Andrius World and Clifford Algebra

[Andrius Kulikauskas]

Kirby,

You encouraged me to look at Clifford algebras at some point. Today I
took a quick look at the Bott periodicity theorem:
https://en.wikipedia.org/wiki/Bott_periodicity_theorem
Which, of course, I don't understand 95% of. Bott periodicity came up
in one of the videos I listend to as something fundamental.

Well, what struck me is that there is an 8-cycle called the Bott
periodicity clock, also known as the Clifford algebra clock.

In Andrius Philosphy World there is an 8-cycle of the divisions of
everything:
* into 0 perspectives (God)
* into 1 perspective (Everything)
* into 2 perspectives (like free will and fate, for Being)
* into 3 perspectives (taking a stand, falling through, reflecting, for
Participating)
* into 4 perspectives (whether, what, how, why) for Knowledge
* into 5 perspectives (for perspectives in time (alternatively, space))
* into 6 perspectives (for Morality)
* into 7 perspectives (for a closed Logical system with some Slack inside)

An 8th perspective would be "all is good and all is bad" which means
that the system is empty, we have a contradiction, and we go back to 0
perspectives.

Now on this 8-cycle there are 3 shifts. They are for reflections that
add +1 perspective or +2 perspectives or +3 perspectives. The latter +3
is consciousness. For example, 2+3 = 5 means that consciousness (+3) of
issues of Being (2) is given by issues of Time/Space (5). Basically, it
says that your mind is like a trolley that moves from one trolley stop
to the next. We use trolley stop 5 to describe your consciousness (+3)
of trolley stop 2. I've spent my whole life mapping this out but this
is just a quick prompt for your imagination.

Well, lo and behold, there are "clock-shift" operators in the
Generalized Clifford Algebra:
https://en.wikipedia.org/wiki/Generalized_Clifford_algebra
Some matrices describe the 8 cycle clock (the trolley stops) and other
matrices describe the 3 shifts (the trolley cars of different increments
+1, +2, +3). These are generalized Pauli matrices:
https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices

Weird.

Just to add more of what I'm looking for. There are 4 representations
of the first four divisions (trolley stops) and 2 representations of the
next four divisions. For example, on questions of Knowledge, we have to
choose between thinking in terms of questions (Whether? What? How? Why?)
like an idealist, with Whether? being the uninteresting ground floor, or
in terms of answers (Whether! What! How! Why!) like a materialist, with
Why being uninteresting. Thus there are 4+2 = 6 ways of looking at the
whole of the division. Also, there are 12 "topologies", ways of
snatching out a single perspective.

So these 8 divisions, 6 representations, 12 topologies are the
fundamental "tables of perspectives" that describe the mind statically.
Well, by coincidence, a cube has 8 corners, 6 faces and 12 edges. In fact,
24 / 2 = 3 x 4 = 12
24 / 3 = 4 x 2 = 8
24 / 4 = 2 x 3 = 6
There also seem to be 3 dynamic languages:
* argumentation taking us from 12 topologies to 6 representations
* verbalization taking us from 8 divisions to 12 topologies
* narration taking us from 6 representations to 8 divisions

These 3 static tables and 3 dynamic languages are "Third-Person
(He/She)" structures. There are also 4 "Second-Person (You)" structures
of 8 perspectives each. It seems that pairs of the latter 4 generate
the former 6. Together they are like the 10 commandments: 4 for loving
God, 6 for not hurting your neighbor.

Each of the 4 "You" structures links a null perspective (God) followed
by a backbone of six perspectives and ending with a seventh perspective
(-1 = slack = good). If we say it is the same God and the same good
then we have 6x4 = 24 + 1 + 1 = 26. So we have the 6x4 = 24 + 2
dimensions which seems to come up in String theory and the monster group.

The 8 fold You structure and the 10=4+6 He/She structures and a 3-cycle
and three states 1+1+1 together combine to give the 24 ways of figuring
things out that I had said I think are at work in each "world".

What I have been working most intensely on is the key to all of this
structure. I call it "God's dance". I start by imagining how God gets
going. God asks, is God necessary? Would there be God if there was no
God? So God removes himself, but being God, has to reappear. But is
the first God (who understands) the same God (who comes to understand)?
Yes because they understand the same God. So these 3 angles on God are
the same God. But that's how it looks like to God #1 when God is "I"
and God loves himself. For God #2, God is "You". This is the case that
we find ourselves, the most unfavorable circumstances for us to be God.
God #2 and God #1 are one through their perspectives on each other, and
they love each other. This takes place through and 8 fold You
structure. Then God #3 sees this same but now from the side,
objectively, so that God is third-person "He/She". Then God #1 and #2
are the same through their non-existence, that is, their circumstances,
which are one. And they love all, either by loving God (4 ways) or
loving neighbor (6 ways). Finally these three unities (of God, of
Person, of People) are united by a human three-cycle of taking a stand,
following through, reflecting. So all together in God's dance we have 3
+ 8 + 10 + 3 = 24 expressions of God in my imagination.

Mathematically, it is a state of contradiction (God) relating with
itself so as to produce a state of non-contradiction, a system of truth
(goodness).

So you can imagine that the same kinds of small numbers that are so
highly restrictive of advanced mathematics 2, 3, 4 are similarly the
same kinds of small numbers that are highly restrictive of my
imagination as well, and I think, of all imaginations. You can see why
I'm optimistic that at the bottom they are rooted in the same. But I
think I have the more powerful way to approach it because I'm working on
a model of human life. My work is defined implicitly (with reference to
the imagination) whereas math is nowadays supposed to be completely
explicit (written down - ignoring the role of imagination in
interpreting it).

But the Clifford algebra clock is a coincidence of higher order. Also,
it points to the importance of very particular structures, some of them
the Lie groups (unitary, orthogonal, symplectic) and others like the
quaternions.

So it will be exciting to choose this as a central point in Math and
show how it is a hub for whatever. And it's that much more motivation to
learn it and try to link it to my own model of human life.

Good night, Kirby and all.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

[kirby urner]

On Thu, May 5, 2016 at 4:51 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

Kirby,

You encouraged me to look at Clifford algebras at some point.  Today I took a quick look at the Bott periodicity theorem:
https://en.wikipedia.org/wiki/Bott_periodicity_theorem
Which, of course, I don't understand 95% of.  Bott periodicity came up in one of the videos I listend to as something fundamental.

<< edit >>
 

These 3 static tables and 3 dynamic languages are "Third-Person (He/She)" structures.  There are also 4 "Second-Person (You)" structures of 8 perspectives each.  It seems that pairs of the latter 4 generate the former 6.  Together they are like the 10 commandments: 4 for loving God, 6 for not hurting your neighbor.

Greetings Andrius --

Interesting reverie, thanks for sharing it. [1]

Your thinking about perspectives / pronouns / tenses relates in
my own thinking to gamification  / simulation in that different games
adopt different tenses, sometimes more than one.

