Archive for the ‘Cool Tricks’ Category

Photo on 2-16-17 at 11.02 PM.jpg

(because violet blue red… but that’s
off-topic [and off-color]…)

three green-peppers; none of ’em green.
it’s even beautiful in its package and,
of course, even more so the more they’re
played with.

as shown here, the tops have been cut off
and served (with their middles cut out)
with ranch. the next thing that happened
was that a “ring” was cut off at the top
of each and the whole set got bagged and
put away.

the rings, in their turn, were opened up
into long slices and split down their
middles, the long way, with a steak knife.
finally (thus far), diced fine and stirred
into a chicken salad (along with some
carrots, also chopped fine, and, obviously,
some chicken—one big american breast).
add mayo to taste; mix; serve. (serving
suggestion: ritz crackers. we’ve got
lettuce & tomato, though, so actual sand-
wiches aren’t out of the question.)

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Photo on 11-28-15 at 10.20 AM #2

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ABCDEFG
(seven chords seven ways, part mercury)

ten for maria.

1 – There isn’t enough user-generated content or “making your own math artifacts.”

equations, most likely, first.

but wait. zero-th.
by-hand copies of the *symbols*
for the material at hand.
“the student learns essentially
nothing until the student’s
pencil makes marks on the page”
is a pretty good first approximation
a lot of the time… or anyhow,
i’m far from the only teacher
given to *saying* stuff like this.

i’ve got plenty to say, too, *about*
this but i’m hoping for a list of ten
in under 2^12 characters (for a little
longer; i’ve begun to despair already
at least a little though if you want
to know the truth).
“unions” should look different from “u” ‘s
as an example more or less at random.

*our medium is handwriting.*

first-and-a-half.
out-loud discussion of and…

second.
…written sentences *about*
those equations. written
at leisure without the
instructor (or fellow student).

third.
similar or exact versions of such equations,
repeated, or, much better of course,
improvised, in a “public” setting
with small or, slightly better i
suppose, large *groups* of fellow
students. oral presentation of
the sentences themselves is not
only okay here but much to be
preferred (the board should not
be littered with sentences).
the “correctness” of the sentences
should nonetheless be at issue
throughout the presentation.
said “correctness” is to refer
explicitly to “code”…
utilizing (hey! ed jargon!)
the symbols from our step zero.

it does not escape my attention
that the “artifacts” created by
the student presentations i here
imagine are scribbles of chalk
on a board, soon erased. so be it.


leaving some out…

sixth
yick, computer code.

seventh
student-designed exercises,
exam templates, lesson plans…

eighth
songs and other verse, games,
comics and other graphics,
something to astonish even me.

ninth
blogs.

tenth
fanzines.

Reply

Sue VanHattumMarch 7, 2010 at 7:08 PM
Maria, I loved your list.

Owen wrote:
>”the student learns essentially
nothing until the student’s
pencil makes marks on the page”

Maybe for higher math, but not at all for young kids. The mathematical issues they’re working on don’t usually require pen(cil) and paper.

My son is thinking so much about what I’d call place value these days. “60 and 60 is 120, right?” “Yep.” No writing – at home, anyway. Lots of mathematical thought.

I’ve long been puzzled by your emphasis on “the code”. Maybe someday I’ll get it… :^)

Reply

AnonymousMarch 8, 2010 at 6:57 PM
My son is thinking so much about what I’d call place value these days. “60 and 60 is 120, right?” “Yep.” No writing – at home, anyway. Lots of mathematical thought.

I’ve long been puzzled by your emphasis on “the code”. Maybe someday I’ll get it… :^)
—sue v.

maybe today!

the “places” of “place value” are
places *in* certain symbol strings!
it sure doesn’t matter that you
*speak* of such strings without
having actual *written* code
in front of your actual eyes…
that’s not what i’m always
going on about at all…

60+60=120
presumably gets its interest
from 6+6=12,
together with, right,
the “place value” concept…
*as it manifests in base ten*.

now of course you and your kid
don’t have to have spoken of
bases-other-than-ten for
the essential *role* of “ten”
in discussions of place value
to have become quite clear
all around.

“what’s so special about ten?”
i can now imagine asking
some kid of the same age
if i were lucky enough to
know any…
and i’d sure enough expect
(maybe with a *little*
stack-the-deck prompting
from me) pretty soon to
start hearing about the
role of *zero* (in, again,
certain symbol-strings).

and when our conversations
*without* written work begin
to break down… and if we
still *care*… why then,
we’ll break out some *pencils*
and take a look:
“what do you mean, *precisely*?”.

we’ve been talking about code all along.

tangent.
calculating with numbers
is the very *model* of
one-right-answer-ism:
3*4=13
is just flat-out wrong.
and this is our greatest strength.

in principle, anything worth
talking about passionately
in a math class should have
the *same* character:
there *is* a right answer
if we could only find it.

