Archive for the ‘Textbooks’ Category

somebody up there likes me

my three top course requests
were offered to me a couple
hours ago. sure, i’d rather
be the lecturer for any *one*
of ’em than grade for all three.
but i look to learn a lot of math
and get paid doing it, so what
the heck.

when i took the course as a student,
3rd edition (and didn’t cost two hundred
american dollars as it now appears to do;
the copy you see here belongs to the
math department of course). the beat-
-up old doorstop (_probability_) i’ve
never seen before. i didn’t even know
i’ll draw on some of the much-more-than-
-ample whitespace… but that won’t
excuse it. awake, awake, U.S.~ignobility.
solomon’s lecture-note packet promises
to be outstanding. i’ve worked with the
second-semester chapters (in a separate
in-house packet) and was way impressed.
also it doesn’t break anybody’s budget
that might want to actually, you know,
*own* the doggone thing.

getting back to work.

rejected repeatedly by “blogger”

written as a comment to
sue v.’s
are (k-12) math textbooks getting worse?
(with particular reference to its link to
annie keegan’s
afraid of your child’s math textbook?
you should be.
)

google trimmed the top off in the preview.
they’ve rejected me for going on too long before
so i tried editing it into two comments. but
then i kept getting the “doesn’t match the
prove-you’re-not-a-robot-gizmo” message regardless
of how careful i was. to hell with it.
i’m never playing *that* game again…

annie keegan’s piece.
math texts have mostly
been worse-than-useless
for some time but the
deterioration continues…
and somehow the process
continues to fascinate me.
good to have another “inside”
source.

the elsevier boycott is a whole
different can of worms… *this*
giant might just get brought down
by the faculty. the motherlode
is this astonishingly-well-researched
journal publishing reform page.

one has about as much hope of fixing
the textbook industry in the large
as of shutting down the war machine
or getting USA on single-payer health…
it would require a *radical* reform
of our entire system of goverment-
-by-global-capital (and probably
wouldn’t be pleasant… cf the french
and russian revolutions and their
subsequent reigns of terror…).

so i’ve given up hope for it
the fact that the rotting corpse
of textbook math-ed *does*
continue to fascinate me…).
is likely and that right early
(i just don’t *hope* for it).

any survivors of the whole ordeal
that for some reason want to create
some math textbooks would do well
to dig up some shaum’s outlines
and dover books and whatnot:
much cheaper and much more useful
than contemporary USA eyecandy-
-weighing-a-metric-ton doorstops.
they’re available even now to faculty having
the power actually to *choose* their texts…

what might one do with this?

waycool video of pendulums by R.~S.~Hilton. spotted by androo.

meanwhile: dy/dan got a sweet publishing deal somewhere along the line.

you could look it up

i didn’t swipe this

when i drew this

… but of course the *idea* has been around for centuries.
i did *learn* a lot from _the_fundamental_concepts_of_geometry_
(bruce e. meserve). it’s a cheap dover reprint
and a masterpiece of clear exposition.
*everybody* oughta have one of these…
just like a bible, a shakespeare, and a dictionary. i’ll bet
the matching algebra book is good too… but i’ve got
*lots* of good algebra books.

Section 5.5: A Manifesto

The post is even more of a mess than usual. That “does not parse” parsed yesterday. I had to cut a piece out (you’ll find the hole) because it was acting downright weird for no apparent reason. Welcome to WordPress.

The text is even more of a mess than usual. Evidently certain forces have led its creators (“the Redactor”—an entity whose exact nature is very ill-understood [and for all I know, incomprehensible], but that we can imagine as a sequence of corporate committees— and “the Author” [typically also, from what I have been able to ascertain, a committee]) to create a display called Steps for Finding the Real Zeroes of a Polynomial Function. And thus far, we are of course in full, sweet, agreement: this is the holy grail of Algebra and as such, one of the most interesting subjects there is or ever could be in Life Itself; let such steps be i-cast on every cel (from the rooftops)… or whatever the kids say these days. (“Factor it if you know how. If it’s a constant, you’re done. If it’s linear, use subtractions and divisions to isolate the variable. The Quadratic Formula tells the whole story in the quadratic case. The cubic presents special difficulties. So first use a change of variable, if necessary, to….“—instead of, say, “buy! buy! buy!“). But now look what they’ve done to the beautiful face of this Alma Mater of problems.