Many games adopt a 3rd person "God's eye" point of view. 
'Civilization' for example, but many others as well ('SimCity',
'SimEarth').  http://www.simcity.com/

"God's eye" is an interesting viewpoint is it's not one any of us
share except in our imaginations.  We're each in a first person
mode.

But in film especially, we take the "omniscient" or "disembodied"

viewpoint for granted.  The camera "eye" is free to look from

just about anywhere, spying on characters in their most intimate

moments.  Sometimes we even here their thoughts (example:

'The Room', in which the five year old's thoughts come to us by

audio track at various points in the movie).

Novelists typically choose a "god's eye" view, and often they

dive into a character's thoughts, in ways no first person viewpoint

so readily allows (unless it's a character with mind-reading

superpowers).

Many computer games are "first person", often "first person
shooters" although shooting may not be the core activity. 

The game 'Alice', which I've played quite a bit [2] allows switching
between first person and third person viewpoints. 

But the 3rd person is anchored around the main avatar, Alice, i.e.
there's no flying off to look from just anywhere.

Once computer games moved to the cloud, including other players

became doable. 

Casting others as adversaries is a no-brainer, but why not allow teams
with a common challenge or objective?  Let players be on the same side
as collaborators.  Several games go that route, with many more to come.

So that's the three "viewpoints" we learn in grade school grammar

(if grammar is taught at all, in my school it very much was):  first,

second and third person. 

Seeing how computer games simulatem all three is an important lesson
in language / logic & grammar.  There's a mathematical aspect to all this.

Kirby

in Champaign-Urbana, Illinois

[1]   http://coffeeshopsnet.blogspot.com/2010/04/smart-bar-lcds.html

[2]   https://en.wikipedia.org/wiki/American_McGee's_Alice

[Joseph Austin]

Kirby, Andrius,

I've stumbled on an interesting presentation of Clifford Algebra, by Steven Lehar.

https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/

It's special appeal is that it is profusely illustrated with color diagrams, some animated,

which is only fitting for a geometric topic. 

So far I have just skimmed it, but I plan to peruse it in more detail.

Lehar identifies himself as "an independent researcher with a novel theory of mind and brain, inspired by the observed properties of perception."

Joe Austin

[kirby urner]

Hey Joe that looks like a really excellent web page, giving lots of history as well, explaining what happened with Gibbs-Heaviside vectors getting to be center stage.

My question about using triangles and tetrahedrons is not about changing any of the operations or numbers, only the diagrams.  The bivector and trivector would be triangle-defining and tetrahedron-defining, rather than parallelogram- and parallelepiped defining.  But not because any numbers had changed, just how we represent these numeric quantities.

Given there's no numeric difference, I see this as an overlay or "skin" we may stretch over the very same Clifford Engine.

You've seen on Youtube how it works with two vectors.  It works the same way with three vectors.  I call it "closing the lid" or "putting a lid on it".

http://controlroom.blogspot.com/2016/06/putting-lid-on-it.html

In other words, if my vectors A, B are length 3 and 4, and at 60 degrees, I draw the third edge of length A-B and say that triangle has area 12.

In other words, take whatever area or volume number we say goes with the square or tetrahedron, and simply apply that to the simpler shape, using 60 instead of 90 as the reference angle.  These were cultural conventions, axioms, not theorems.  We're free to play around.  Legally.

Outside of Clifford Algebra it's easy enough to show that we have a choice:
http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html
(shows 60-90 equivalencies for 2nd- and 3rd power whole numbers, as but the equivalence carries over to any A x B or any A x B x C).

Thanks for that link.  I'll be reading it more.

Kirby

PS:  more PR for quadrays #4D on Twitter today, sharing my latest Jupyter Notebook:
http://mybizmo.blogspot.com/2016/06/legal-in-oregon.html

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[kirby urner]

On Wed, Jun 15, 2016 at 7:25 PM, kirby urner <kirby...@gmail.com> wrote:

<< SNIP >>
 

In other words, take whatever area or volume number we say goes with the square or tetrahedron, and simply apply that to the simpler shape, using 60 instead of 90 as the reference angle. 

.".. with the square or cube" I should have said. 

Then map the same area or volume number to the topologically simpler shape in each case:  triangle or tetrahedron.

Of course this is what I do in Martian Math, but with some caveats.

If one were planning to turn one's back on the "old currency" as it were, there'd be no need to come to grips with some "conversion constant". 

One unit replaces another so there's no need to inter-convert?  That's not realistic.  XYZ is not going anywhere. 

It's well-established and under no threat. 

If I say what sound like critical remarks about it, that's merely to tease open the imagination for a different gizmo.

It's more a matter of finding a place in the sun for these 4D IVM whole number volumes (RD 6 and so on), as a part of some curriculum somewhere, a matter of religious freedom maybe.  It's a big world.  That at least a few schools would include this stuff is hardly unrealistic.  Of course a few do.

Anyway, "replacing the cube" (English:  "qyoob") was not the plan.  That would have been quite futile.

An overly defensive reflex is to think whenever we propose an alternative reading or model, that we're trying to *replace* as in "displace" the accepted reading.  People give lip service to multiple axiomatic systems and already make room for two namespaces using 4D (timeless and time-free), but they don't realize what this means in practice.  It means maths is *not* one giant castle of knowledge, a single edifice, but is rather many sandcastles along a roomy (spacious) beach.

That kind of either / or thinking is what I try to avoid.  You seem mentally nimble enough to avoid it yourself, so take these as broadcast reminders, including to self.

Cyberspace (Cyberia) has plenty of room for multiple sandcastles.  It's a very roomy beach.

Martian Math is all about how to keep both models of multiplication alive and well and inter-convertible, as well as accessible. 

Why anyone would need to access Martian Math in real life?  That question has many answers. 

Learn more Art History and Architecture. 

Learn more Literature. 

I don't have to resort to "you'll learn more math" as my chief appeal, as I learn from The Math Myth by Andrew Hacker that many people tend to think they've had plenty of math already.  They're not wanting to be force fed.

I want to reach a broader audience than just professional mathematicians and math hobbyists anyway.  So I focus on poetry (P) and history (H) in PATH, intersecting with STEAM as we've discussed.

Kirby

[Joseph Austin]

On Jun 15, 2016, at 10:25 PM, kirby urner <kirby...@gmail.com> wrote:

You've seen on Youtube how it works with two vectors.  It works the same way with three vectors.  I call it "closing the lid" or "putting a lid on it".

I'm confused.  I've read that the tetrahedron cannot fill space. But the Fuller figure suggests that it does.

(I'll try to build it with my straws and pipe-cleaners when I get home and see how it works.)

BTW, where do you get the "tinkertoys" for the structure you are holding in the photo on Bizmo Diaries?

Joe

[Peter Farrell]

Fascinating presentation! Thanks for posting it!

