## Archive for November, 2015

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

seven signs in seven positions.

PALEGAS and MRBGPYO (aka ROYGBIV)

and days-of-the-week and ages-of-

-man and whatnot… the trivium &

quadrivium… may be superimposed

in various interesting ways.

the seven principles of UU will

eventually be invoked if this

is ever (again) the basis of a

sermon by me. “window crayons”

on cardboard box; 2015.

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.

that 7-space has seven-way symmetry is obvious

(rotate the drawing by 2\pi/7—about 51.43\degrees—).

but the *three*-way symmetry isn’t so obvious.

here it is in its 7-color wonder.

we’ve chosen to fix the “Mud” point at the top.

we then chose the “secondaries” line

{Green, Purple, Orange} and permuted;

the “primaries” permute accordingly;

voila.

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

a 2-way symmetry can be displayed by “swapping”

each primary with its “opposite”:

(RG)(BO)(YP).

i call this one “reflection in the Mud”.

(one may also… of course… reflect

in any of the other colors;

for each (there are three)

“line” through a given color,

interchange the positions for

the other two colors.

the lines-on-blue are

{bgy, bmo, bpr}, so

“reflection in the Blue” has

(GY)(MO)(PR)

as its permutation-notation.)

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*.

or, the fano plane presented symmetrically.

each of the three triangle-edges

found along any of the “long lines”

(joining vertex-to-vertex

on the biggest 7-point “star”)

is a “line” of rainbow-space.

check it out. the “points” are

Mud Red Blue Green Purple Yellow Orange

the lines are MRG, RBP, BGY, GPO, PYM, YOR, OMB.

here’s a version from last year.

this new one’s much cooler.

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!

consider ordinary (x,y,z) space.

co-ordinatize a “unit cube” in the all-positive octant.

put “primary colors” on the axes. next slide.

from the edges of the cube, “cut out” the three that

pass through the Origin of our system—i.e., (0,0,0).

next.

distort the resulting diagram so that the “top face”

(and the “missing” bottom face) remain *square*. i’ve

shown this “flattening out” in two steps: once as a

truncated-pyramid in a “3-D” view, and then as a

fully-flattened 7-vertex “graph”.

meanwhile, introduce the secondary colors… in the

“natural way”.

all this is pretty old hat around here. you could

look it up.

the novelty here is the stick-figure iconography

(each of the “icons” has the “top of the cube”

represented by the square-in-the-middle; the

three nonzero vertices of the “bottom” of the

cube appear along the top and right-hand “sticks”

of a given icon).

each of these 7 icons now represents a

*linear equation*; these are precisely

the equations of the 7 2D-subspaces-

-through-the-origin of the vector-space

{(0,0,0), (0,0,1), … , (1,1,1)}

having exactly eight vectors.

one can calculate directly on the icons

(rather than the triples-of-numbers or

the colors) using “set differences”.

but that’s it for today.

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.