being fruit of work from the preceding years.

(But maybe not too strange. A long parenthetical digression giving context could be placed here. Let it suffice to say that even the notion of a quotient group was only formalised in the 1920's, and one can hardly state the “fundamental theorem of Galois theory” as we know it today without understanding group-theoretically what a normal subgroup is.)

I don't know what I'm doing in the fall but there's a certain chance I'll be teaching a section of this linear algebra class I've been grading – and I surely will be doing so sometime before I graduate – so some things Artin has written, and one passage in particular which I'll quote presently, have been a little on my mind. How do you tell people about linear algebra? At heart all I can answer is: Really, the same way as you do for anything else. Karl Jaspers thought that the problem of communication was one of the fundamental problems of philosophy. But we needn't feel abstractly pessimistic or overburdened: there are plenty of fundamental problems we manage willy-nilly to cope with every day. We have twenty thousand purely practical facts to draw on. And in this case, one of them is Artin's legacy.

Artin's book Geometric algebra is curiously organised: he deposits in the first chapter, prior to the main subjects of the book, all the external tools and apparatuses he'll need in the sequel. ("Curiously?" Well, normal people would put this in an appendix.) In the very thoughtful short preface labelled “Suggestions for the use of this book,” he explains:

The most important point to keep in mind is the fact that Chapter I should be used mainly as a reference chapter for the proofs of certain isolated algebraic theorems. These proofs have been collected so as not to interrupt the main line of thought in later chapters.

He goes on to say that “the inexperienced reader should start right away with Chapter II,” which to me reads like an agreement that Chapter I ought to be adjacent to the other cover. (Is he saying that the experienced reader shouldn't start right away with Chapter II?) He continues on, to give some of the best advice possible for reading mathematics, namely,

This skipping [of “a few harder algebraic theorems” in “a first reading”] is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal [!] to the reader. In most cases he will be able to go on and later on he may return to the parts which were skipped.*

Probably the students will object: there is hardly time for all this, to first skip and then to come back. Perhaps so. It is certainly unrealistic to think that the students will understand something that's unlike anything they've ever seen before in time to give clear and concise solutions on the weekly problem sets. But by the end of the class there should no longer be any mystery about the material of the first week, and the gap between the end of lectures and the beginning of the exam period (in Princeton's fall term, this is a gap of a whole month!) is enough time to start to put the entire course into perspective. On this time scale, the advice is not only reasonable, it is the only sound thing to do, if one operates according to the principle that no one ever learned a thing the first time he saw it. (Well, how could he?)

I want to convince you that Artin is super-cool. For that purpose there is at bottom only one thing to do: namely, show you that he's a rebel. A wild, wild rebel. Just thirteen pages into this book, not far into his appendix-at-the-beginning**, he has stated a theorem whose content is that when you fix a basis (of some vector space -- you're skipping that link, right?) there's a correspondence (“isomorphism,” in the vernacular) between linear transformations and matrices, (and change of the choice of basis corresponds to conjugation of matrices). He goes into a lamentation/screed for a page and a half, (emphasis added):

Mathematical education is still suffering from the enthusiasm which the discovery of this isomorphism has aroused. The result has been that geometry was eliminated and replaced by computations. Instead of the intuitive maps of a space preserving addition and multiplication by scalars (these maps have an immediate geometric meaning), matrices have been introduced. From the innumerable absurdities – from a pedagogical point of view – let me point out one example and contrast it with the direction description:

Matrix method: A product of a matrix A and a vector X (which is then an n-tuple of numbers) is defined; it is also a vector. Now the poor student has to swallow the following definition: A vector X is called an eigenvector if a number λ exists such that AX = λ X. Going through the formalism, the characteristic equation, one then ends up with theorems like: If a matrix A has n distinct eigenvalues, then a matrix S can be found such that S^-1AS is a diagonal matrix. The student will of course learn all this since he will fail the course if he does not.