in order to have this happen,
we have to agree on things.
we *can’t* agree… and be
*sure* we agree… and be *right*…
without certain so-called “rigorous
definitions”: marks on paper
(generally; otherwise
*verbatim verbal formulas*
memorized syllable-for-syllable
[mostly… i don’t seek a
“rigorous” definition of “rigor”…
“one is *this* many”
and its ilk (so-called “ostensive
definitions”) are all the rigor
we can *get* sometimes]).

generally the “rigor” one speaks of
is… i think… pretty *close* to the
being-able-to-calculate-it-out-like-a-computer
thing i spoke of (with reference to
elementary arithmetic) a moment ago.
and this comes from “code”.

again. our power in mathematics
comes to an amazing extent from
being-able-in-principle to emulate
some doesn’t-know-anything-*but*-code
*machine*.

now i’m as much of a luddite as the next
guy, if the next guy figures the wrong turn
was somewhere around “domesticated animals”.
but one *glaring* benefit of computers
in math ed is that students will work
for *hours* on getting code letter-perfect
(if they know no human being can see
their failures happening), that wouldn’t put
in five *minutes* of homework on paper
without getting so frantic about each
“move” that they fall apart before even
getting started. it’s that “interactivity”.
this used to break my heart but it’s true.

if schools were for clarity,
command-line programming
would begin in about first grade.
it’s much *easier* than almost
any other thing you can do
with a computer (which is why
it emerged much *earlier*
than the hugely-user-unfriendly
[from a “code” point of view]
*graphical* interfaces that
erased it from the national
consciousness in around 1984).

(somebody mention “logo”.)

math *is* hard.
but it’s much easier than anything else.
because we’ve got *all* the certainty.
(programming on this model
is of course a subset of math).

ot

borromean guitars

Photo on 11-27-15 at 11.33 AM

three newish guitar-stands arranged in such a way
that any two will fall down without the third.
and madeline’s “three women” statue, having a
similar property. blessings from our happy home
to yours if you’ve got one; double blessings if
you’re doing without. happy “black friday”.

the linking and not-linking rings
(blogpost of 05/03/014).
w’edia.
flickr shot.

the orange blend

Photo on 11-24-15 at 2.28 PM

exercise: draw the other six lines.
hint.

about time these guys got some names.
ROY (this one) is obvious.

let’s see.
green blend.
bgy byg gby gyb ybg ygb;
GABY, then.

secondaries.
gop gpo ogp opg pgo pog;
PEGGIE-OH, then, maybe.

the orange blur.
bmo, bom, mbo, mob, obm, omb.
JIMBO comes to mind.

submissions welcome. yes, it’s the great
name-the-lines contest of ought-fifteen!

i’ll be forced for the sake of my own peace
of mind to type out the rest of the exercise
and make up three more names. “the purple
blur” isn’t much of a *name* for a line.
more like a *secret identity*.

left-handed G chord

Photo on 11-20-15 at 3.32 PM

formed by my right hand. i got a new shipment
of strings today… thanks, madeline!… so i
(finally!) tuned up one of my guitars “lefty”.

if i try to *play* it that way, it sounds like
day one. but if i play it *right* handed, i
can make it sound like music. the chords and
the dynamics are different, though. so, cool
trick, it sounds like somebody else playing
(while still sounding a lot like me).

or maybe that’s the cough syrup talking. whee!

Photo on 11-17-15 at 10.46 AM

saturday night i colored in the corners of this cube.
the underlying black-and-white is based on a work of
the great dutch artist m.~c.~escher. the cardboard
cut-out version is from a collection by the american
mathematician doris schattschneider.
(_m.c._escher_kaleidocycles_).

anyhow, i’ve had the whole “5 platonic solids” set
from this work on display in the front room at home
for a while. the others are in color already, right
out of the book. i’ve had *another* set of these,
too: it’s a great “book” and might still be in print
for all i know. i had two editions, from years apart,
years ago.

i took this one to church on sunday and used it in my talk.
there wasn’t time to explain why i’d colored it the way i
did. but i *did* count the symmetry group of the cube,
two ways. any talk by me should have a theorem in it;
i’m happy to count that as a theorem.

24 because any of the 6 faces can be “face down”,
and each such choice-of-face allows for any of 4
remaining faces then to “face front” (all but the
“face-up” one *opposite* to our “chosen” face).

but also 24 because
1 identity
6 180-degree “flips” that fix two edges
(one for each pair of opposite edges)
8 120-degree “corner-turns”
(fix a pair of opposite corners;
there are 4 such; one may “turn”
right or left)
6 90-degree “face turns”
(fix opposite faces—3 ways; again,
one can go “right” or “left”)
3 180-degree face turns.

and this messy version is actually quite clear when
one is actually holding up an actual cube and pointing
at the drawing on the board. or in this case, at one’s
own sweatshirt. canvas makes a good whiteboard.

you can do it all without even mentioning “group theory”…
and i *sure* didn’t get to prove that this set of 24
“moves” gives a version of “the symmetric group on 4
objects”. anyway, part of the point is that one need
not have introduced any “math code” into the discussion
at all to arrive some some *very* useful results.

i learned the symmetry-groups-of-solids trick from an
old master
. i mentioned this book in the service.

but mostly i talked about stuff like bible studies and music.

stuff that people actually show up at church *for*. bring what
you love to church and share it. food and money are particularly
welcome.

