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


  1. brutus12

    nice post and good lecture today! the steps were clear and understandable. The divsion can be confusing but when i write it clear i can do it.

    as for the whole staff blaming the students, i wish more staff felt the same way. As a salesman (self-proclaimed) if i dont sell a company pizza boxes, its my fault. alot of guys give alot of excuses but at the end of the day, they are still using the competitor and we didn’t EARN their business. I understand that sometimes you are not selling a product that is as good as what they are already using or the cost might not be as good, but i still hold true to that fact that if your a good salesman you can sell them. I think teachers should feel the same way. The whole I am just going to blame the students, they didn’t understand what I was saying, well my friend explain it differently. Vlorbik this is the point where you rise from the competition. I’ve never had a teacher (in the math dept.) willing to explain something in every detail so one can understand! In my experience with the math classes the “faculty” (full-timers that consider themselves Gods) are terrible teachers and the “adjunt” (those that are over worked, under paid, and care about their students) are the ones who you learn from. The whole department needs reconstructed, with one thing in mind, do NOT move on until its understood. Okay, have methods and equations you must be able to solve to show “compentency” but for goodness sake, to hell with standardize testing and required material!!

  2. ellie

    I never had a good math instructor until last quarter for 104 and i had already taken 102, 103 and 135. No well, stats was good too.

  3. some of ’em may’ve been better
    than what it looked like–and, yes,
    i do quite realize that this claim
    verges awfully close to
    “wagner’s music is better than it sounds”.
    the inappropriateness of the material
    for the audience in 102, for example,
    makes this 148 stuff appear tailor-made
    for you guys by comparison.
    *nobody* can teach beginners about
    functions in general when they haven’t
    even studied (a) linear functions
    (in particular) and (b) basic set theory.

    and this fact is actually pretty obvious
    to anybody who’s tried. but it’s hard
    to get people to understand
    what their livings depend on
    them not understanding …

    and so we beat on, boats against the current …

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