## Archive for the ‘Rants’ Category

by quotiant rule

EXHIBIT A

but this is *senior level* stuff…

“analysis” *not* freshman-calc.

and even *there*, one seldom encounters

such an *explicit* version of the

natural-log-equals-reciprocal fallacy

.

(the “quotiEnt rule”…

or rather, *the* quotient rule…

our subject has fallen into the

“leaving out articles makes it

harder to understand and so is

in my best interest” trap…

has *no bearing* on this [mis]-

calculation [we aren’t diff-

erentiating a fraction (or any-

thing else for that matter)].)

here’s the whole sad story.

math is hard and everybody knows it.

what they *don’t* know is that it’s

nonetheless easier than anything else.

*particularly* when one is trying

to do things like “pass math tests”.

there’s always a widespread (and *very*

persistent) belief abroad in math

classes that trying to understand

what the technical terms mean (for

example) is “confusing” and should

be dodged at every opportunity.

we teachers go on (as we must)

pretending that when we say things like

“an *equation*” or “the *product* law”

we believe that our auditors

are thinking of things like

“a string of symbols

representing the assertion

that a thing-on-the-left

*has the same meaning as*

a thing-on-the-right”

or “the rule (in its context)

about *multiplication*”.

but if we ever look at the documents

produced by these auditors in attempting

to carry out the calculations we only

wish we could still believe we have been

*explaining* for all these weeks?

we soon learn that they have been thinking

nothing of the kind.

anyhow. the example at hand.

calculus class is encountered by *most* of its students

as “practicing a bunch of calculating tricks”.

the “big ideas”… algebra-and-geometry, sets,

functions, sequences, limits, and so on…

are imagined as *never to be understood*.

math teachers will perversely insist on

*talking* in this language when demonstrating

the tricks.

well, the “big ideas” that *characterize*

freshman calc are “differentiation” and

“integration”. for example “differentiation”

transforms the expression “x^n” into the

expression “n*x^(n-1)” (we are suppressing

certain details more or less of course).

this “power law” is the one thing you can

count on a former calculus student to have

remembered (they won’t be able to supply

the context, though… those pesky “details”),

in my experience.

anyhow… long story longer… somewhere

along the line, usually pretty early on…

one encounters the weirdly mystifying

*natural logarithm* function. everything

up to this point could be understood as

glorified sets, algebra, and geometry…

and maybe i’ve been able to fake it pretty well…

but *this* thing depends on the “limit” concept

in a crucial way. so to heck with it.

and a *lot* of doing-okay-til-now students

just decide to learn *one thing* about

ell-en-of-ex

(the function [x |—-> ln(x)],

to give it its right name

[anyhow, *one* of its right

“coded” names; “the log”

and “the natural-log function”

serve me best, i suppose,

most of the time]):

“the derivative of ln(x) is 1/x”.

anything else will have to wait.

but here, just as i said i would,

i have given the student too much

credit for careful-use-of-vocabulary:

again and again and again and again,

one will see clear evidence that

whoever filled in some quiz-or-exam

instead “learned” that

“ln(x) is 1/x”.

because, hey look.

how am *i* supposed to know

that “differentiation” means

“take the derivative”?

those words have *no meaning*!

all i know… and all i *want* to know!…

is that somewhere along the line in

every problem, i’ll do one of the tricks

we’ve been practicing. and the only trick

i know…. or ever *want* to know!… about

“ln” is ell-en-is-one-over-ex. so there.

it gets pretty frustrating in calc I

as you can imagine. to see it in analysis II

would drive a less battle-hardened veteran

to despair; in me, to my shame, there’s a

tendency… after the screaming-in-agony

moments… to malicious glee. (o, cursed spite.)

because, hey, look. if we all we *meant* by

“ln(x)” was “1/x” what the devil would we have

made all this other *fuss* about? for, in

your case, grasshopper, several god-damn *years*?

did you think this was never going to *matter*?

in glorified-advanced-*calculus*? flunking fools

like this could get to be a pleasure.

but how would i know. i’m just the grader;

it’s just a two-point homework problem.

and anyway, that’s not really the *exact* thing

i sat down to rant about.

next ish: *more* bad freshman calc from analysis II.

so. a poor workman blames his tools

and i’m about to get right back to it.

recap: i’m just as guilty as anyone else.

