Archive for the ‘Blogs’ Category

clean-dishes montage

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domestic arts in the age of digital distribution.  RIP, vlorblog.  here are some posts mentioning “clean” and “dish”: https://vlorblog.wordpress.com/?s=clean+dish (courtesy of the “search” feature from that blog).

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… but i’ve lost control of that one.
so here it is again. with a photo
from “vlorbik unstrung”.

the music isn’t by me, of course. “stealing”
already-well-known tunes was good enough for
joe hill and woody. and dylan. so it’s good
enough for me.

i easily figured out the main “trick” in playing it
but haven’t practiced it enough (even now) for
public performance.

Photo on 6-3-16 at 10.59 AM

negatively fifth street (2015)

a drifter escaped from a boxcar/ denouncing obviously jive believers/ has a zine about it in the catalog/ but nobody can work the damned randtrievers/ and the cats are praying in the alley/ and the pool shark is chalking up his cue/ an’ i’m out lookin’ for my lady/ down on kirkwood avenue

jesse, he’s round the corner/ buskin in front of the bird/ doesn’t bother him if no-one stops to listen/ doesn’t bother him if they don’t like the words/ and some violence boys might come and beat him down/ and he’ll forgive ’em more than i could ever do/ but that’s nothin compared to what’s goin on/ back on kirkwood avenue

a melancholy cougar/ buys a hoagie from a clown/ there’s a tempest brewing somewhere/ and there’s panthers on the other edge of town/ and the goddess of gloom and the jester/ are doin’, that thing they love to do/ in a video montage/ at the parking garage/ on kirkwood avenue

a cloud of marijuana/ is obscuring the people in the park/ or maybe i’m just losing my eyesight/ or maybe it’s just getting close to dark/ an’ the tournament game was a victory/ so now it’s turnin’ into a zoo/ an’ the riot squad and the thunder god/ are on kirkwood avenue

second draft of 7S_0.0–4
~1500 words.
(the first draft started here.)

Seven Stories: Part Zero

☉  Sun
☽  Moon
♀  Venus
☿  Mercury
♂  Mars
♃  Jupiter
♄   Saturn

On Monday, when Al first shows up for the Symposium,
he finds Betty already there writing in her notebook,
seated nearer to the aisle (up the middle of the room)
than to the door, in the front row.

“All that stuff was already on the Board when I
first got here.  Astrological signs, apparently.
But I wonder why those particular signs
were singled out.”

“Well, I had first-year French five times,”
Al replies.  And he sits down, just to Betty’s right
(Betty appears to be left-handed… but is writing at
a standard right-handed desk of the kind that can
be pulled up [for note-taking] or pushed back down
between seats [for easier moving around]).  From this
position, Al can follow along if Betty should continue
working in her notepad.

“… and it looks to me like ‘Lundi, Mardi, Make-Ready…’,
the days of the week.  We’ve got something like
it in English:  ‘Monday’ is more-or-less obviously
‘Moon-day’… and ‘Sunday’ is even-more-obviously,
the “Day of the Sun”.  But Tuesday and Wednesday
don’t fit in English.”

“Oh, I know that one!  English uses Norse
gods!  So ‘Thursday’ is Thor’s day… just like
that hunky hero in the movies.”

“Oh, I love those movies.  Not so much Thor,
really, but Avengers was amazing… and it’s just
been incredible having ’em make all this stuff about my
favorite comics from when I was a kid and wanted to be
Stan Lee… before your time, obviously.”

“I’m more into the whole ancient-warrior kind of thing,
myself… but some of the superhero stuff is OK.”

“Anyhow, you’re right.  ‘Wednesday’ is ‘Woden’s day’…
they call him ‘Odin’ in the comics… but ‘Frigga’s day’
and ‘Tiu’s’ day are evidently from godesses’
names.  This probably ought to be better-known…
but I sure don’t know much about it…”

When Gemma shows up, she can see that they’re talking
about the symbols on the board.