[kirby urner]

On Thu, Jun 16, 2016 at 9:34 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Jun 15, 2016, at 10:25 PM, kirby urner <kirby...@gmail.com> wrote:

You've seen on Youtube how it works with two vectors.  It works the same way with three vectors.  I call it "closing the lid" or "putting a lid on it".

I'm confused.  I've read that the tetrahedron cannot fill space. But the Fuller figure suggests that it does.

(I'll try to build it with my straws and pipe-cleaners when I get home and see how it works.)

Good questions Joe.

The Fuller figure accommodates the fact that regular tetrahedrons fill space in complement

with octahedrons.  That's what I've been calling the IVM, in architecture known as the octet-truss,

for which Fuller at one time held a patent (these expire).  Alexander Graham Bell was into the

same scaffolding.  Great minds think alike I guess.

The National Council of Teachers of Mathematics (NCTM) used to have an IVM for its

logo, a layer of tetrahedrons and half-octahedrons, like in that Fuller figure.  They swapped

this out for an infinity symbol in the 1990s sometime, maybe signifying a shift in emphasis

from discrete to continuous (digital to analog) math.

The octahedron has a pleasant volume however:  4 (same edge length). 

So when you think of a regular tetrahedron doubling along every edge, think of four
tetrahedrons the size of the original at each tip, and an octahedron at the center. 
4 + 4 = 8.  8 times the original volume.  It all works out.  Volume changes as a

3rd power of change in linear distance (area as a 2nd power).

For something to be a "unit of volume" it's not required that it fill anything in particular i.e.

cubes do not often exactly pack what they measure (a sphere or whatever). 

The target shape will not have a nice whole number volume, or even if it does, the
cubes don't precisely fit.  Just fill them with liquid instead then, to show the relationships.

Works the same with a tetrahedron.

Also, tetrahedrons do fill space, just not regular ones.  Apologists for Aristotle like to

point that out, when defending his proposition that "tetrahedrons fill space" as I gather

in the ancient Greek, the word "tetrahedron" does not imply "regular", same as in

English.

An irregular space-filling tetrahedron is called the MITE for Minimum Tetrahedron in

Synergetics and consists of 2x what we call A modules + 1x what we call B modules.

Eight MITEs make a unit volume oblate octahedron shape that also fills space, known

at least since Archimedes' time (though maybe not with that nice volume).

You'll find these A and B modules listed in the volume chart on Wikipedia (Synergetics): 
both have a volume 1/24, meaning our canonical space-filling MITE has volume 1/8.

I was just tweeting this morning to ask if any of the MIT Open Courseware (@MITOCW)
shares about these A and B modules (also T, E and S modules) in the tradition of
Dr. Arthur Loeb, who used to teach at MIT and Harvard.  Crystallographer.   Knew
about A and B modules (I know because I interviewed him myself on that question).

Were these called something else previously, these As and Bs?  People sometimes
get upset with Fuller for not sticking to the accepted nomenclature but then sometimes
he's zooming in on shapes others have not found worthy of a name?  Given others

started with the cube, it's not surprising they missed naming the A and B modules.

Anyway, we have this slogan:  "Aristotle was right, remember the MITE".

 

BTW, where do you get the "tinkertoys" for the structure you are holding in the photo on Bizmo Diaries?

Yeah, that's an older toy called "Rhombagon" or something like that but I can't find
it on Google and no longer have any boxes with branding on them, if I ever did.

I have quite a nifty little geometry toy collection.

Weird scenes in Photostream today, show more of what's happening.

Example:

Icosahedrons galore, spring-interconnected:  https://flic.kr/p/JdNn5F

Kirby
 

[Steven Lehar]

Dear Andrius,

Thank you very kindly for alerting me to the existence of the mathfuture group! And thank you also for promoting my visual introduction to Clifford Algebra / Geometric Algebra.

You and the group may also be interested in this paper of mine on the Double Conformal Mapping, an extension to David Hestenes' Conformal Geometry extension to Geometric Algebra, which relates directly to my theory that the origins of mathematics lie in the laws of perception.

https://doubleconformal.wordpress.com/

I also have a book in progress (not yet complete) titled The Perceptual Origins of Mathematics.

https://slehar.wordpress.com/2014/09/12/the-perceptual-origins-of-mathematics/

As with my Visual Introduction to Clifford Algebra, I prefer to explain math in pictures rather than equations, wherever possible, to clarify the connection to perception.

Indeed the extraordinary Grand Unification of math accomplished by Clifford Algebra stems from the discovery that all of algebra is a branch of geometry, and that most mathematical operations can be represented as spatial operations on spatial structures. This makes my writing immediately accessible to the non-professional mathematician.

I intend one day to write a book that explains all the most interesting aspects of math in simple intuitive terms that most anyone can understand.

Thanks again for making contact with me!

  Steve Lehar

[Joseph Austin]

The mighty MITE might indeed fill space.

So indeed would the "pyramid" (4 MITEs or irregular pentahedron) which is the word used in the translation of Aristotle that I read.

As for instability of the rectangle, I inadvertently demonstrated that by building a shelf unit with two "ladders" formed by bolting 2x2 cross-braces to 2x4x.

(Using plans published in a handyman magazine.)

In the course of time, the joints became pivots.  Fortunately, no living beings were in the way when gravity eventually prevailed.

BTW, I was reading about Quaternions, and note the Hamilton was concerned with division (ratio?) of vectors.

Which he accomplished by factoring out the magnitude from the directions.

Which fits right in with my intuition that "volume-producing product" is not really a "natural" operation.

I suggest that volumes exist a priori; the "real" operation is "projection" of a multi-dimensional volume onto a lower-dimensional entity.

Granting 3 dimensions. we choose a regular or equilateral projection as something of a normal form,

but there in nothing inherent  in the nature of "volume" that determines the shape of the boundary.

In physics, moreover, there is not even a restriction that each of the factor dimensions be commensurate, or of the same physical meaning.

E.g. we can have "square seconds" but also kilogram meters, as well as quotients of kilogram meters and square seconds.

We call these vector operations "multiplication" because of the way they relate to magnitude,

but the directional components, as Lehar points out, behave more like addition or even subtraction.

From a physics perspective, what concerns us is that the mathematics has an interpretation consistent with the measurements of experiments.

We care about how an electron moves in the vicinity of a magnet, (which once had practical application in putting pictures on TV screens, or saving bits of data in a computer); whatever mathematical formalism allows us to relate measurable observables of the trajectory to measurable of the magnets and electrons, will do.  Apparently Hamilton's quaternions worked well enough for Maxwell, until Gibbs came along with his vectors.  And I passed courses in E&M with Gibb's vectors before ever hearing of Hestenes' blades.

Einstein did "thought experiments" or conceptual models; his students (perhaps his wife) did the math.

Perhaps I will need to read Hestenes' Mechanics before I get a truly intuitive idea of what a "bivector" or "pseudo-scalar" is,

in relation to physical observables.  But I think I'm happier with the idea that a "product" of two quantities is an area rather than a normal vector.