when there’s stuff i think i’d *like* to know…

but that i *don’t* know… there are probably

*reasons*.

for example: (1) maybe i *don’t* want to know

(“does she *really* love me as much as

she says?”, e.g.).

or this: (2) the possibility of consequences

too horrible to accept (“is this guy just

another macho windbag or is he actually

tough enough to *hurt* me?”, let’s say).

or it’s just (3) flat-out too much trouble

(“what’s my credit rating and who’s

looking?… and who should i believe?”)….

and one could of course go on.

but in our context. and, again, i’m just

as guilty as the next guy.

i might “want” to know…

without having to admit

that i’ve been *wrong*.

wanting what can never be: life on one’s own terms.

including, in this case, an apology from the universe-

-at-large for having pretended for so long that one

hadn’t long since *really* known the true heart

of the matter. (stubborn universe. oughta know better.)

well, it’s pretty pathetic stuff, heaven knows.

but like i say, i’m just as guilty as the others

(that i’ve been working with right in here) seem

pretty consistently to be. give or take.

your milage may vary.

so, now. damn it.

when you hate and fear mathematics.

what then are we to do?

************************************************************

well, *avoid* the SOB if you get a chance, obviously.

i hate and fear fighting with fists and have done

fairly well so far running away from every chance…

too well to learn anything *about* that subject

in any realistic way.

but that’s what *you* should do.

what then are *we* to do… when, in some strange

power’s employ… “you” and “i” have agreed to

work together on clarifying some of your ideas

about maths?

well, with any luck, *i* will be sensitive to

some the emotional baggage that typically goes

with *your* position. in dealing with some

intricate system of rewards-and-punishments,

you will have come up again and again against

some authoritative force telling you “you need

more math” (to acheive such-and-such a goal).

and, with any luck…

*you* won’t take it for granted that i’m just

another con artist out to get over on you

by pretending that making-it-look-easy

does anybody any service.

this is not fucking gym class.

maybe i’m willing to try to understand your point

of view; maybe i’d really like to help you get

closer to *your own goals*.

so, now. why? why? why?

why, *for the love of god*, will you look away

the second the words “do the same thing to both sides

of the equation” escape my mouth?

why, when i make it a point to get your attention

back to the work on the page and say it again,

will you *openly complain* that i appear to be

changing the subject (from “what’s my next move

in this particular problem” to “how does one

actually go about *solving* problems like ours”).

why, when i fucking *beg* you to take me seriously

(this *one* fucking time) and get over your infantile

insistence that *you* know better than every teacher

of the subject worth taking seriously (alive or dead),

must you have some fucking *problem* with that?

because, sure, *i’m* an emotional basket case.

but *i* passed this fucking class easily and

have gone on to learn *much* more about it,

whereas *you* are going to fail it badly

(and deserve to) because you don’t want to

consider that an expert’s opinion *might*

sometimes be more valuable than the ill-

-formed fantasies of some bare-beginner.

(your smug self-righteousness won’t weigh

much on the exam, most likely…)

go ahead, *hold* me in contempt.

you’ve paid for the privilege.

but, geez. this trying to *understand*

why you do it has me just about worn out.

so, you know what?

just *humor* me.

“do the *same* thing to *both* sides of the equtation.”

do the same thing to both sides of the equation

*on my authority* if it makes you feel any better.

(and *remain* a true-thinking math-is-nonsense

committed-to-ignorance equations-are-confusing

“normal” person… *blame me* if you have to…

“i’m not *really* pretending to be a math-head;

my teacher just *makes* me do it [like *all* teachers]

just to prove he can push me around”)…

i haven’t got the strength even to talk

about this any more.

because the *right* answer to “what do i do next?”,

after you’ve taken some perfectly good code

and munged it up by changing only *one* side

of an equation, is, and can’t ever *not* be,

“fix up this equation so it’s right”.

and if… never *mind* your reasons…

you’re not *having* it? well, it’s just

another one of those (many, many) problems

whose *solution* is “owen leaves the room”.

at the foundation of (an earlier version of)

this blog i ranted and rambled about

a tendency on the part of (lower-division

college math) textbooks to hamper the work

of the teacher by (deliberately!) suppressing

correct technical language.

of course things have continued to deteriorate.