“Is it OK if I erase all these planets here?”
*
“Well, hang on,” says Al, “and let me copy
the symbols, over here.” And he picks up
a red marker out of the chalk-gutter and
moves over to the the upper-left-hand corner
of the board.

☽♂☿♃♀♄☉
MTWRF sS

“Now, wait a minute,” says Betty. “How come
you put Monday first?”

“Well, that’s how it is on my calendar.
With Saturday-and-Sunday at the end.
Anyhow, that’s how I think of it:
Sunday’s the day-of-rest from the Bible;
‘on the seventh day He rested’ and all that.
What I wanna know is: what the heck do you
mean planets? The Sun’s no planet…
and neither is the Moon, for that matter.”

“Well, in ancient times, there were only
these seven objects that moved around
in the sky against the background of the
so-called ‘fixed stars’,” says Gemma.
“What’s that got to do with days of the week?”

Just then Del shows up with his rainbow-painted
guitar and big blue backpack. He raises his head
to glance once at each of the others and finds a seat
way in the back. Then puts his guitar into one
and moves it around until the desk holds it into
the chair. Next he slides his pack off and puts
it into a chair two-over from the one with the guitar,
then sits between the pack and the guitar.
He digs around in the pack for his colored pens
and drawing pad. Saying not a word, Del begins
to draw in the pad without looking again at any
of the others.
*
Meanwhile, Gemma’s erasing part of the board.

☽ Moon ♂ Mars ☿ Mercury ♃ Jupiter ♀ Venus ♄ Saturn☉ Sun
monday mars-day make-ready jove-day friga-day saturn-day sunday

Al is simultaneously drawing the various symbols
at the other end of the board, along with their
Gods-and-Days identifications.

“You see,” says Gemma, and she looks around the room
trying to make eye-contacts, “it must have had something
to do with the number seven itself: there were
seven celestial objects wandering around in the night sky…
‘planet’ means ‘wanderer’ (or so I’m told)… and seven
days in a week… so the fortune-tellers decided that
maybe each planet should ‘rule a day’, so to speak…
whatever that might turn out to mean, as it were…”

“But wait a minute. No, before that, even.
What’s your name? I see you haven’t got your tag
on…” Al grabs a corner of the nametag hung around
his own neck and wiggles it a little.

“Oh, hi… Al. I’m Gemma. I’ve got it here with me
somewhere.” As she roots around in her purse.

“The thing is, why should there be seven days in a week?
Are you saying they did that because of these
seven so-called ‘planets’ or whatever?”

“I always heard it had something to do with the Moon,”
says Betty. “A month is about 28 days, and”

Al interrupts her with “OK, seven goes into 28. That
makes sense, I suppose.”

“Right.”

And Al sits down again next to Betty.
*
Gemma finds her nametag in the purse
hanging from her left shoulder and
loops its string over her head. Then she
finds a seat in the second row across the
aisle from Al and Betty. She puts her purse
on the desk.

Betty’s got the notes from the board in her
spiral-bound notebook.

☉ Sun
☽ Mon
♂ Tue
☿ Wed
♃ Thu
♀ Fri
♄ Sat

“It doesn’t make sense this way, though!”
According to Al. “Those are symbols for objects
in the sky, not for days-of-the-week!”

“I disagree”, Betty replies, “the Sun and Moon still
match up with the days even in English, and, like you
said, in French they match up better still.”

“But the genders are wrong, for heck’s sake!
I want it to at least match up to the comics!
Now this
♂ (drawing in her notepad)
I know means Mars… a wargod, even.
That oughta be Thor, on my model: thundergod,
wargod… pretty close fit. But whoever this “Tiu” is,
that Tuesday’s supposed to be for, I am pretty sure
she’s a goddess: wrong sex altogether.”

“Don’t you mean gender?”