Which gets to my fundamental concern that, in a physical sense, multiplication is not "closed", in the mathematical sense;

that is, the product of two whatevers, of a this and a that, is not another whatever, is neither this nor that, but is a conglomerate, a  "thisthat" of different dimensionality altogether.

I see hints of that idea in GA, but I'd like to see it more developed by putting "units" explicitly into the formalism.

That is, the bases e1 e2 .... eN are not just magnitudes, or directions, but also "types".

Or perhaps the "types" are "bases" of the "magnitude" part, and we preserve the existing bases of the "direction" part.

Scalar multiplication, or multiplication of a measurable by a number, is then just repeated addition.

But even so, it is not closed: "three" times "apple"  gives neither pure numbers nor just fruit; it gives a "quantity" of 'things"

I'm not sure how all that would map into tetrahedra, but I suspect it would "skew"er therm!

Joe

[kirby urner]

On Thu, Jun 16, 2016 at 6:34 PM, Joseph Austin <drtec...@gmail.com> wrote:

The mighty MITE might indeed fill space.

Indeed it does. 

A mathematician named Sommerville studied which tetrahedrons fill space in the 1920s, with no left or right reflections (he wasn't counting those).  He came up with the same ones Fuller did, but of course used different nomenclature.  I have no idea if either was aware of the other's work.

The 8 MITE space-filler, an oblate octahedron, called a Coupler by Fuller, plays a strong role in what are called Archimedean dual honeycombs.  Again, it's a matter of nomenclature (namespace) that keeps the Synergetics concepts walled out.  There's an established academic nomenclature that covers these shapes and a tacit agreement to keep the 4D IVM namespace out of the wiring diagrams (i.e. out of the curriculum).

In declining to adopt / invest in A and B modules of volume 1/24, the drawbridge is kept raised, if that's the right metaphor.  It's OK to talk about the dome a little, but not about these irregular tetrahedron-shape capsules of equal volume that aggregate in such and such a way. 

That all got shelved after Dr. Arthur Loeb's tenure at MIT apparently?

Fuller's works get a lot of of lip service.  Nature Magazine circled Operating Manual... as one of the most influential [1]. 

His positive futurism and pronouncements about a potentially upbeat future (as in 'Utopia or Oblivion') get cited, the dome is shown.  But when it comes to the unit volume tetrahedron and an alternative not-cubic model of 3rd powering, that takes more concentration, plus is sixth grade level stuff (very accessible). 

No self respecting individual wants to revisit sixth grade level concepts by the time they're an adult, a working professional.  That's to late in the game.

What's on our side is math itself, which never claimed to be monolithic. 

Think of how many board games stack up in some closets.  Each one has rules. 

Once the rules are agreed to and applied consistently, exceptionlessly (whatever that means), we have the nucleus of a potential math, a growing crystal.  So many.  The axiomatic beginnings yield their logical consequences.

I've also drawn quite a bit on an essay by Karl Menger, a dimension theorist.  He suggested a "geometry of lumps" that'd be non-Euclidean in not requiring the same "dimension talk". 

This proposal of his fit well with Synergetics in making anything definite (like a point) a tetrahedron at minimum, a topological maneuver, a kind of axiom in graph theory (a tetrahedron is the wireframe with the fewest nodes and edges that defines a volume with a convex outside and concave inside).

Everything is a "lump" in Menger's proposed namespace.

What's important, in part, is to remind us of our freedoms. 

We're not locked in to just one set of axioms, and it's not either / or.  Pretending we're locked in and can't afford to think outside the box a little, is just that:  pretending.  Thanks to math allowing many axiomatic systems, we get to jump around among the many sandcastles, not the prisoner of any one of them.

Kirby

[1]  http://mybizmo.blogspot.com/2015/12/global-data-global-matrix.html

(this blog post talks about Glenn a lot.  He's the gent in recent Photostream pix working with "flextegrity", invented by Sam Lanahan.  Here's a picture of Glenn and Sam holding a piece of Flextegrity from earlier today:   https://flic.kr/p/Hie7g4  )

[Bradford Hansen-Smith]

Joe,

I'm not surprised your are confused.  Why do we want to fill space in the first place? The traditional tetrahedron as a unit will not fill space. Kirby spoke well enough to this. In spherical packing of the same size the tetrahedron and octahedron are inseparably one, and will fill space as polyhedra within the isotropic vector matrix. In the closest packing of spheres the center of the space between the spheres fill that space as a pattern of cubic relationships. Stretch a bit further and consider when you fold a circle in half you have created from movement a dual tetraheron relationship from four points (two imaginary points, mark them if you want to see the points, then touch them together and crease.) There are now 4 points and 2 tetrahedra as the circle folds on the diameter axis 360 degrees; positive/negative, front/back, top/bottom. Don't think just straight line when you have a circumference, the complete movement of 2 tetrahedra fill the entire spherical envelope. You can't get away from it, temporal reality is a duality, two which is also three. Math being an abstract generalization does not tell us that one in recognition of one (itself) is three, the function of recognition is movement to the other is the third part, three. It goes back to folding the circle in half creating a duality of two areas and the line of division making three. The tetrahedron itself is a relationship of four vertexes forming triangles. It can be described as two sets of two points but the nature of the tetrahedron is not 2 it is 3, it is structural. Two tetrahedra balanced symmetrically in a compound form a cubic pattern of eight vertex points. Now we see a dual form of the tetrhedron as an all-space filling patterned unit seen in the spherical matrix.

There are many ways to approach your confusion, "tinkertoys" is only a linear form, one of many that can give sense to the confusion of information. All five regular polyherdra are not centered systems and can be formed by the tetrahedron in multiples through transformations. any center is an abstract assignment of location and has no spatial reality. The vector equilibrium is the primary centered system. When Fuller developed the Vector Flexsor he removed the center allowing it to collapse through the five regulars that have no center. You will not understand all this and it will probably confuse you even more, but know there is much depth in geometry we have yet to discover.

As has been noted, math has been discovered not invented. It is the abstracting of relationships and functions that are everywhere in nature that were first recognized in geometry. That given we have to go back to geometryalgebra to clarify the confusions that has grown from the incestuous math logic that breeds distorted truths. Unlike all polygons the circle is whole, it is both compass and straight edge. We do not know that because math is given to us in line with focus on the constructed straight edge; it is easier to measure is my guess. We could not have discovered either  geometry or abstracted math as we have if it were not in the circle/sphere in the fist place. My suggestion is start with folding the circle and look for as much as you can recognize as a place to start. As far as I can tell that is where it all is, in one place, before fragmented thinking takes hold; as I tell students, look for what you don't see then we will talk about what we do not know to gain clarity on what we do know.

Brad

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[kirby urner]

On Thu, Jun 16, 2016 at 7:32 PM, kirby urner <kirby...@gmail.com> wrote:

On Thu, Jun 16, 2016 at 6:34 PM, Joseph Austin <drtec...@gmail.com> wrote:

The mighty MITE might indeed fill space.