Instead one should argue like this: Given a linear transformation f of the space V into itself, does there exist a line which is kept fixed by f? In order to include the eigenvalue 0 one should then modify the question by asking whether a line is mapped into itself. This means of course for a vector X spanning the line that f(X) = λ X. Having thus motivated the problem, the matrix A describing f will enter only for a moment for the actual computation of λ. It should disappear again. Then one proves all the customary theorems without ever talking of matrices and asks the question: Suppose we can find a basis of V which consists of eigenvectors; what does this imply for the geometric description of f? Well, the space is stretched in the various directions of the basis by factors which are the eigenvalues. Only then does one ask what this means for the description of f by a matrix in terms of this basis. We have obviously the diagonal form.

I should of course soften my reproach since books have appeared lately which stress this point of view so that improvements are to be expected.

It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out. Sometimes it cannot be done; sometimes a determinant must be computed.

He then re-enters the stream of the exposition. “Talking of determinants,” he says, “we assume that the reader is familiar with them.” And we're off.

But, by the by, and coming back to our concrete problem, is that prophecy correct, that future books will move toward Artin's geometrical view? I have a vast number of linear algebra books, and on inspection it's not so rosy. Some do exactly what Artin decries, without any apparent shame. Some even make linear transformations into second-class objects by introducing hideous notation for the matrix of a transformation with respect to such-and-such bases (not necessarily the same basis for the input as the output – good grief!). Some try to “motivate” the problem with differential equations, which from the point of view of an engineer may not be so ridiculous as it seems to us on the face of it (and anyway systems of first-order linear DE's are a classic application in such a course). Bretscher, the book they use here, is actually not so bad. It claims to emphasise geometry, and seems to do so pretty well, for the level of the class.

But there's the rub: Artin's approach is too difficult to teach to the students and still expect them to also master the matrix material they'll need to, you know, actually do some problems and not fail the course. In the end it is not a way to teach “something that's unlike anything [the students have] ever seen before,” because its emphasis on geometric character presupposes some geometric intuition to which one can appeal – in other words, some underlying familiarity not necessarily with linear algebra but, absent that, with some other and really more difficult mathematics. Bretscher's compromise, and it seems a reasonable one to me, is to give examples of matrices with special geometric meanings, transformations we've seen before (rotations, reflections, projections), and ask what their eigenvectors are. The students should be able to answer right from the geometry they already understand, without ever writing down a matrix, (although they could write it down if they wanted to, or felt the need to).

This balance between the conceptual and the formal the would-be instructor must maintain with care and deliberation. Maybe I won't stake my infant career on Artin's throwaway comments. I could just photocopy that page as a handout. If that handout wouldn't confuse anyone. But in that case I can always give it out to them, on my didactic authority if they don't feel comfortable judging it for themselves, that it's all right to skip it.


Endnotes.

* I am reminded of an English teacher from high school who wondered why it was, or how it came to be, that everyone thinks they should read a book by starting from page one and continuing to read page-by-page. Clearly there are many more ways to read the book, although most make no sense. Probably this habit is out of respect to the author, who presumably (though this belief is often well-characterised by the negative, skeptical connotation of "presumption") has put his industry and his learning into crafting a well-structured book. A reader inexperienced in some subject hasn't necessarily the knowledge to know what parts he needs to read to do whatever. But if all this is so it merely makes us wonder instead (1) why a well-structured book means a linearly-structured book; and (2) why more authors couldn't write such helpful "suggestions for the use of this book."

It is a custom, I should mention, in many corners of the textbook world to outline a couple of different options for the use of the book in a one- or two-semester class: cover these chapters but not those sections, and so forth.


** We need a good archaeologism, but for that one needs good Latin. “Precedix” is tempting, coming fairly directly from Latin “praecedere,” but doesn't carry quite the right meaning: it is “a thing coming before,” whereas Artin's appendix-at-the-front is more like elementary material. We could try “fundix,” from fundere, (cognate with to found: “found a city,” “a foundation,” and such), but it sounds ridiculous. Maybe “precedix” is better; after all, an appendix in English doesn't literally mean “a thing hanging on,” either.