“Do I? And Odin-to-Mercury doesn’t make much sense either;
Odin’s the bigshot-in-chief like Saturn ♄ or something;
let’s see… Thor’s his son, so I guess Thursday
oughta be Jupiter ♄, not Mars… okay, good… I
always thought that was game day somehow…”

But Betty isn’t quite following all of this; Eppie and Zed
have come in and are watching the goings-on in Betty’s note-
book.

“What’s going on?”
*
“What’s going on?” asks Zed. Eppie has
produced a cel-phone meanwhile, and now turns
his back on the seated pair, apparently intending
to photograph the notes on the board.

“Hey, I thought we weren’t supposed to bring any
computers to this thing.” Zed to Eppie.

“We haven’t even started yet. Anyhow, the
judges have made their decision; now let ’em
enforce it. Does anybody else actually object if I
use my phone?”

“Oh, right. Here comes mister do-what-thou-wilt-shall-
-be-the-whole-of-the-law again.” Zed again.

“I sure don’t mind” says Betty, and takes her
phone of her pocket. “It doesn’t seem right to me;
of course we can be quite certain that they’re taping us“.
She gestures at the slide-projecter on the ceiling
at the back of the hall. “I’ve got lots of notes
and things on mine.”

Al throws up both hands in a don’t-ask-me! gesture
and turns his upper-body and neck around looking
behind him for Gemma and Del.

“Well maybe we should wait until we get started
before we commit to anything. Does anybody know
who’s supposed to be presenting today?” Gemma says.

Del doesn’t look up at first. But he can somehow
tell anyway that everybody’s looking at him.
“Not me, man.” Still not looking up. “Unless I
get to use this guitar over here.” As he
touches the neck of the guitar and finally glances
quickly at each of the others’ faces. Then looks
directly back at his pad and picks up a marker.

“But what about this cel-phone-usage thing?” Zed.

“Hey, I don’t care either way. I can’t work
the damn things, of course.” Del finally looks directly
at Zed and they size each other up silently for a second.

Betty gets up, pointing at her phone and looking at Eppie.
“The signs of the seven ancient planets were on the board
when we all got here. Al and I were talking about how it
might have something to do with days of the week.”

By the time Etta shows up a few minutes later, Al has moved
to the aisle seat of Gemma’s row. “I thought you were
going to be our speaker when you came in.”

And Zed’s in the back talking to Del.

econ 101

i’m browsing as hard as i can.
but i can’t find any images
as pleasing to my eyes as these.
and yet these are sloppy first takes.
it’s almost as if reality
were more beautiful than the net.
this belongs in the other blog.
who said life is fair.
Photo on 8-10-15 at 3.57 AM

Photo on 8-10-15 at 3.56 AM

Photo on 2014-07-12 at 10.42

Photo on 2014-07-12 at 10.47

VME: Vlorbik on Math Ed

2007
june
06/07/07 First Post hello, world
06/08/07 Why I Teach Such Good Classes (2003) short zine piece
06/12/07 Eliminating The Middle, Man “and” versus \cap
06/22/07 {X:X is full of baloney} abuse of notation
06/28/07 Contradictions of Ed Reform (1999) presented at joint meetings
06/28/07 Tutors and Teachers (1998) tale of two reforms

july
07/13/07 Tom Lehrer great songwriter
07/20/07 Con Report (2001) t-cubed calculator show

august
08/02/07 Learning Center easy money
08/08/07 Carnival of Mathematics XIV no “facts about 14”
08/15/07 The Pencil Rant why make things harder?
08/16/07 Jazz Math Ed go man go
08/23/07 Please Lie More Carefully impossible conditions

september
09/28/07 Maybe If I Ignore It… the statistics coup

october
10/11/07 ZBox Considered Harmful calculator (FTI)
10/12/07 Fisking Fennell NCTM
10/16/07 JAWOPPA just another way…
10/31/07 On Finally Teaching Calc II at long last

november
11/27/07 LEGO® Links very popular post

december
12/04/07 Open Letter To A Course Co-ordinator more calc ii stuff
12/06/07 I’ve Been Saying This For Years and i plan to go right on