Indeed it does. 

A mathematician named Sommerville studied which tetrahedrons fill space in the 1920s, with no left or right reflections (he wasn't counting those).  He came up with the same ones Fuller did, but of course used different nomenclature.  I have no idea if either was aware of the other's work.

The 8 MITE space-filler, an oblate octahedron, called a Coupler by Fuller, plays a strong role in what are called Archimedean dual honeycombs. 

On this chart by Guy Inchbald you'll see the MITE and Coupler both depicted:

http://www.steelpillow.com/polyhedra/AHD/AHD_Chart.pdf

The Mite is the "trirectangular tetrahedron" 2nd row right, and the Coupler is the oblate octahedron towards the bottom.

In addition to Mite, Fuller defined the Syte, Rite and so on. 

http://www.rwgrayprojects.com/synergetics/s09/figs/f86427.html

http://www.rwgrayprojects.com/synergetics/s09/figs/f86429.html

The 1/4 Rite is one of the Sommerville's.

The Inchbald chart makes no mention of A or B modules as that dissection / analysis is just not part of the Archimedean game.

These concepts are not "hard" though in the sense that animations / cartoons make it easy to follow.  Sure, there's extended Euclidean geometry of N dimensions sharing the road here, with lots of jargons.  It's not either / or right?  We're free to pitch a tent and share our lore as well, correct?  N-D Euclidean geometry is not the only language game in town.  There's non-Euclidean.

Kirby

[Joseph Austin]

As a mathematician, one can play math for fun.

As a natural scientist, one wants the math to "mean something".

I could say that the "Electromagnetic Field" is a Quaternion space, or a Gibbs vector space, or a GA multi-vector space.

But what it "really" is is the way currents and magnets and electrons, etc. manifest themselves.

Actually, those terms themselves are just "theories" of whatever the the "reality" may be that manifests the observable behavior.

But we know how to shoot "electrons" from a gun at a phosphor screen to make TV pictures,

and we know how to wrap wires around iron and connect them to batteries to make "magnets" and record data on iron-coated discs, etc.

And we know how to make circuits that send "electricity" up and down a wire and broadcast signals the other circuits can detect miles away.

And some of those things we didn't even know could be done until somebody put the "theory" of electromagnetism into a mathematical form and discovered that the equations had solutions that suggested possible modes of behavior that we hadn't previously expected.

On the other hand, mathematicians had not developed calculus until science had a phenomenon that needed an representation that didn't yet exist.

So there's a synergy between math and science.

Today we have the elementary particles and entanglement and neutrino oscillations and dark matter to challenge the existing math models.

GA seems to be successful in consolidating existing theories, and may even suggest alternative conception of studied phenomena,

but who knows what new math may prove useful in addressing the frontiers of our observations?

A world made of tetrahedra may be useful in art and architecture, maybe organic chemistry, 

but I haven't seen claims that it is useful for constructing the nucleus.

That's not to disparage tetrahedra, any more than saying drills are not the the best tool for driving nails disparages drills.

Do tetrahedra have something to say about programming?

Topologically, we've been using the 4-pointed diamond to diagram decision trees and loops in program flow-charts since the beginning.

More generally, the "math" of topological structures for representing programs and organic compounds seems a fertile field today.

Compared to what we understand about functions, I've seen very little on structures.

For example, I would expect there to be a structural "grammar" for molecules and for computer programs,

and an "algebra" for reactions of molecules and transformations of program structures,

An "algebra" for programming could be useful for combining chucks of different programs to create a new one, perhaps semi-automatically.

It could change programming language design from an ad-hoc exercise to a rigorous process.

Joe

[kirby urner]

Greetings Joe --

I appreciate your meditation.

In terms of tetrahedrons having relevance, I think we've seen a marked shift in recent decades from a more exclusivist 90-degree sensibility to one that includes hexagons, including when tiling a sphere surface.  Perhaps the architecture of the virus and discovery of buckminsterfullerene both had something to do with this, and the emergence of the "hexapent" into almost-popular consciousness.  I've found it a strong meme in the computer gamer community in any case.

I associate the tetrahedron with both 60 and 120 degree angles, whereas XYZ is rectilinear (90 degree, orthonormal).  In ball packing we have patterns named BCC, HCP, CCP and SCP, for body-centric, hexagonal, cubic and simple close packing. 

Those are not household names, but are native to STEM and come up when writing 3D graphics generators, patterns such as these become the target for exercises.  3D versions of kites and darts enter in.  Quasicrystals -- a hot area for new patents, some with Princeton's name on them (the Board of Trustees).

You've seen how I use quadrays to map these spaces, as a way to keep my programming skills in shape.  Jupyter Notebooks.  Whatever.

If you dig into Synergetics you'll find like a Feynman diagram with neutrino and anti-neutrino in a dance between proton and neutron and their transforming into one another.  It's a tetrahedron, just another way to draw a graph (a simple one).  The suggestion that energy by quantized using Mite accounting is intriguing but there's a barrier before we might further speculate with the newer tools.

The barrier is we're short on art historians who know enough to tease apart our own cultural lineages.  They could use help from philosophers I'd hazard.  Is Princeton yet offering much using Clifford Algebra?  Maybe.  I don't claim to be omniscient, even about Princeton matters, just because I'm an alum.

As I mentioned to Andrius, our home base, where we lived was 2 Dickinson Street.  John Baez, whom we both admire, lived there later (I was Class of 1980).  He's into quaternions and octonions and all of that.  Quaternions actually feature in three.js, the JavaScript library I've been looking at, in case I want to do more with polyhedrons using JavaScript.  I've made some progress.

All of which is to say:  I think we happen to be in an era when 90-degree based thinking is gradually giving way to something smarter.  The 90-degree idea came from being so tiny relative to the Earth and thinking perpendiculars to the land were parallels to each other.  We learned there's convergence in the down direction and divergence in the up direction.  This asymmetry helped us break free of the hegemony of rectilinear thinking.  Synergetics is just a very explicit manifestation of that, whereas with Bell's kites, it's more inarticulate, less worked into the basic language.

Kirby

[Joseph Austin]

On Jun 17, 2016, at 10:13 PM, kirby urner <kirby...@gmail.com> wrote:

All of which is to say:  I think we happen to be in an era when 90-degree based thinking is gradually giving way to something smarter.  The 90-degree idea came from being so tiny relative to the Earth and thinking perpendiculars to the land were parallels to each other.  We learned there's convergence in the down direction and divergence in the up direction.  This asymmetry helped us break free of the hegemony of rectilinear thinking.  Synergetics is just a very explicit manifestation of that, whereas with Bell's kites, it's more inarticulate, less worked into the basic language.

So then we get into polar coordinates. Riemann sphere. Homogeneous coordinates.

Generalize to N dimensions.