Introduction
(Lines 1—40)

God (as I choose to call my higher power)
Grant me an audience for half an hour
And I will, if I can, do all the rest.
My subject is the story I know best;
I mean my own. It starts in a motel,
The night of my divorce. I felt like Hell.

Think of a pilot, learning how to fly,
Who, though he should know better, flies too high,
Then falls in the Atlantic and is drowned.
His body and the plane are never found.
There’s something like our marriage in that story,
The way it shoots to misery from glory.
The similarity might not be strong,
But, as to suffering, I’m never wrong:
Divorce is brutal. Trust me when I say
I’d rather be that pilot any day.

Lisa, in a voice that tore my heart,
Had told me, “From now on, we’ll live apart.
I’ll keep your stuff till you’ve got your new place.
The First Street house is mine. I want my space.”
And so, a stranger in my own home town,
I left my room to have a look around.

Across the street, a Big Red liquor store
And Waffle House. A porno shop next door.
“The restaurant then. For now, I’ll do what’s right.
I’ve got no strength for sex and drugs tonight.”
The waitress, call her Ruby, perked me up.
I never saw the bottom of my cup.
A refill and a smile, and off she’d glide;
She wore her sixty years with grace and pride.

“Why look upon myself and curse my fate:
I couldn’t stand to only serve and wait.
I’ll bet that woman’s life is harder still
Than mine, by far. And, if I only will,
I could throw all my misery away
And love my life the way it is today!”

If that was true—and I don’t think it was—
I proved myself an awful fool, because
For years I didn’t love my life at all.
The story starts with my decline and fall.

dick’s exegesis.
inspiring visions of an inspired visionary.

6.1.5
Let a, b, c, & d
be integers satisfying
b > 0, d > 0, and
ad – bc = 1.
Let n = max{b,d}.
Then a/b and c/d are consecutive
fractions in the n^th Farey Sequence.

Proof:
Since ad-bc=1, we have
ad/(bd) = (1+bc)/(bd), hence
a/b = 1/(bd) + c/d and so
a/b > c/d.

Note that (cf. Corollary 6.2)
ad-bc = 1 also implies
(a,b) = 1 and (c,d) = 1;
our fractions are in
“lowest terms” (and hence
appear in F_n).

Now suppose c/d & a/b are *not* adjacent.
Then there is at least one fraction,
which we may assume is in “lowest terms”,
“p/q”, say, with (p,q) = 1, such that
p/q is *between* c/d and a/b
in the Farey sequence F_n:
F_n = { …, c/d, …, p/q, …, a/b, …}.

Then
q = (ad-bc)q
= adq ___________ – bqc
= adq – bdp + bdp – bcq
= d(aq-bp) + b(dp-cq)
(*).

But a/b > p/q implies
aq – pb > 0, and so
(since the variables are integers)
we have that
aq – bp is greater-or-equal to 1
(aq – bp \ge 1).
The same computation with different
letters shows that
[p/q > c/d] \implies [dp – cq \ge 1].

These inequalities together with (*) give
q = d(aq-bp) + b(dp-cq)
q \ge d + b.

Recall that n = max{b,d}.
It follows that d + b > n.
This gives us q > n, a contradiction
(“p/q” cannot be in F_n with q > n).

Hence a/b and c/d *are* adjacent in F_n; done.

\sharp \{ (v_1, v_2, \ldots , v_k) \in (\{ j\}_{j=1}^{n})^k \mid [a \not= b] \Rightarrow [v_a \not= v_b]\}=
{{n!}\over{(n-k)!}} = \null_n P_k.
— the number of “permutations of k things
from a set of n” (“en-permute-kay”, in its
most convenient say-it-out-loud version);
the number of ways to make an (ordered)
*list* of k (distinct) things chosen from
a set of n things. one has known this from,
well, time immemorial. but never written
it out in straight-up *set* notation
until today.

now it’s on page 5 of my “bogart”.
(_introductory_combinatorics_, kenneth
p.~bogart, 1983; mine’s a recent acquisition
from the math complex, once of the
“bertha halley ross collection” according
to its bookplate. that turns out to have
been mrs. arnold ross, late of the OSU
[an all-time giant in recruiting new talent
into mathematics, founder of the “ross program”].
you can see from the edges of the pages that
somebody read it up to chapter six and stopped.
they didn’t write on the pages though.
[but that’s being put right now.])

and cranking out suchlike “set code” *is* fun
and easy… for me. showing somebody how to
*read* it? fun but *not* easy. showing some-
body how to *write* it? well… how would i
*know*? (how would *anyone*?)