2008

february
02/01/08 Fractions and Student Incredulity trust no one
02/07/08 “Clearing Fractions” Clarified move your pencil
02/19/08 Radical Notations don’t overgeneralize

april
04/22/08 On Teaching Math Ed At Long Last boy it was great

june
06/05/08 Last Day Of Classes the beginning of the end?
06/06/08 188th Post one year old

july
07/16/08Section 1.1: Natural Numbers (1995) much my most-read post
07/17/08 Section 1.2: Modular Equivalences (1995) more from the same lecture notes

september
09/25/08 Wheee! back to work; textbook grumble

november
11/20/08 They Are Standing Still calculator code
11/24/08 Trust The Code. annuities

december
12/02/08 Fear And Trembling scheduling; politics
12/18/08 Why Live (x^4 + x^2 + 1) tricky factoring
12/24/08 Jeremiah 6:14 peace, peace

2009

january
01/02/09 Introductory Remarks math 148 precalc
01/06/09 {(0,0),(0,1),(1,0),(1,1)} graph transformations
01/08/09 Capital-Script-D-of-f domain & range
01/13/09 What The Heck, Some Math piecewise functions
01/22/09 Big Day For Former Math Professor self-publishing
01/22/09 Message: I Care notation for transformations
01/23/09 It’s Not The Cheat, It’s The Futility needless case-divisions
01/27/09 Quadratic Formula Lore “complete the square” etcetera
01/28/09 Our Story So Far summarizing 148

february
02/09/09 Section 5.5: A Manifesto textbook rant
02/18/09 Bricks Without Straw compositions
02/20/09 And Into The Black rational functions
02/23/09 One Must Imagine Vlorbik Happy moebius transformations
02/24/09 I Quit: A Clarification mathblogging considered harmful

Math 148: Precalc too late to stop now
Community College Calculus (Spring ’09)

MATHY (BUT POST-VME/CCC)

august
08/09/09 Open Letter At Random e-mail reprint
08/20/09 nobody writes to the kernel ramble of ex-math-teacher
08/20/09 this will all make sense later pedagogical content knowledge
08/23/09 the equality meaning of the equal sign try it you’ll like it
08/27/09 why we fight definitions are hard

september
09/21/09 look over there our medium is handwriting

november
11/09/09 notes for chapter zero blahblahs
11/12/09 classes, laws, and associations associativity
11/12/09 And Left Me In Reputeless Banishment “A” and “a”; page layout
11/13/09 testing testing boolean functions
11/14/09 in media’s rays ) cross-products
11/15/09 use the source luke permutations
11/16/09 never get outta the boat four 4’s
11/17/09 there is no frigate like a boat trees
11/18/09 the chinese room more permos

december
12/03/09 exercises at random S^ := S \cup {S}

2010

january
01/27/10 Photo 96 MEdZ #0.1

may
05/07/10 pascal’s triangle (so-called) drawing
05/20/10 it said there’d be some thunder at the well i ching & binary
05/20/10 stop me if you’ve heard this more binary

november
11/23/10 trying again dammit P_2(Z_5)
11/23/10 perhaps *this* will refresh your memory P_2(Z_2)*
11/23/10 now here’s something you’ll *really* like more P_2(Z_2)
**************************************************
Math 148: Precalculus
Archive
(links to the following; i’ve lost control
of the actual code…)
Archive
As I Was Saying
The Natural Log: Eli Maor plugged.
In The Arena: “inflexible knowledge” at KTM & Lefty on Devlin.
exp & log
Monday Linkage: CO9′s on ANT.
Math Newsbite: “Square root day”.
Introduction to Logarithmic Functions
The Veil of Maya: simple, compound, & continuous interest.
When There Is No Peace: links.
Stretching A Point: doubling time etcetera.
Why Was I Not Informed?: MT@P, Spirograph.
Catherine Nails It: 21st C. skills.
Remarks Toward The Dissolution Of This Blog: review.
Forests And Trees: links.
Herculean Sense-Making: Garelick on discovery learning.
More Review
Formula Scratchpad
3.14
Last Post
**************************************************
Community College Calculus
alc I (151) and Calc III (153)