I think your "space" has M directions in N dimensions, M > N.

Or if it's really M dimensions, there's a curious geometry where "orthogonal" is less than Euclidean 90º,

or it's some kind of degenerate 4D, like representing 3D with isometric drawings in 2D.

In my mind, the strongest argument for the tetrahedron is the carbon atom, the "Tinkertoy connector" for organic molecules.

There must be a set of "axioms" for the construction of molecules, which would be a mathematical system.

And packing options would have to be a consideration in such a system.

At one of the turning points in my life, I passed up an opportunity to go down that path.

When we stand on the frontier, too many roads diverge, and we don't know where they will take us!

Lehar says it's all waves. Which at some superficial level at least agrees with quantum mechanics.

So suppose we have waves in tetrahedral directions resonating around a 3-d spherical space?

What are the eigenstates of waves on a sphere?  Could we use Lehar's Chlandi figures to visualize them?

Joe

[Joseph Austin]

Brad,
I've just built a straw-and-pipe-cleaner model of one tetrahedron stacked on three, and now I clearly see the octahedron in the middle.
I did't recognize it as such from the RBF diagram, which didn't clearly show the interior.
It's much easier to visualize three dimensions in three dimensions!

The reason for filling space with a unit volume is to determine how many unit volumes fit in a given space, e.g. compute the integral.
My "stake" is that "volume" is conserved. Now I realize that swiss cheese conserves volume as well as cheddar, but it's easier to compute the volume without the holes.

I've noticed an asymmetry in hexagonal sphere packing in another context: the hexagonal arrangement of keys on a musical instrument such as a button accordion. There are three row directions of adjacent keys on a 2D surface, but if one row is parallel to the edge, neither of the other rows will be perpendicular, and vice versa. This means inter-button distances on parallel and perpendicular rows are different, and it makes a difference to the musician which way they are laid out.

Joe

[kirby urner]

On Sat, Jun 18, 2016 at 1:06 PM, Joseph Austin <drtec...@gmail.com> wrote:

In my mind, the strongest argument for the tetrahedron is the carbon atom, the "Tinkertoy connector" for organic molecules.

 

I think for me the main argument is topological:  of all the Platonics, it's the simplest and self dual to boot.  It "haunts" the cube in oscillating back and forth twixt two positions, the stella octangula on steroids.

I've gotten to where the cube just looks bloated, it's just not as aesthetically pleasing as its proponents make it out to be.  Their defensiveness is perhaps their least attractive feature, but it masks awkwardness and instability.  This is all just street corner psychology anyway, obviously, as with Platonic shapes there's no soap opera.  That's a happy family with many progeny.

I'm just saying, I don't need to look in nature to show myself the cartoon I find convincing.  We say "one point is no dimensions" (forgetting about the viewer somehow, thinking the room or camera has gone away), then "two points is one dimension" (a line segment), then three make a plane (a triangle), and then four, our first true Volume, a container, a cave, a tent. 

Call that "three dimensions" if you wish (as a first rung in N-D geometry, that's what it is), but it's still four points and a tetrahedron, no need for any cube.  We go off with our unit volume from there, happy to be using an Eulerian primitive, a minimum graph.  Let the orthodox keep to their cubism, which by now has huge libraries.  I'd be foolish not to want backward compatibility.

Lets think about calculus and div and curl, Maxwell's equations.  In animating curl in electrostatics, like in a cartoon or animation, is there any harm in using a hexpent at that juncture?  Must we use little trapezoid-squares?  Surely some handwaving will make it XYZ-sympatico and we won't have to waste a lot of time introducing an alternative visualization.  I'm not for reinventing all the wheels, only some of them.

Kirby

[Bradford Hansen-Smith]

Yes Joe it is easier to experience 3-D in 3-D. It is impossible to have a 3-D experience in 2-D. Straws and pipe-cleaners are a good place to start. Next stack four spheres and look for similarities and differences. Next cut out triangles and tape them together in same arrangement.  Then fold two points on the circle together and crease.  Draw lines connection all four points to see the kit shape that already is there. Look to identify the tetrahedron properties; 4 points, 6 relationships between them and 2 solid planes and 2 open planes. Find both both of them, one reciprocal of the other. Then fold the tetrahedron following instruction at:
 http://wholemovement.com/how-to-fold-circles .
Make two tetrhedron and follow instruction for the octahedron to give you another idea about the tetrhedron/octahedron relationship.

To the octahedron net add 2 more open tetrahedra to complete a tetrehdron patterned net that will form the regular icosahedon. From there the cube and dodecaheron relationships are possible adding more tetrahedra to the octahedron and to the icosahedron. Yes Kirby we do have one happy family of many progeny, but from multiple tetrahedra. This is not traditional anything but anyone can do it. It does tell me that there is no question about the primacy of the tetrahedron, it comes first, after the circle-sphere compression, which itself reveals triangulation. No need to throw away the old, much will drop away when updated by a new world view that favors unity over linear thinking and fragmentation.

Brad

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[Joseph Austin]

On Jun 18, 2016, at 4:31 PM, kirby urner <kirby...@gmail.com> wrote:

I'm just saying, I don't need to look in nature to show myself the cartoon I find convincing.  We say "one point is no dimensions" (forgetting about the viewer somehow, thinking the room or camera has gone away), then "two points is one dimension" (a line segment), then three make a plane (a triangle), and then four, our first true Volume, a container, a cave, a tent.  

Consider that the product of length * length is not length, but area.  Likewise length * length * length is volume.

So if we can replace "square length" or "cubic length" with "triangular length" or "tetrahedral length,"

we could replace it with any other units as well.

So consider two vectors P,Q in polar coordinates:  P = p e^i𝜃,   Q = q e^i𝜑

The product of the two vectors is: p*q e^i(𝜃 +𝜑)

But if there is no need to preserve units between the vector "length" and product "area", 

we could write p,q also as powers of  e:  p = e^u, q = e^v;  where u = ln p, v = ln q.

Giving:

P = e^(u+i 𝜃), Q = e^(v+i 𝜑)  

 P*Q = e^(u+i 𝜃) * e^(v+i 𝜑) = e^[ (u+v) + i(𝜃+𝜑)]

 Thus we can represent the product of two vectors as sums in "exponent" space!

The new polar coordinates would replace the traditional polar coordinates by using log scale for the radial direction.

Joe Austin

[kirby urner]

Good creative play.

We might also keep the math exactly the same and just *show* the area differently.  Ditto for the tetrahedron.  That's what that figure was about (that I linked to a buncha times).

There's no computational link between the square visualization of 2 x 2 and the computation.  If you show 2 x 2 as a triangle instead, that's not lying or misrepresenting, that's just using different units of area.  Free country, many ethnicities.

I've got enough in the Storyboards album to fuel any number of animations.  I'm not expecting to do them all myself.  Indeed, I think I've gotten my part mostly done, torch has passed, but I'm happy to keep helping the teachers who continue the 4D lineage (which was already well established when I took it on in the 1980s).