JoaLDG: Negative Reinforcement (or, Sin and Redemption in Mathematics)

In his book Innumeracy John Allen Paulos, the veteran professor and exegete of mathematics and statistics, recounts a story about some pilots and their instructors.

This phenomenon[, regression to the mean,] leads to nonsense when people attribute [the regression] to some particular scientific law, rather than to the natural behaviour of any random quantity. If a beginning pilot makes a very good landing, it's likely that his next one will not be as impressive. Likewise, if his landing is very bumpy, then, by chance alone, his next one will likely be better. Psychologists Amos Tversky and Daniel Kahneman studied one such situation in which, after good landings, pilots were praised, whereas after bumpy landings they were berated. The flight instructors mistakenly attributed the pilots' deterioration to their praise of them, and likewise the pilots' improvement to their criticism; both, however, were simply regressions to the more likely mean performance. Because this dynamic is quite general, Tversky and Kahneman write, "behaviour is most likely to improve after punishment and to deteriorate after reward. Consequently, the human condition is such that ... one is most often rewarded for punishing others, and most often punished for rewarding them." It's not necessarily the human condition, I would hope, but a remediable innumeracy which results in this unfortunate tendency.

He goes on to give two paragraphs' worth of further examples about movie sequels, music albums, baseball players (what is it with Americans and baseball?), and stock markets, but the lesson for would-be pedagogues is clear: don't treat your subject with statistical rigour and suffer the fate of all pseudosciences that came before.

The contrary view alluded to in the first quoted sentence, namely, the ascription of intentionality and significance where there are only the capricious mechanisms of probability, is, I believe, a symptom of a whole another and different ontology of the physical world than the scientist/naturalist's, one which is surely mistaken. And it is one which is far too vast to take on in the screed of a single evening. So we'll consider ourselves admonished on two levels, namely with regard to our evaluation of our methods, and with regard to our concepts of praise and blame, and press on.

(By the way, every living person in the developed world, I kid you not, should read one of Paulos' books. It doesn't really matter which one, they're all more or less the same. Innumeracy is good and A Mathematician Reads the Newspaper is too. Go read!)

Not to put it in too maudlin terms, but I think I believe in positive reinforcement. As a very general principle this is connected to the Nietzschean optimism that (on our weblog's happier days, if I have succeeded) is our spirit here. (I keep saying that's our spirit, anyway.) If one "doubts with well-founded suspicion" that good things are possible, I want the courage to build a world, speaking literally or of the world as a metaphor for my ontology, which those good possibilities populate. As a principle of pedagogy it is maybe a reaction to something, some grousey grouch in my past, or maybe a recognition that the undergrads of today are the colleagues of tomorrow -- put another way, that the aims of a class are largely but not entirely about the students' command of the syllabus material and I am judged not merely by the standard of review for an educator-as-mechanism.

On the face of it, there's something very suspicious about negative reinforcement. I don't insist on being loved or liked -- I'll settle for respected -- and I don't repudiate Burrhus Skinner (entirely). It's just unclear how this negativity is going to get any desired result, particularly when many students are already very anxious about math, or about their academic position. What intermediate steps have failed that we need to take out this big club with its dangerous and indeterminate consequences? I fear that its advocacy comes from exasperation, but my perverse optimism tells me it could still lead to good things.

Because there's exasperation to be had. I read an article, "Teaching Freshmen to Learn Mathematics," whose author, Steve Zucker, (a professor of mathematics), took the following extreme (?) position. "We shouldn't," he argues, "overlook the power of negative reinforcement." He goes on to describe two mathematical errors often committed (in the past -- he tells us that their frequency has since dropped off) by his Calculus II students which he calls the "ultimate sin" and the "penultimate sin."

(They are, for the interested and mathematically inclined, respectively (a) the belief that a series, say the harmonic series, converges if its terms vanish; and (b) the computation of a limit of a sequence whose terms are given by some expression in n, like (1 + 1/n)ⁿ, by selectively letting instances of n go to infinity -- in this case, inside and then outside, concluding that the limit is 1, when, as everyone knows, it is in fact e.)