Step 1: Use the degree of the polynomial to determine the maximum number of zeros.
Step 2: If the polynomial has integer coefficients, use the Rational Zeros Theorem to identify those rational numbers that potentially can be zeros.
Step 3: Using a graphing utility, graph the polynomial function.
Step 4: (a) Use eVALUEate, substitution, synthetic division, or long division to test a potential rational zero based on the graph.
(b) Each time that a zero (and thus a factor) is found, repeat Step 4 on the depressed equation. In attempting to find the zeros, remember to use (if possible) the factoring techniques that you already know (special products, factoring by grouping, and so on).
Why not skip the Rational Zeros Theorem altogether? Or, not that I’m proposing to do it here at Home Campus Community College, omit the calculator?

I’d probably love to do a “drill-and-kill” version of the course (where I have of course used the industry code for “those who establish a routine of doing lots of routine exercises, set by the instructor, should flatten the exams”)… but it’s just not an option, not with this many topics on the schedule (and us, mea culpa, so far behind it): a lot of teachers really like this “synthetic division” thing and there’s a pretty obvious reason: if you’re gonna crank out dozens of divisions-by-monic-linears, this is your tool.

In such a course, one would—naturally—ban calculators (and check by-hand homeworks for completeness, and much else besides). Certain Computer Gods of Texas have made certain unholy alliances with the local Management Gods to decree that ours shall be a calculator-driven version. In this context, I’m even ready to pretend to accept this decree: this very section is, for me, the first really essential use of the doggone graphers in the whole 102-103-104-148 sequence. There’s no time for lots of polynomial divisions, that’s for sure… so we’ll only do divisions if we have to… which means we won’t use ’em to find zeros (R‘s,say [“roots”])… but will use ’em to “divide away” the corresponding factors ((x-R)‘s, say).

The mostly-unspoken absurdity here is of course that, once you’ve decided to use a computer, why should you limit yourself to one of these expensive handhelds that do very few things compared to more modern electronica (and mostly do those badly)? I dropped a link to a free polynomial-factoring page into my homepage recently; any goodsize class will include some students who can access such programs on their telephones. Why should the line for “what computations the human should do” be drawn at “what such-and-such no-bid-contracting Behemoth decides they can sell”? But as I say, I’m pretending to accept this state of affairs (in order to speak to other issues).

Returning to the text. The “human computation” version should omit Step 3, together with the reference to “eVALUEate” (which, besides being twee, is bad calculator advice: one of course actually uses TRACE here). And then you just put in a “calculator” version saying what to do if you’re using a graphing calculator (which by the way is not a fucking “graphing utility” [utilities are programs not hardware except in edu-babble]). Instead of this neither-fish-nor-fowl thing that nobody will ever do (that isn’t crazy, or stupid, or, what is obviously the most likely case, simply following orders).

Steps 1 and 2 I’ll accept as they stand. Note that the Rational Zeros Theorem can be considered part of the “calculator” version of the process (RZT gives us a bound on rational zeros and a darn good set of hints as to where to “guess-and-test” [between integer values as observed on the grapher, say]; this avoids using intersect [or, worse, root] to determine certain rational values).

Step 3 speaks for itself; put it in the one version, out of the other.

In Step 4 we’ll find most of the trouble, then. So here’s a scholarly crux right off the bat: VME (to [selfindulgently] use “impersonal third person authorial” for a moment) is here using the Fourth Edition while knowing full well that Step 4 has actually been changed in the Fifth. But, a poor workbeing blames its tools, the Fifth is downtown in the office, whereas Fourths, their cash value having fallen suddenly to nothing a short time ago, are promiscuously littered about in various remote VME locations.

This much is known to me of the new Step 4 as of now: they made it worse. Because now it has the nerve actually to say “Use the Factor Theorem to determine if the potential rational zero is a zero”, when, goddamnit, the Factor Theorem has precisely nothing to tell us about whether a rational number is a zero until we already have the factored form—which is essentially the problem we’re supposed to be trying to solve. The Redactor has swallowed its own philosophical tail here and entered some new dimension of incomprehensibility.