Hey, I started a new "media campaign" today.  Too many Portland Public Schools have been discovered to have lead in their pipes, plus CommonCore is waaay too base-10 oriented for the Silicon Forest. 

If the industries here are gonna pay more taxes (Ballot Measure 28), build some new schools, then they're not gonna sit on their hands about this CommonCore strict focus on base-10.

The schools are jeopardizing their operations as we could just offer free content on-line that's way better (we already do, but next we can offer testing and assessment, no need to burn all the fossil fuel commuting). 

Putting the schools out of business is not the goal however.  Helping the math teachers climb out of the stone age is the agenda. We'll get more of a footprint online for our Oregon schools, and we'll teach hexadecimal.  I'm upbeat about our prospects!

Kirby

Learn Hex! 
http://controlroom.blogspot.com/2016/06/learn-hex.html

[Joseph Austin]

Kirby,

I wish you well.

It will be interesting to see what happens to schools when parents discover that their kids can learn in spite of them!

My concern is that "poor schools" will be replaced with "digital divide",

and the kids that need the most help will still get the least.

Joe

[kirby urner]

On Wed, Jun 22, 2016 at 4:00 PM, Joseph Austin <drtec...@gmail.com> wrote:

Kirby,

I wish you well.

It will be interesting to see what happens to schools when parents discover that their kids can learn in spite of them!

Yeah, it'll be interesting. 

In light of the toxic lead levels being discovered in Portland Public Schools, it's really up to the District to provide on-line courses for credit. 

Students should not be forced to attend a toxic building, where mental retardation is in the water. 

Besides, wasting hours a day going to and fro, with the better bandwidth at home, makes no sense either. 

"Socialization" will happen with or without all that fuel burning, plus "socialization" too often just means learning to submit to capricious authority in any case, so I'm not sad for those students who opt out. 

Those who opt in and voluntarily go to a school building (we hope not toxic or sick), can still go, but the equal access curriculum needs to be as much online as not, and we're not talking about some theoretical future. 

We're talking about right now. 

We're paying for these equal access educational services right now (over $3000 a year in property taxes in my case, with no family members attending).
 

My concern is that "poor schools" will be replaced with "digital divide",

My focus is having enough of a home to be able to study at home, which includes Internet access.  I call that a personal workspace (PWS).  "Fixing the schools" is not nearly as important as making homes livable.

A lot of the "poor schools" are irreparable, just like a lot of the housing stock is past salvaging.

Each generation needs to accept the challenge of building and maintaining infrastructure.  In his 'Critical Path' Bucky Fuller gives designs for whole cities, including Old Man River City which I know the folks of East St. Louis were excited about (I went to some meetups).

Just sitting around watching buildings decay, while people drink toxic waste, is not the point of education.

Judging from how people are behaving today, education of the current generation has not been effective.  TV was probably a big part of it.  Too much fiction.  People grow up believing what they've viewed.  So sad.

Not their fault really.  Lots of inertia.  Karma some call it.

 

and the kids that need the most help will still get the least.

Joe

At the same time I'm for upgrading the math curriculum, I'm in favor of not using it as a filter such as by making calculus mandatory for college, or anything like that.

I thought The Math Myth by Andrew Hacker was effective and will be interested if we get back to that.

The way math is abused today, it's no wonder that (A) so many hate it and (B) it's in the pits.

But I'm not able to by myself make much happen, I know that.  I've got my little "media campaigns" and what not.

Today I was asking @IEEE on Twitter why it appears to support CommonvCore Math when the latter is so watered down when it comes to number bases.

Only base 10 is to be considered.  Yet Silicon Forest, our local economy, is built on hex numbers.

I think of Common Core Math a delay tactic, i.e. a political compromise aimed at keeping the CS side of things from being too disruptive.  Keep CS at arms length.  Keep math just like it used to be.

This setup of making something named "math" mandatory with something named "CS" playing second fiddle, is a symptom of deeply flawed thinking, a dying civilization (good riddance).

Integrating numeracy and computation, as well as composition and calculation, is probably not something our present setup can accomplish. 

Not a problem.  We're setting up new systems as we speak.

We have the Internet, and more and more people staying home to really learn something, to study. 

The mandatory form of day care we call "school" is more and more just another lifestyle option and many serious scholars will choose a stronger path, including among the least advantaged.

Anyway, what happens in the US is not necessarily what's most important or interesting. 

There's a lot of complacency that comes from this pathetic belief in "being a superpower" and/or "the wealthiest nation" (I gag on the immaturity of these placebo cliches).

As Nietzsche (not a German, unlike Heidegger) once wrote:

One pays heavily for coming to power: power makes stupid. The Germans -- once they were called the people of thinkers: do they think at all today? The Germans are now bored with the spirit, the Germans now mistrust the spirit; politics swallows up all serious concern for really spiritual matters. Deutschland, Deutschland uber alles -- I fear that was the end of German philosophy.

-- Twilight of the Idols.

American philosophy is about as pathetic today.  I wish @Princeton could do something about it.  Maybe there's some courage left in 1879 Hall?

Kirby

Kirby

[Joseph Austin]

On Jun 22, 2016, at 8:14 PM, kirby urner <kirby...@gmail.com> wrote:

my comments below to selected points.

"Socialization" will happen with or without all that fuel burning, plus "socialization" too often just means learning to submit to capricious authority in any case, so I'm not sad for those students who opt out. 

I'm not so sure it will happen without conscious effort.  When I was young, I had health problems that kept me out of school for a good bit of the year

I did my lessons at home, but I'm convinced my "socialization" suffered.  I could relate to adults but not to peers.

Parents electing "home school" will need to provide explicit opportunities for socialization.

Besides that,  no social change can endure unless a method of propagation is included in the system.

My brother found in the industrial setting that "learning together"  or "each one teach one" is a more effective way to learn.

What he would do is present the skill, then those who "got it" on the first pass would be enlisted to coach the rest of the class.

"Teams" would not receive credit until ALL members of the team could demonstrate competence.

The "stars" at first resisted [why should my grade get pulled down my somebody else?], 

but changed their mind when they discovered, as has every teacher, that the best way to learn something it to teach it.

Of course, the traditional school system doesn't really encourage "cooperative learning" either,

because in reality the system is not structured so much to teach as to rank--to separate the classes.

My focus is having enough of a home to be able to study at home, which includes Internet access.  I call that a personal workspace (PWS). 

"Fixing the schools" is not nearly as important as making homes livable.

You're fighting our "getting ahead" culture!  The only way for some to "get ahead" is to insure "most children left behind".

We pay lip service to diversity until "one of them" gets ahead of my kid (or if my kid doesn't catch up,

depending on which side of the divide you start on.)

I believe the initiative for making homes livable must come from those living in it.

As teachers, we may be able to open people's eyes to what is possible, but we can't make them want it,

or want it enough to give up something else (time, money, effort) to get it.