He says that the demonstration of these sins by his students argues some very dire things about the relationship of the student to the course and even to the instructor. (It's clear at any rate that the students don't understand what they've been taught if they make such errors, but naively we might wonder about the value of introducing eschatological language to describe these mistakes.) He informs his students that commission of the penultimate sin on any submitted work will immediately earn zero credit for the problem, and commission of the ultimate sin will earn negative credit.

I have to admit that I'm a little envious. I'm pretty sure I couldn't get away with that.

Anyway, since we're all here to hear about linear algebra, and not about calculus, which no one understands anyway, I have a couple of candidates for confusions that drive me up the wall. Following precendent, I would like to propose, as an "opening bid," the:

• Penultimate linear algebraic sin -- phrases like "the vector is linearly independent" or "the vector is linearly dependent." I would also like to take on the concept of a "redundant vector," which to my view is simply pedagogical lemonade encouraging a nonexistent intentionality, but them lemons are in the textbook, with a footnote about how "redundant vector" isn't actually a linear algebraic concept but the author has found it "useful" in teaching, if you can believe that, so I'll save that one for another post.


• Ultimate linear algebraic sin -- any sentence of general type, "a basis for the kernel of matrix A (or another space) is the span of such-and-such vectors."

I'm not wedded by iron chains to these as my two least favourite things to read on students' papers, but it's hard to take a shot at things like co-ordinate chauvinism or implicit inner products which are, among other things, too far advanced errors to strenuously indict. (We'll see if I still feel the same way about co-ordinate chauvinism in the next two weeks, when eigenvectors and similarity of matrices are the topics of discussion.) The two cited transgressions show, roughly, that the student is confused about the relationship between vectors and vector spaces: the role of linear combinations in general and their significance to the concepts of independent sets and spanning sets (and bases) in particular.

And besides, these not-silly-mistakes (as Zucker would put it) -- especially the ultimate linear algebraic sin -- show up with depressing regularity. So I think I can summon a roughly analogous frustration to our poor calculus instructors -- though, mind you, I'm just grading these papers, not teaching the class. So how do I feel now about negative reinforcement?

I'm giving it a try. As I said I don't think I could get away with negative credit for problems, and anyway it would be a completely unfounded move for someone who doesn't even have any interaction with the students to tell them why. I mean, I can't even explain to them verbally what's wrong with what they've written, but commission of the ultimate linear algebraic sin is cause for loss of two points from five, no matter what the problem was about, and no appeals, damn it. It's not right for this nonsense to get written and for me to say nothing: that doesn't do anyone at all any favour.

How's it working? It's a little tough to evaluate, as I said, not being the instructor. I don't recall reading an instance of that sin lately. It drives my blood pressure up a notch to see so I'm pretty sure I'd remember if it had been there -- and, hey, I have been privately calling it the ultimate linear algebraic sin for a couple months now. Forgetting that would be like forgetting another fall of man. On the other hand, the students haven't been asked too many times to give a basis for the kernel of some matrix, and when they do, now that I think about it, they've been saying things like "the kernel of A is the span of such-and-such vectors," and not even using the word basis, as though they'd like for me to draw an inference, just answered like they would have in the first weeks of the class.

I...

I think my plan may have had some unintended consequences. Like apostasy.

Negative reinforcement, huh --

At this stage I'd like to remind everyone that it's not my fault, I didn't do anything, it's just regression to the mean. Somehow.

I'm going to abruptly end this post on the pretext that it's already long enough and, umm, further data must be collected to continue the discussion.

Journal of a Lower-Division Grader: Introduction (and if you warned me,...)

"I despise your despising," spoke Zarathustra. "And if you warned me, why did you not warn yourself?""

There's something very dubious about work-related blogging. It seems to get people into trouble (cf. brief remarks here under "the end of credibility," or elsewhere at more length and erudition), for one thing. But more than that, and in its pervasity worse still, it seems all-too-often both unprofessional and morosely self-indulgent. How often do we read only glib snark cooked up from the catalogue of the daily travail? Endow it with vices and malfeasance beyond its station, and speak of it with the dryly clownish condescension of sufferance yielding to exasperation! But sad occasions are only sadder still when they command the service of powerful prose.