but, by some miracle, i’m still earning

the random crust of bread by helping students

learn to *read* these ever-more-horribly flawed

documents. so far so good, then, i suppose.

anyhow, i’m grading linear algebra again

right in here (nothing *but* linear algebra

for something like a *year* now)…

and i’ve only recently become vividly aware

that this “tendency” has penetrated deeply into

the textbooks at this “higher level of the game”.

specifically, i hereby announce that some

satanic force has somehow (even here) replaced

*the sign of set-membership* ()

with its mindbendingly-wrong “plain english”

equivalent(s). the perfectly-correct

(and altogether-necessary) symbolism

(“x is an element of S”)

is now to be replaced, by the edict of

invisible (and mostly unimaginable)

entities, with “x is in S”.

[

this is a good place to skip ahead.

i’m going to geek out slightly here.

you *don’t* have to be an adept to follow.

i’m hoping to make a point that can be

at least *partly* understood by math laity.

is the set of natural numbers (more here…

much my most popular post here and probably

my best-read production of all time) “in” the set

of real numbers? loosely, yes. more precisely,

.

i can easily imagine myself talking to, say, another

teacher about, say, some “property” (like commutativity-

-of-addition; x+y=y+x [for all x & y]) that applies

in the natural numbers. “how do we *know* it applies?”,

i might say. and the answer might come: “because

the naturals are *in* the reals, and the *reals* enjoy

the property of commutativity”. “good answer!” i would

then reply, and move on to whatever i *really* wanted

to talk about.

again. are all possible probabilities *in* the reals?

well, yeah, (duh)! in “code”, one has ;

rephrasing, “all the numbers from zero to one (inclusive)

are *in* the reals” (but also, more precisely,

“the [closed] unit interval is a subset of the set of real numbers”)…

so. now i’m talking to some grad-school dropout (say): “is

the-interval-from-zero-to-one *in* the reals?” she asks;

“heck, yes” say i, and we get on with whatever we’re really doing.

is “pi” *in* the real numbers? sure! .

“pi is an element of the reals”.

but wait! being-a-subset is an *entirely different* relation

from being-an-element! pi is simply *not* “in” the reals

in the same way that is!

who cares? well, me and a few hundred thousand others or so.

if *you* don’t care? well, that’s why i invited you

to skip this part! read on!

]

the biggest problem from a practical standpoint

(if “how can we make this material better understood”

is a practical question) is simply that students

*hate writing* and at *every opportunity* will

replace “plain english” with (typically very

ill-understood) bits-and-pieces.

*nobody*… no student, no lecturer, no pro

mathematician… is going to write out the phrase

“is a real number” a whole lot more than

twenty or thirty times (in a given sitting-down)

without wanting *some* abbreviation for that

phrase.

and likewise for “is a subset of”… indeed,

*any* sufficiently common phrase *begs for

abbreviation* even in “plain english”

so there it is. mathophiles also… in some sense…

“hate writing”. anyhow, we *love abbreviating*.

“algebra is the science of equations”, i once

heard someone say (explicitly repeating something

he had learned “by rote” from a public-school

teacher during his own schooling… it went on

for another few lines but i didn’t learn that

part from listening to this guy say it three

or four times that one night). and i consider

this to’ve been very well said.

so i’ll hope to return to it.

but first. this history of elementary algebra

at w’edia summarizes the standard dogma of its subject (as i

understand it) well. the evolution from “rhetorical” algebra

(describe *everything* in plain-language words) through “syncopated”

algebra (where “shorthand” symbols [many still common today]

began to replace the most common techical terms… but the

actual *reasoning* was still natural-language based [and so,

by contemporary standards, “informal”]) into

“symbolic algebra” (the “science of equations” as we know it

today: a study of “formal” properties of [carefully-defined!]

symbolic “objects” [“variables” and “equations”, for example]).

what w’edia *doesn’t* make much of… but what matters to me

a great deal… is that the emergence of algebra pretty

closely *coincides* in (so-called) *western* history with

the (so-called) renaissance and the (no sneerword necessary)

scientific revolution.