march
03/15/09 First Post and 1st busted link
03/31/09 153:Day One could’ve been worse

april
04/14/09 The Sky’s The Limit (At Infinity) students at the board
04/15/09 Aspects Of A Topic arclengths parametrically
04/16/09 They Put You On The Day Shift dijkstra. comments.
04/23/09 MT@P verse
04/25/09 The Best Lack All Conviction minty, lenard, & nowak
04/30/09 Down At The Bottom Of The Garden polars & parametrics

may
05/01/09 I Love The Smell Of Chalkdust In The Morning 153 homework
05/05/09 Parabolic Language hard “focus” problem
05/07/09 Buy Conditionally d/dx of a constant; epsilon-delta
05/19/09 Last Quarter’s Final 151 calc i (totally cool)
05/29/09 Triangles, Too, I Bet math circles; i quit

june
06/01/09 Caviar For The General binomial theorem
06/04/09 Last Year’s Final. Partly. Sort of. 153
06/05/09 More Of The Same 153 test
**************************************************
posted in ’09 (overlaps VME &c; then mostly OAV)

january
02 Introductory Remarks math 148 precalc
08 Capital-Script-D-Of-f notations (“domain” e.g.)
13 What The Heck, Some Math piecewise functions
22 Big Day For Former Math Professor uploaded lecture notes
27 Quadratic Formula Lore completely square

february
09 Section 5.5: A Manifesto rant on 148 book
18 Bricks Without Straw compositions
23 One Must Imagine Vlorbik Happy moebius transformations
24 I Quit: A Clarification blogging is too hard
26 Put Me Back In The Mud madness and poverty (intro)
28 Stay-At-Home Bookrun

march
13 The Triage Lecture life during wartime
17 A Decent Interval from short-lived 103 blog
18 The Crack-Up just like francis fitz
19 Targeted Tutoring extra paperwork
31 As Usual office politics

april
07 Morbid Navel-Gazing me me me
23 Won’t You Light Up And Sit For A Spell? verse (& more madness)

may
15 listening the most precious thing sociology
19 Last Quarter’s Final for calc I
21 First There Is A Mountain voluntary tryouts
28 nothing in his life became him i have to quit

june
01 I Quit academic lumpenprole RIP

august
04 All Uphill From Here driving test
09 Open Letter At Random introductory ramble
15 Assume A Can Opener If You Have It Not “acceptance” ramble
17 We Don’t Need No Stinking Badgers professional ethics
19 Lies Of The Program teach the whole student
23 the equality meaning of the equal sign (!)
27 concerning that of which we cannot speak analytic/romantic split

september
02 Letting I Dare Not Wait Upon I Would barre chords (guitar)
16 meanwhile in a distant comment thread drawing ramble
21 look over there no big deal philosophically

october
31 off in the distance walking & running

november
18 the chinese room yet another permutation ramble
22 It’s Been Too Long Since We Took The Time data-manipulation milestone
27 spam buy! buy! buy!