How often have I jotted notes on nothing-in-particular only later, a day, two days later, to find that I no longer had any interest in the subject? It is not that I am fickle or capricious, though perhaps I am. It's that the fundamental cost of all art, all media is viewer attention. The literature of forgettable pettiness only worths its price when it seasons itself with bile and hyperbole. It's the thing-of-the-moment that mistakenly universalises itself in a search for absent significance when it in truth has no such power. It's Guildenstern stabbing the Player King.

And it's really bad writing. Maybe not bad prose, maybe often not, but badly misspent. I think we ought to command better from ourselves. Call that a re-statement of our mission at Sun-Drunken.

If you remember the last time, or the time before that, or --, that I went on a tear immediately prior to doing, or while doing, exactly what I'd just railed against, I'm sure you know what comes next. It's not just self-conscious irony, the plainest special pleading, a seeking to convince the reader that really nothing wicked passes here by sending him through loops and whirls until he's too dizzy to figure out whether it's satirical or just crass. It was never the despising that was at fault. Returns are still being earned when Guildenstern furnishes his "art to creating suspense" with a "law of diminishing returns."

Here, however, Zarathustra interrupted the foaming fool and put his hand over the fool's mouth. "Stop at last!" cried Zarathustra; "your speech and your manner have long nauseated me. Why did you live near the swamps so long, until you yourself have become a frog and a toad?... Why have you not gone into the woods? Or to plow the soil? I despise your despising; and if you warned me, why did you not warn yourself?

"Out of love alone shall my despising and my warning bird fly up, not out of the swamp.

"... What was it that first made you grunt? That nobody flattered you sufficiently; you sat down to this filth so as to have reason to grunt much -- to have reason for much revenge....

"But your fool's words injure me, even where you are right. And even if Zarathustra's words were a thousand times right, still you would always do wrong with my words."

Anyway, when the preface puts the body of text into the smallest corner of its shadow -- it could be bad writing, or it could be the preface was the point.

* * * * *

Way-back-when at UBC I took a class in Euclidean geometry. After the midterm a friend told me a sad story about the grading of one of the problems. It had asked for a statement of some theorem or definition or thing like this. As always in Euclidean geometry a good diagram was central to the issue, and she had included a helpful exegetical diagram. Alas, the instructor had not found it so helpful.

"I didn't label it a circle," she explained to me at some length. "That's why he took points off. You tell me, what else could it be?"

Obviously I couldn't say. I'd labelled mine.

"He said, 'Maybe you thought it was an ellipse'! What? What?! When in this class have we talked about ellipses?"

Well, it wasn't the most circle-like circle. They're not the easiest thing to draw.

"When has Instructor-Man even mentioned the word 'ellipse'?!"

Never, of course; if you have neither co-ordinate geometry nor solid Euclidean geometry at your disposal, and this was a strictly plane-geometric class, it's a bother even to say what an ellipse is.

So she went to work on him again, but it didn't help. He was the kind of person who helps earn a reputation for eccentricity for mathematicians. (It's a math pun; click the link to convince yourself it might possibly be funny.)

In truth, although I wished I could back up the professor's take, I really couldn't figure it out. I dislike unlabelled diagrams as much as the next guy, but this ellipse business couldn't be a serious criticism.

I have since recanted my naivete. A term or two grading papers has given me a somewhat more storied perspective. My understandings about what other people might mean when they speak what they say or write what they scribble are no longer Earthbound -- heavy, constrained by the gravity of consequence. Rather they roam between the sky and the space which is beyond, fantastic, undreambound. It's the fog of the Nothing, of the confusion-of-being, of the not-understanding and the not-communicating. It's sadness on a page.

And that's why I couldn't give examples. It would take a callous knowledge-worker not to empathise. Or at least it would if the clever critters didn't try their damn'dest to dress it up. And yet, heavens guard us, maybe now and then I've only rouged it -- I am on a deadline.