“modern times”, then, *began* when certain humans (*finally*!)

figured out how collections of symbols-on-paper (representing

certain abstractly-defined-objects), produced according to

various “rules”, could be interpreted to reveal previously-obscure

*laws of nature* (so-called). this *was* the “scientific revolution”.

and how does it work? *equations* is how.

so one of the *first* things in understanding

what’s going on the contemporary philosophical

environment is to find out *what an equation is*

(for all literate people): “the equality meaning

of the equals sign”.

when we’re being sloppy, we can confuse “=” with “is”…

but when we actually get to work *using* equations,

we have to *much more precise* to get any value from

the procedure at all. plain-english “is” is *always*

in some sense metaphorical (except in empty utterances

like “it is what it is”)… whereas the equal-sign

rightly-used is as far away from metaphor as we know

how to get.

how does algebra work? (equations and *what* else?)

by *the method of substitution* is how.

“in a context such that A=B is taken as ‘true’,

a properly-written piece of code

including (the symbol) B

*does not change its truth-value*

when (the symbol) A is *substituted* for B.”

this “method” *characterises* algebra.

i first became aware of some its awesome power

in about seventh grade.

and *what* else? “doing the same thing to

both sides of an equation”. and what else?

that’s about it. that’s algebra. the rest

is commentary.

now, for *set theory*, two of the main ideas

are caught up in set-inclusion and set-containment:

and , for example.

and one must be every bit as careful in the use

of these symbols when studying sets as one must be

in the use of the sign-of-equality in studying,

say, polynomial equations (i.e., pretty much,

in algebra).

but about forty percent of the class already don’t

take *equations* seriously. and they’re morally

certain that “sets” are meaningless traps designed

to distract them from “how do i get the *answer*?”.

and the textbook does a great deal to encourage them

to *maintain* this view.

and i don’t like it. please stop.

*most* of the exercises i’ve been marking

are… pretty obviously… somebody *else’s* work.

it becomes very tedious making corrections when

the alleged readers haven’t even taken the trouble

to try to understand whatever source document

they’re copying from.

a “2” turns magically into a “3”. what was mere

sloppiness in the source becomes *contempt for the

grader* in the BS-artists who unthinkingly copy it

over. how stupid do you think i *am*, just because

i fucking work here?

sure, i get it. everybody *else* is trying your patience.

they’ll pile on work just to see how devoted you are.

college is fucked; i get it. but this is the math

department and lies just won’t fly.

get with the program or get gone.

the procedure for TA’s administering quizzes:

log into the “secure site” with your university password;

print out hardcopies and run off copies (one does *not*

have direct access to the copying *machines*, however…

run ’em off at your own expense or wait your turn

at the department’s copy center).

so far so good if the system actually worked.

but no such luck. when i got the email with

the link to the originals over the weekend,

i went and glanced at the link; okay.

there appears to be a quiz here. but i don’t

own a printer so i waited until i was on campus

to begin creating the actual paper documents.

whoops. locked out of the system. no time

to fix it. gotta postpone the quiz.

naturally there’s some resentment from the class;

naturally i’ll take a great deal of the blame.

(fortunately for me there were others suffering

the same problem or the *administration*

would presumably go ahead and blame me, too.)

in the twentyfirst century economy there’s *usually*

a robot between me and whatever i want.

and the more i want it? the more of my time

the robots will burn up uselessly while i try

to get it. telephones? your service *will* fail

and if you try to get it fixed, prepare to wait

on hold for a long time before you can get

to the menu with no option remotely like

serving your needs. home internet connection?

same thing, but worse (on my model because

i actually *want* to be online but not to talk

on the god-damn telephone). and don’t even

try to get me to *think* about health insurance.

look, i like breathing in and breathing out

as much as the next guy. but when does

this become unacceptable?

(here’s last year’s why i don’t

live at the p.o..)

3.Let be the Universal Set (for this problem) and let , , and .Compute the given sets. (Remark: we will have shown that set difference is not associative).

Again (sorry for the child’s play). A poor workman blames his tools; this doesn’t make him wrong. Obviously X – Y is (the set) {1, 2}. It follows then that (X-Y) – Z is also {1, 2}.

Whereas, in light of the fact that Y – Z = {3, 6}, one has X – (Y-Z) = {0, 1, 2}.

Anybody (outside a math class) pretending to want an explanation of what (if anything) has gone wrong is, pretty reliably, actually looking for some *weakness* whereby they can *undermine somebody’s authority*.