december
19 m’aidez union stuff
31 the man went to earth and looked again madeline made bread
**************************************************
posted in ’10

january
01/02/10 posted in ’09 partial contents
01/07/10 lovely dark and deep math ed zine debut
01/19/10 prehistory of MEdZ, part i really of the ten page news
01/27/10 Photo 96 Math Ed Zine #1

february
02/02/10 dekist’s hot shelves photo
02/07/10 January 2010 rambling self-annotation
02/08/10 the domestic arts in the age of digital distribution part i first of the DAADD photo series.
02/11/10 daadd part ii photo of madeline

march
03/04/10 more than 4K characters for sue v. math eddy ramble
03/05/10 threehundredsixtyfive days music update
03/09/10 yet another crosspost: from adj-l autobiographical snippet
03/11/10 like it’s twenty-eleven verse
03/25/10 why i don’t live at the p.o. anti-gov’t screed
03/27/10 i’m not there “reimagining economics” thread

april
04/04/10 a petri-disch community my life in schools
04/07/10 the swirling beachball of doom more “life in schools”
04/10/10 madeline’s front room (with guitarist) selfportrait
04/10/10 not for auction this machine kills fascists

may
05/07/10 pascal’s triangle (so-called) MEdZ #0.8 p.7
05/20/10 it said there’d be some thunder at the well i ching
05/20/10 stop me if you’ve heard this more pp. from #0.8

june
06/26/10 i’m not there imminent death of the net

september
09/24/10 university hall basement get back to work

november
11/06/10 midterm report discrete maths at big-state-u.
11/23/10 now here’s something … and more “lectures without words”.

december
12/01/10 community college calculus contents page for short-lived blog
12/29/10 more than 4K characters for k. nowak comment on “log laws” thread
*************************************************
the end of open a vein (2011–2012)
january
01/02/11 posted in 2010 partial contents

february
02/10/11 …in front of a mic… announcing my soundcloud page

march
03/09/11 we are not data metaphysical ramble

may
05/25/11 more four-string guitar liner notes

june
05/03/11 not dark yet wisdom & fame
06/12/11 how public like a frog link to video of me

july
07/05/11 poster by me still up today
07/27/11 miracles of the web r. shaffer painting spotted

august
08/15/11 go tell it on the ‘net broken security

2012

january
01/06/12 quarterly report after long hiatus

april
04/03/12 many dollars short day of higher ed

june
06/24/12 de profundis listening as punishment

gainfully employed

grading linear algebra again.
so far this quarter, i’ve (1) returned
the text from last quarter (late) and
i’ve (2) gone back and got a different
copy of the same text a week later.
and that’s it… a week and two days
into the semester.

but starting today there’ll be bigfat
envelopes full of homework piling up
in my mailbox. so very likely i’ll be
posting less over in the cooking show
(where i’ve had a pretty good run going
for about a week).

m’aidez

math stood tall: edusolidarity recap at JD’s.

the anti-social network

Mathblogging.org looks pretty promising.

(i couldn’t put this in a comment
for this “log laws” post
at f(t); too long.)

it’s the old principles-versus-procedures problem.
students hate general principles.

i’ve had tutees pay me hundreds
to supervise their homework
who very clearly “tune out”
whenever i try to explain
what’s going in in a general way.

it’s worst when it’s about a topic
they’ve been exposed to before…
the famous “math phobia”. panic
sets in when you realize that
you’re going to expose your ignorance
yet again… about fractions, variables, or
vocabulary (“equations” and “expressions”,
for example)… to name some common
weaknesses for *beginning* algebra.

the tendency is to believe that whatever
this “general principle” thing is that
teachers keep wanting to come back to
is secret-math-head *code* for something
that, if we’d only present it in plain english,
then they’d know how to work the problems
on the test (which had darn well better
be *just like the ones we practiced*).

so we give in and show ’em the “procedures”:
FOIL instead of “repeated distributive law”,
for example… and they *pass* the tests
but get *further behind* in the grand scheme.

“i’ve done OK this far by disregarding
all the math-brains-only *theory*…
and now i’m failing. but i’ll *never*
be a math-brain… good heavens,
i’d have to go back and understand
an *awful lot* of stuff i’ve been
given a pass on all these years!…
it’s the pedagogy! it’s my learning
style! it’s my teacher! math sucks!”

the phenomenon persists at all levels
(that i’ve passed through in learning
math up to my published thesis…
when, sure enough, i still felt myself
an imposter compared to the “real
mathematicians” i’d been working
with in grad school).