Returning to the edition at hand, then:

In (a) I’ve complained of the calculator slang already; separating the p-and-p (paper-and-pencil, natch) methods from the FGC (graphing calculator) methods.

In (b) we have another of the Redactor’s masterpieces: we are told to “repeat Step 4 on the depressed equation”. But the depressed equation is available to us only if we have used a p-and-p method in step 4a. The calculator version here requires an explicit declaration to the effect: “Use the root to depress the equation (by either division algorithm)”. Anyway, the depressed equation appears to have popped out of the thin air here: in part (a) it hasn’t been mentioned even as a side-effect of the “test a zero” process. And yet this process is the very heart of the matter: in practical terms, it’s very much what this section is about. (Where, to be perfectly explicit, by “practical terms”, I’m referring to “terms of ‘how do you do the exercises?’ “.)

Speaking of which. Does anybody here not lay out all the possible p‘s and q‘s out across the top and the LHS of a table to form all of the “potential roots” supplied by the Rational Zeros Theorem (RZT)? Because if you’re in a big hurry and there’s this overwhelming amount of stuff that you’re given to cover in a couple meetings that oughta probably take months, this is an exercise you can essentially train students to do, in a pretty short time, and I’d never dream of doing this other than by laying out a table … oughtn’t that be in the book somewhere?

Even the statement of RZT resists comprehension (as I guess… it’s clear enough to me…): “${p\over q}$, in lowest terms, is a rational zero of f, then p must be a factor of $a_0$ and q must be a factor of $a_n$” is perfectly clear in its context; don’t let anybody tell you any different. (In particular, $a_0$ and $a_n$ have been displayed with their usual meanings right there in the statement of the theorem, as good taste requires.) But one should darn well put it in words as well, as if people are actually going to talk about it: “the numerator (of the zero) divides the constant term (of the polynomial)”.

“Numerator divides constant” is much more memorable to at least some minds than “pee divides a-naught”, and is anyway more meaningful (since, in another context, my own lectures for example, one may have, say $A_i$‘s in place of the $a_i$‘s or $n\over d$ in place of $p \over q$). That the verbal translation of a formula should appear somewhere near its display looks like a simple corrollary of, what is taken by at least some people as a basic principle for Math Ed, the “Rule of Three” (or of “Four”).

I’ll go ahead and add that “p must be a factor of a_0, and q must be a factor of a_n” gets old pretty fast when you’re writing on the board (or notebook or what have you); one soon discovers a crying need for some such symbolism as the (completely standard and easily understood) $p | a_0$ and $q | a_n$; moreover, we’ll be able to use this notation quite a bit in other contexts (like
$f(R) = 0 \Rightarrow (x-R) | f$
this is of course the statement of the Factor Theorem (as it appears, not in the book, but in The Book).

One more thing here. Students will of course take $p | A_0$ and $q|A_n$ as facts to be memorized. And some will inevitably mix them up: it was for situations of exactly this type that the phrase “minding one’s p‘s and q‘s” must have been coined. But of course, by merely contemplating, say 2x – 3 = 0 for a few seconds, one easily reminds oneself of what’s going on: $3\over 2$ is the root … it must be the numerator that divides the constant… (and so on). We darn well have to keep showing students how to do this kind of thing (use small examples to remind oneself of the details of big generalities). Whenever there’s some easy way to avoid memorizing something, we should at least mention it.

I wouldn’t mind so much—I kind of like being the “good guy” who gets to come in and say, “You know that passage of the text that doesn’t make any sense? Well, what they’re trying to tell you is this…”—but I’ve got reasons to believe that some of my colleagues are even more clueless than I am about stuff like this; it’s safer to just put it into the text in the first place.

The theorem of the very first display of the section—mysteriously called an “algorithm” there—is that polynomials “divide like natural numbers” … a fact summarized in the equation f/g = q + r/g.

