Archive for the ‘Rants’ Category

bad freshman calculus

log_a(x) = ({{ln x}\over{ln a}}) = {1\over{x ln a}} by quotiant rule

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
ln(x) = {1\over x}.
(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
(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

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”.

midterm report

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* (\in)
with its mindbendingly-wrong “plain english”
equivalent(s). the perfectly-correct
(and altogether-necessary) symbolism
x \in S (“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,
{\Bbb N} \subset {\Bbb R}.

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 [0, 1] \subset {\Bbb R};
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 \in {\Bbb R}.
“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 \Bbb N 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

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:
\pi \in \Bbb R and {\Bbb N} \subset \Bbb R, 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.

bricks without straw

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 U= \{0, 1, 2, 3, 4, 5, 6\} be the Universal Set (for this problem) and let X=\{0,1,2,3\}, Y=\{0, 3, 6\}, and Z=\{0, 5\}.

Compute the given sets. (Remark: we will have shown that set difference is not associative).

Y^c \cup Z^c
X \cap X^c

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?)

I Relaunch(01/05) as MathEdZineBlog at the site…… of my quit-in-a-fit-of-frustration Vlorbik On Math Ed of ’07 to ’09. I offer subscriptions to the then-purely-imaginary MathEdZine (and, though of course this isn’t revealed in the post itself, I get no takers. I am not at all surprised. Heck, I’m not at all discouraged, even. “Disappointed” might not be going too far.) In Where Did That Amazing Blogroll Go? (01/05) I offer a backhanded apology for having deleted the blogroll for the relaunch; this was probably the feature I’d’ve found most attractive about VME if somehow I’d found one just like it done by somebody else before I’d ever started blogging. A fellow mathblogger with better net chops than mine… “sumidiot”… had already modified it for the new feedbased formats anyhow, so my work was finished (“feeds” are part of what I’d been so frustrated at when I’d quit, I’ll go ahead and mention). Also WDTABG? was the first in what promises to be a pretty steady stream of links lists. Probably I shouldn’t drop links here in this post for all of ’em… here’s the WordPress Archive for Jan ’10.

In Songs For Beginners: Pascal’s Triangle (01/05) I hinted at one of my ideas for the first issue of the zine. “Lectures Without Words” is a phrase I’d been dropping here and there already… for example in this post from Open A Vein (my “homepage” blog) back in December. I have indeed gone on to create four zines on this theme and have convinced myself I could probably do one a week or so… on the model of the hugely-popular-though-nobody-knows-about-it cartoon diaries of recent years… indefinitely and love doing it. Not that I will. I’ve got plenty of other ideas about math, and ed, and zines. Unless this is some monster hit, I’ll move on pretty quick I imagine. One never knows. Where was I.

in lovely dark and deep (1/07)
i slip into my “ramble” style for some
discussion of early drafts of issue zero.
i crossposted this
in order, i think, to cause a link to show
up in twittr or facebook or some other
“social” site that i’d caused to be linked
but had lost track of (this problem has
of course grown even worse in the
ensuing weeks).

then in comment thread reprint (1/08)
i continued my very-intermittent conversation
with the frighteningly-successful dan meyer
about sizzle and steak; i followed up immediately
with “continued from…“, yet another
in a never-ending line of luddite ravings.
what am *i* gonna do with it?

Vlorbik On Math Ed(1/13) is the title of this post; its contents are actually a list of links to selected posts from the blog of that name… much like this one though much better organized. I’ve revised it substantially from the version of its posting date. Usually I avoid doing this but in this context it looks like plain common sense to go ahead and over-ride that “house style” rule.

prehistory of MEdZ, part i (1/19)
tells, rather, the prehistory of the ten page news
(volume iii), with some math ed thrown in.
editorial license.
prehistory of MEdZ, part ii (1/21)
is a dismal failure wherein i tried to put
a photo in and it was there briefly and
even drew a comment from JD (one of
VME‘s most faithful readers;
thanks jonathan). further fiddling…
specifically this photo-post (1/25)…
clobbered the picture; to heck with it.
i’ve gotten a little better at moving
pictures around with flickr and tumblr
but it’s infuriatingly “intuitive” and
i can still barely work any of it at all.
the picture that stuck… not at all by
the way… is of madeline’s scanner/printer.
i’ve made some mighty cool zines with
this thing already but i’ll probably
have to beat it to death with a goddamn
hammer before too much longer.
when it can be made to work at all…
at random as far as i can tell…
it makes gorgeous copies but
gohd knows what they cost and
anyway i can’t stand the aggro
and when it breaks soon i’m done.

continued from…

… 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.

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

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.