“logs” are a particularly interesting case:
otherwise-well-prepared calculus students
are often very weak on logs. hell, every
tenth student has decided they’ll just
go ahead and pretend “ln(x)” is to be
replaced with “1/x” somewhere along
the line no matter how many times
it’s marked wrong.

when “should” students learn about
the abstract, proof-heavy style
of math-rightly-so-called?
the typical student will never
*feel* ready for *any* new idea.
(this should be considered in
any discussion of “developmental
barriers”).

do-the-same-thing-on-both-sides.
logs-are-inverse-exponentials.

here’s a common-log equation:

log(4) + log(25) = log(x).

and here’s its solution.

10^[log(4) + log(25)] = 10^[log(x)]
10^[log(4)] * 10^[log(25)] = x
4*25 = x
x=100.

now, kate’s worksheets are great
and i’m not suggesting that anyone
make my approach into their
exhibit A for day-two-of-logs.
so don’t get me wrong.

the point i set out to make when i started…
if i can remember that far back… is that
*no* approach can overcome the simple
fact that our courses are designed to
go *much too quickly* for median-level
students. the good news is that we
can spot talent pretty easily this way:
anybody who can keep up typically
hasn’t “hit the wall” where math
gets *hard*; give ’em an A and
pretend we’ve done something
to be proud of.

my “discrete mathematics” students
last quarter were all calc 2 vets:
future programmers and engineers
and whatnot. and *almost* all of
’em learned quite a bit about
writing proofs. the *easy* thing
here is (of course!) the logic.
(i mean the symbol manipulation;
what all the p’s and q’s have to do
with the “logical structure” of
a given proof is much more obscure
than it appears to non-teachers.)

if *anybody* i’ve ever worked with was
developmentally ready to take on
“show using the definition of logs
that log(a) + log(b) = log(ab)”
as an exercise, it’ll’ve been these
talented hardworking students.

but the resistance was palpable and i folded.
there was… as always… much too much
material (i also had to skip stuff like
“manipulations with sigmas” and
“complete induction”… the class
emerged ready to take on maybe
a third of the exercises in the sections
covered [mostly the easiest ones]).

there *is* no developmental appropriateness
in “real world problem solving”: one needs
a technique that *works* wherever it’s found.

our task as math-teachers-in-schools,
though, is pretty much “preparing students
for yet another class”… and one technique
that “works”, alas, is *hide your weaknesses”.

that idea of “apply some inverse-thingy
on both sides of the equation”? one of
a handful of big ideas in algebra
(along with “variables”, “graphs”,
and “roots of polynomials” and…
well, not much else…)

“look: to undo an *addition*, we do
a *subtraction*… to undo a multiplication,
we do a *division*… and these ideas
were *hard* when we tackled ’em
back in some earlier course.
a little later, we decided that
we wanted to ‘undo’ squares,
and introduced ‘square roots’;
more generally (and about half
the beginning-algebra folks
get lost around the bend in
here somewhere) we’ll need
to undo powers-of-the-variable
like x^K, and introduce Kth roots.
well, then, doggone it, why shouldn’t
the same undo-the-operation
strategy work for K^x?”

i’ve ranted out this mini-lecture
not just to get it out of my system
(for today), but to demonstrate
that one need *not* have any
*formal* understanding of
“inverse function” to use the
never-stop-harping-on-general-principles
method that i’m more or less advocating here.

(indeed, inverses-treated-formally
is a classical crux. “never mind
*why* i solve x = f(y) when i want
a formula for f^{-1}(x)… just show
me *how*!”… one of the most
glaring examples of the
i’ll-never-understand-the-reasons
phenomenon known to me.)

okay. this was fun.
haven’t i got a *job* or something?