I’ll remark here on the fly that $(\exists q, r) (\forall f,g (g\not=0)), \delta(r)<\delta(g)$ should precede this equation (in some dialect—I hope it’s obvious that I don’t dare indulge in such straight-up set-theory with my live audiences… in part because one would also need to make it clear somewhere that $f, g, r, q, 0 \in {\Bbb R}[x]$ and that the inequality under the $\forall$ quantifier states that g is not the zero polynomial) and that $\delta$ here represents “degree”. [There’s a lost passage here that caused WordPress to freak out utterly and set the whole page wrong. The Tex code parsed OK, but then … blooey. It wasn’t essential. Just me playing around with the degree function.]

But that was just me playing around. The fact that quotients and remainders can be computed (for ordered pairs of polynomial functions) deserves such prominent placement. It also deserves the name of a theorem; “the Division Algorithm”, rightly so-called, is the process defined in the proof of the theorem (and used in actually computing the polynomials r and q—we’re speaking of a constructive proof). What does not deserve such prominent placement is the next thing: the dreaded Remainder Theorem (RT).

Not in my course anyway. RT is hard to understand (of course I can’t prove this … but I can say that I’m pretty sure I didn’t understand it until about Abstract Algebra or so …) and is used only in proving the “hard” direction of the Factor Theorem (by us; those ever-so-fortunate p-and-p classes use it a bunch; I’m guessing here). Moreover, the authors have just gotten through admitting that they stated The Theorem Called “Algorithm” without proof; this theorem is of course quoted in the proof of RT (so it ain’t much of a proof at that).

And I walked into a trap here and caused myself to deflate right out in front of a class when I suddenly realized I wasn’t willing to try to really explain—I mean “explain so as to be understood” (with all the necessary give-and-take)— what was going on with this part of this section (and so I oughtn’t to have brought it up in the blackboard notes at all): you can lose a lot of hard-won trust in a moment flat by just giving up.

The Theorem for Bounds on Zero is omitted campuswide; good. I’ll go ahead and mention that this omission sort of hints that the creators of the local version of the course are aware that this might not really be the text we should use. While I’m at it, they’ve also changed the order of the sections in this Chapter. This might very well be contributing to my difficulties. If there can be said to be an intended audience for this treatment, then that audience will have had more experience in graphing rational functions before getting here (and so would have seen lots more examples of the Factor Theorem at work before its statement here, for example).

As much as I’ve been complaining about the text, I ought to make it clear that what I’m really fighting is the lack of time to talk about it. There’s enough here for a whole ten-week course as far as I’m concerned (and I’d love to teach that course, with students just like the ones I’ve got now). Meanwhile, there’s this completely demented parody of an industry standard to the effect that “This is College! It’s supposed to be hard! Let ’em learn how to study!” and so on. This is far from a majority opinion in most departments according to my wild guess. But when the committees start making up the rules, all of a sudden those with this opinion speak up plenty loud and nobody wants to appear like the weakling. (“Well, my students seem to need about twice the time on this topic than what’s allotted” can very easily be twisted into “I don’t know how to teach this stuff properly”, so it’s just easier to keep your mouth shut. And another invisible 800-pound gorilla is born.)

And then, and this is the most frustrating phenomenon of all, you get together in the group-office-cum-teacher’s-lounge and all anybody ever wants to complain about is the students. “They keep wanting to do this, no matter how I tell ’em to do that!”—and I keep trying to change the subject to “We’re telling ’em to do that, in the wrong way!”

Because once new students are seen to make the same old mistakes, that’s information: knowing the most likely mistakes tells us where to put up the warning signs (even Bourbaki, whose indifference to pedagogy was legendary, did this). The fault, dear colleagues, is not in our students but in ourselves.

So why do I always feel like I’m the only one complaining about textbooks and syllabi and stuff that’s actually somewhat under the control of people right here in our department (instead of the lack of math maturity found in math students, which is not)? OK. Rhetorical question. Because disrespect for the helpless is free, but fighting the power is dangerous, is why. To which I can only say, sure. But at least it’s interesting

It’s Not The Cheat, It’s The Futility

The transformations section of the text begins, to my predictable chagrin, with a graphing-calculator “Exploration”: adding (or subtracting) a constant value at the end of a function (in the Function Editor [i.e., the “Y=” screen] of the grapher) produces the by-now familiar (to any student in regular attendance) vertical shift. “We are led to the following conclusion:

If a real number k is added to the right side of a function y = f(x), the graph of the new function y = f(x) + k is the graph of f shifted vertically up k units (if $k > 0$) or down |k| units (if $k < 0$)."
And this is about as clear as can be expected: this stuff is hard. But, doggone it, we’re adding to the right side of an equation, aren’t we. And shouldn’t we take a hint from the doggone grapher and refer to $y_1$ and $y_2$ (the authors go on, in effect, to call $y_1$ by the name f—but what’s the name for the new graph?)?