The classic blunder at this point is to refuse to take such a student seriously. And yet, they’re doing what *I* would’ve done; what I’ll go out on a limb and guess *you* would have done. Students didn’t just become stupid because you got an advanced degree any more than kids got to be some inferior order-of-being just because you decided to call yourself a growup.

*How are they to know* that, here in math class, we *mean what we say*? When, in every other arena (and for all we know, this one), *authority* rules?

(Yes, yes… they *call* it “opinion” and blame it on “the people”. Are we bored yet?)

… the comments to my last post here.

what “we” do with math; right.

i’ve never done *anything* with a large

data set. what i *have* done is coached

a senior computer-science student in SQL

(a “database” program based on ordinary

p-and-q symbolic logic in a straightforward way)

*without my ever having seen* SQL.

she was pretty helpless in it for a day or so

because she hadn’t looked at the tiny little

“easy” problems quite hard enough to make

certain things-obvious-to-experienced-users

available to her consciousness *at all*:

math is full of (what turn out to be)

“simple” things that are either trivially easy

or difficult-to-impossible (with no

middle ground; there’s some

not-at-all-quantum-like “leap”

where one “gets it”; this is widely

understood actually…).

i can read “logic” in any of many flavors

after all and even write it if i absolutely must.

and *that*’s what i’d like

*math teachers* to be able to do

and even to spread around.

dan’s made WCYDWT…

“what can you do with this?”…

famous all over town. good.

here’s a shaggy-dog story (lack-of-spoiler alert)

about one of my favorite lines in one of my

favorite movies. it’s in _fisher_king_ and

robin williams has gone nuts from witnessing

some horrific violence and lives in a squat

and hangs with the street people. jeff bridges

played a part in perry’s triggering episode

(robin williams is called perry; there’s a

character in the “fisher king” legend called

percival and one is left to draw conclusions

or not). so jeff feels guilty and wants to do

something for this perry guy and hunts him

out down in the derelict street to give him

some money. robin williams immediately

turns around and gives the money to

another crazy street person. “i gave it

to *you*” sez jeff; “what am *i* gonna

do with it?” sez robin. gets me every time.

(yes, that was the punchline. move along;

nothing to see here.)

“What Can You Do With This?”

well, if you can’t do it with

*tools at hand*, probably nothing.

what am *i* gonna do with it?

what can i do with SQL?

nothing until somebody else wants

to understand it; then i can

*put it aside* and look at

some much simpler ‘underlying

structure’ that i actually understand.

then i can talk about it.

same with, pretty much, any other “application”.

i don’t interest myself much in the question

“how can i generate student interest?”.

all the other teachers in all the other subjects

plus mass media, the church, and the wolf

at the door are my competition if i enter

this arena and i’ll lose.

my material not my presentation is

the best possible. my presentation

is just what “we” who are present

at the time are able to make out

of the materials at hand. going out

and working on my presentation

is distasteful *to me* because

i know i’m going to get my ass

handed to me by richer better looking

smarter people with better connections.

no fun.

so. at long last. a vision of what math is?

well, it’s sort of like 42nd street.

where the underworld can meet the elite.

respect and attention will be lavished upon

those who best understand equations

(and, okay, certain, yick, drawings).

much more so if they’re able and willing

(rare in combination) to *explain* them

to interested parties of a wide variety

of levels of previous experience.

best of all like everybody knows

is working with small groups

(scribbling symbols and *talking*).

you can’t buy it or download it.

if you need this explained

you’re never gonna get it.

jazz math ed.

*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.

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?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).

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*…): “, in lowest terms, is a rational zero of *f*, then *p* must be a factor of and *q* must be a factor of ” is perfectly clear in its context; don’t let anybody tell you any different. (In particular, and 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 ‘s in place of the ‘s or in place of ). 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) and ; moreover, we’ll be able to use this notation quite a bit in other contexts (like

—

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 and 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: 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 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 and that the inequality under the quantifier states that *g* is not *the zero polynomial*) and that 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*…

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 numberAnd this is about as clear as can be expected: this stuff is hard. But, doggone it, we’re adding to the right side of ankis added to the right side of a functiony = f(x), the graph of the new functiony = f(x) + kis the graph offshifted vertically upkunits (if ) ordown|k| units (if )."

*equation*, aren’t we. And shouldn’t we

*take a hint*from the doggone grapher and refer to and (the authors go on, in effect, to call 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.