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!
while you still can.
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.
the livingston review 