As for the cases on the sign of k (and the use of absolute value), well, it’s not the way I’d handle it but it’s not obviously worse. I usually say something to the effect that the graph of [the equation] y = f(x) + h is obtained by shifting the graph of y = f(x) “up” by k units with the understanding that “shifting up by a negative number” is interpreted—in a very familiar way—as shifting down. It can be helpful to mention, say “y = Mx + B” here—the point being that we don’t need to know the sign of B for the formula to be valid. (Or the words: for the case of y = f(x-h) we add h to each x co-ordinate of the graph—regardless of the sign of h.)

Again, it’s not obvious that treating the “up” and “down” cases explicitly right there in the display that (effectively) defines “vertical shifting” was a bad idea … but I believe it was a bad idea (sort of): anyway, by the time a student reaches the Conic Sections stuff (next quarter) some of the “formula” displays have become much more complicated than they have any good reason to be (and goodness knows, Conics are hard enough already). The sign issues have to be discussed somewhere; using inequalites and absolute values to do it is probably very much the right idea … what I’m after here is that the definitions be definitions. Which would mean, first of all, identifying ’em as such; and then, as concise as you can be while getting the job done.

It would appear that somebody made an editorial decision not to discuss the kind of “understanding”s I mentioned a moment ago; the results are disastrous. This is particularly true for students inclined to the so-called “rote memory” strategy (learn the formulas by heart before even beginning to work at “seeing the big picture”). Of course, most of us probably consider it something of a duty to try to talk such students out of relying heavily on this strategy … but this hardly seems like the right way to do it. And yes, there sure does seem to be quite a bit of resistance (anyway at the level just below 148) to the idea that, say, “dividing by a number is just multiplying by its reciprocal”—this comes across as “mumblemumble” to many a 102 student (for example).

I call it “teacher talk”: the student somehow knows for sure that this technical stuff you’re always so careful about saying cannot possibly have anything to do with their existing ideas about how to solve equations; they’ll move heaven and earth to try to find some list of “rules” they can memorize if only they can avoid using the word “reciprocal” (or, of course, what’s worse, “multiplicative inverse”). And if you’d only stop trying to pretend any of this means anything and just tell them what to do, the scales would probably fall from their eyes in a minute flat. Of course it’s damn-right a duty to try to disabuse these poor souls of their misapprehensions as to the nature of our art. And this duty doesn’t fall on me alone.

Overlooking the power of the symbols until we end up with different formulas for, say, ellipses with vertical and horizontal major axes feels to me like pandering to the worst instincts of our weakest students and, I keep having to apologize (“Well, if they’d done this right, it’d be much easier … well, look: forget it. Here’s what they want you to do …”).

So. Let’s try and make this very difficult task as easy as it can (reasonably) be. Thinking of, say “stretching” and “compressing” graphs as two aspects of the same process (requiring only a single formula), is just a flat out good idea: the notation is beginning to do some of our thinking for us. It’s not quite the Heart of the Matter—an awful lot of people have learned about Transformations and Conics (for example) with the kind of overly-detailed treatment of cases I’m complaining of here, after all. But for me it’s pretty close. And, believe it or not, I don’t necessarily want to do constant battle with textbooks.

Big Day For Former Math Professor

Turns out you can create PDF files with the Xerography device down the hall. Here at long last are 67 pages of Numbers, Sets, and Logic by yours truly. You couldn’t really print it all out and use it with your own classes—yet; it’s got scribblings by me from classroom use on some early pages and breaks off in the middle. For that matter, the drawings are exactly as ugly as my blackboard work. But I’ve wanted to put some version of this up on the web for years (without having to reformat everything by hand) and am thrilled to have suddenly broken through to here. I posted re-formatted versions of section 1.1 and 1.2 in this very blog back in July.

Conspiracy Theory

The text for my 148 class says

A relation is a correspondence between sets.
The boldface type (implicitly) means that this is given as a definition. But of course this won’t do: what’s a “correspondence”, after all? To go on to say that “If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that y depends on x, and we write $x\rightarrow y$” isn’t helpful. In fact, it proves very clearly that our authors don’t believe their own words: if a “relation” is a correspondence between sets, then how the Heck can a “relation” exist between x and y (which are [again implicitly] numbers, not sets)?

There’s more—a lot more—to complain of even in the two sentences I’ve copied out here (and it stays this bad for at least a few paragraphs), but let me leave my analysis at that for a moment.

Actually, of course, a relation is a set of ordered pairs. Our authors know this of course. They’ve even tried to tell us so: an earlier edition had “A relation is a correspondence between two variables, say, x and y, and can be written as a set of ordered pairs (x, y).” … but this was evidently too clear for somebody at Prentice-Hall, so now we’ve got the handwaving.

“Oh, this is hard to understand”, we can imagine somebody saying … “so let’s make it impossible to understand instead.”. Somebody out there prefers imprecise “definitions” to the real ones. I’m convinced that they don’t want students to know how simple the truth can be: math is supposed to be hard for these villains.

Why Live

It so happens that $x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1)$. And this, all by itself, far from being a cause for despair, is kind of a cool thing to know. What depresses me more than I can safely say is that “factoring trinomials” is a very big deal around here (I blog out of the office even during breaks because I don’t have the social skills needed to establish an internet connection at home) and that this particular equation very clearly has no place in our curriculum.

So. What then are we to do? Well, first of all, obviously, verify it (take nothing on authority in mathematics). Maybe you’ll notice that the RHS (“right hand side”, natch) can be parsed as $((x^2+1) + x)((x^2+1) - x)$ ; for me this was the easiest way to check (because the Special Product Formulas of our 103 class are second nature to me; I can do the expansion to $(x^2 + 1)^2 - x^2$ [a “difference of squares”] mentally and mentally use the “square of a binomial” formula to expand further; one $x^2$ “cancel”s; done). Or not; it’s fun to watch six terms of the longer expansion wipe themselves out. Then what? Well, then ask, “how could I have found this out if I didn’t know it already?”.

And the awful truth there is: Google it. This online rootfinder was pretty easy to find and gave back our result more-or-less correctly (I’m quibbling with some notations and suchlike conventions here; nonexperts needn’t be troubled by my weaselwordage). “Google it” is the right answer to a lot of questions!

No, really. Stick with me here. Lord Satan appears to you and says “Factor x^4+x^2+1 over the integers and I will spare your tribe; fail and I will annihilate you all without appeal.”. What do you do? Well, if you’re good in algebra, you solve it (and check it very carefully); if not, well, if you’ve got any sense, you get up on the internets and find a page that’ll do it for you. The last thing you’d do (if you’ve got any sense) is take our 103 course or anything like it: not only won’t they tell you how to find this out, they’ll probably leave you with the impression that you know for sure that x^4+x^2+1 is prime or something; alas, your very teacher will probably think it’s prime.

So what are we spending so much time and money for, trying to teach these poor devils to do badly what an easily-accessed robot slave can do well? And, assuming for the moment that there are good reasons for doing so, is it at least possible to be honest about it?

Because, check it out: if we rewrite the question as “How could I have found this out without a computer?”, we’ll encounter some interesting math, along with some insights into the Remediation Congame that has replaced Mathematics Education rightly-so-called with its putrefying corpse.

First of all, notice that Lord Satan said we had to factor our polynomial over the integers. But if he’d said “over C” (i.e., over $\Bbb C$; this typesetting interface is free but otherwise far from perfect), one would (do I get to say “of course” here?) have a different answer: $x^4 + x^2 + 1 =(x-\zeta)(x-\zeta^2)(x-\zeta^4)(x-\zeta^5)$, where $\zeta = e^{{\pi i}\over 3}$. My point here, for now, is simply that the domain over which we are to factor a given polynomial matters: it’s part of the problem and needs to be part of the problem statement (again, assuming we’re interested in trying to be honest). I’m not proposing here that 103 students need to learn about the complex number field before learning about factoring polynomials (and neither do I say that that it’s necessarily a bad idea for them to do so); one could point out that x^2 – 2 is prime over the rationals but $x^2 -2 = (x+\sqrt2)(x-\sqrt2)$ over the reals (and, indeed, seeing that our texts and courses seem determined not to make this point, one is led to wonder how—if at all—they expect to be taken seriously in making such a fuss about the distinction between the reals and the rationals in an earlier part of the course … every chance to actually use the distinction having apparently been rigorously suppressed).

OK. Once we do allow ourselves to mention C, we can make a great deal of progress. It’s probably not wildly optimistic to hope that the median-level students of a 104 class could use the quadratic formula to write down the four complex solutions to $x^4 + x^2 + 1 = 0$, namely $x = \pm \sqrt{{-1\pm\sqrt{-3}}\over2}$. It probably is wildly optimistic to hope that they could then write down the four roots separately. I’m not kidding. A few “A” students might be able to produce the equation
$x^4+x^2+1 =$
$(x +\sqrt{{-1+\sqrt{-3}}\over2})(x-\sqrt{{-1+\sqrt{-3}}\over2})(x+\sqrt{{-1-\sqrt{-3}}\over2})(x-\sqrt{{-1-\sqrt{-3}}\over2})$ (which is of course equivalent to the C-factorization I gave earlier). A student that could produce this equation with no other instruction from me than “Factor x^4+x^2+1 over C” would gladden my heart more than I can say even though I don’t approve of square-roots-of-negatives written without the “plus-minus” sign. Bitter experience convinces me that most 104 students, even given the “x=” solution I provided above, couldn’t produce the factored-form equation with me right beside ’em telling ’em how to do it. It’s like they feel themselves physically incapable of writing down anything so messily algebraic. Not that I blame ’em (much); you’ve gotta walk before you run.

OK. But a 148 student could probably do it. Still couldn’t multiply out the result and check it, though: for that we’ll need a better notation. So let $\zeta = {{1+i\sqrt3}\over2}$ and take it from there: $\zeta^6 = 1$, $\zeta + \zeta^5 = 1$, and so on (inscribing a hexagon in the unit circle is helpful here). But we sure as heck don’t do this in 148 around here; it’d take a week. And nobody cares. Not the students, not the publishers, not the department, nobody. Just me.

What am I even supposed to be doing here? Does everybody else involved actually want me to get up in front of the room and say, “Look, in real life you’re going to let the computer take over whenever anything gets at all complicated, so never mind trying to understand the big picture: that’s for the experts anyway, and the creators of this course don’t want you to learn to think like an expert …”? Because I don’t need this god-damn job that bad. I can beg on the street.

Wheee!

I’ve met each class twice with no major snafus.

I didn’t have the book for 130 until after class yesterday so I did the “Linear Inequalites” section (1.2) before the “Applications of Equations” (1.1)—it’s a whole lot easier making up inequalities off the top of my head (that’re essentially the same as the exercises) than it’d be to make up “applications” (i.e., word problems). Now that I’ve got the book, I discover that there’s a boxed-off “definition” to this effect.

An inequality is a statement that one number is less than another number.
This isn’t just wrong or even just obviously wrong: the silly son-of-a-bitches know it’s wrong: the line before this box says “The next definition is stated in terms of the less-than relation (<), but it applies also to the other relations (>, ≤, ≥).”.

How is this anything but worse-than-useless to anyone? I’m more or less convinced as of right now that I’d’ve given a worse lecture yesterday if I had’ve had the book—I’d very likely have felt obliged to do some version of this rant right there and students are already pretty weirded-out on the first day of class without the instructor going off on some random angry tirade.

It may be even more to the point to mention that the “Rules for Inequalities” that follow could hardly be presented in a more confusing style: the heart of the matter (“change direction when you change signs”) is given using esoteric terminology (and also in code) and appears third in a list of Rules mostly having nothing to do with working out the actual exercises. The first thing I’d do if I had a free hand is throw out this book.

• (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)