  In his 'Critical Path' Bucky Fuller gives designs for whole cities, including Old Man River City which I know the folks of East St. Louis were excited about (I went to some meetups).

I grew up in St. Louis.

Judging from how people are behaving today, education of the current generation has not been effective.  TV was probably a big part of it.  Too much fiction.  People grow up believing what they've viewed.  So sad.

I believe TV has precipitated a profound de-socialization.

When I was a kid, on a summer evening the parents would sit outside on the steps and the kids would play together up and down the block.

Then came TV.

As each family got one, they would stay inside and watch instead of going out and mingling.

Within a decade, the only people on the streets at night were up to no good.

I believe another effect is the dis-incentivizing of "ordinary" talent development.

Via TV, (now internet) we are exposed to "the world's greatest" whatever.

The ordinary performances of ordinary people (unless it's your own kid) is passed over in favor of the mediated performance of a world-class star.

The schools are perhaps the last venue of amateur talent.

We'll go see the school play if one of our kids is in it;

we may to to the community theater if one of our friends is performing,

but do we support home-grown sports or arts "for art's sake"?

Not their fault really.  Lots of inertia.  Karma some call it.

 

At the same time I'm for upgrading the math curriculum, I'm in favor of not using it as a filter such as by making calculus mandatory for college, or anything like that.

Ah, but the "filter" aspect is what industry and prestige colleges want!  It doesn't matter that it's calculus so much as that it's "hard,"

that only a few are able or willing to do it.

If I have a job to fill, it may be than any HS grad could do it. But if there are 3 college grad applicants and 30 HS grads,

my decision process (and EEO justification)  is a lot easier if I say "college required."

What we see today is twice as many college grads as "white collar" jobs.  So I predict within a few years, it will become "masters required".

It's the basic logic fallacy: if people in good paying jobs have college degrees, then if I get a college degree, I'll get a good paying job.

I thought The Math Myth by Andrew Hacker was effective and will be interested if we get back to that.

I need to check that out!

The way math is abused today, it's no wonder that (A) so many hate it and (B) it's in the pits.

I think of Common Core Math a delay tactic, i.e. a political compromise aimed at keeping the CS side of things from being too disruptive.  Keep CS at arms length.  Keep math just like it used to be.

In NC they are doing that literally. The legislature is allowing schools to go back to Algebra 1 Geometry Algebra 2 instead of the integrated Math 1 2 3.

The problem isn't that the students can't learn it so much as the teachers can't teach it.  And the parents can't help because the parents learned it the same way the teachers did.

I think the major flaw in our education system is curriculum development. We academics discover stuff. We know stuff. But we don't know how to teach stuff. That's why we need the filter.  Students have to prove they can learn in spite of our "instruction" or we won't be able to continue to pretend that we are "teaching" them.  In grad school, at least, we get to go thru the motions of 'teaching" the elementary concepts.

My daughter majored in secondary ed.  She came home from practice teaching and said, Daddy, I love teaching.  But I never get to do it.

I have to spend all my time "reasoning" with parents and doing paperwork.

So we need to focus on the parents.  If we want better education, we need better homes. We need to read to our kids, take them to libraries and give them books. But we also need to prepare them for math and computing.  We need to do more counting, more quantitative comparison, more quantitative prediction.  E.g., if it took us 12 minutes to bake this one sheet of cookies (or mow this side of the lawn), what time will we be finished baking all three sheets?  [We do this stuff in our head all the time, but the kids can't see what we are thinking."]  Get them a stack of paper places or coasters and fold circles.  Get them straws and pipe cleaners and built polytopes. Get them a tablet and spend an "hour of code" with them and build some games.

When I was a kids, we played board games with dice, and card games.  We were doing comparisons, ordering, at least adding, even multiplying (5 points for each this and 10 points for each that). We went to the store and had to count change and see how many 2 cent candies we could get for our nickel.  Now the game machine does all the calculating for them.

This setup of making something named "math" mandatory with something named "CS" playing second fiddle, is a symptom of deeply flawed thinking, a dying civilization (good riddance).

The kids themselves will revolt--at least, the ones who have computers.

How ya gonna keep 'em learning their times tables after they've played Minecraft?

How ya gonna keep 'em using pencil and paper after they've done "hour of code"?

I've seen it happen in other fields.

To the consternation of many a piano and violin teacher,

the kids of my generation all learned to play guitar instead.

They didn't need teachers; they taught each other.

I remember reading some "advice to young people" that said guitar was a poor choice of instrument because there was so little repertoire for it.

Integrating numeracy and computation, as well as composition and calculation, is probably not something our present setup can accomplish. 

Not a problem.  We're setting up new systems as we speak.

We have the Internet, and more and more people staying home to really learn something, to study. 

Yes, they will survive us.

Joe

[kirby urner]

On Thu, Jun 23, 2016 at 7:20 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Jun 22, 2016, at 8:14 PM, kirby urner <kirby...@gmail.com> wrote:

my comments below to selected points.

"Socialization" will happen with or without all that fuel burning, plus "socialization" too often just means learning to submit to capricious authority in any case, so I'm not sad for those students who opt out. 

I'm not so sure it will happen without conscious effort.  When I was young, I had health problems that kept me out of school for a good bit of the year

I did my lessons at home, but I'm convinced my "socialization" suffered.  I could relate to adults but not to peers.

Parents electing "home school" will need to provide explicit opportunities for socialization.

Yeah, it's different in 2016 when your PWS has bandwidth.  Crowd sourced meetups, parents who self organize, all these ways to use the school building without committing to the old regimen.

I taught a class of homeschoolers at Free Geek.  They learned about how to code vectors in Python. 

See Python for Homeschoolers at my Oregon Curriculum Network website:
http://www.4dsolutions.net/ocn/

It's not like if you're a homeschooler you can't go anywhere. 

You just need to be able to log in to the public school system which offers courses at all levels e.g. if a parent (not so much an English speaker?) wants to take the same high school English course her kid is taking, and get it on the record somewhere, why not? 

Public school is for learning a living throughout life.  This idea that one's studying a "high school level topic" means one is under 18 (or "should be") is just bad design.

The idea that public high school courses are for a specific age range should go away pronto, with the advent of public schooling online in a big way.  Guardians can take courses with their wards, also for credit.  Pythonic Math could be for the dad and mom as much as for their kids, why not?

Shows like Sesame Street were pioneering, as the writers knew a stay-at-home guardian was probably watching.  Indeed, many parents got hooked. 

What's nice about Youtubes is one assumes "world viewable" and therefore a more age-agnostic approach is normal, although "geared for the very young or very old" remain as genres.

 

Besides that,  no social change can endure unless a method of propagation is included in the system.

E.g. the Internet.
 

My brother found in the industrial setting that "learning together"  or "each one teach one" is a more effective way to learn.

What he would do is present the skill, then those who "got it" on the first pass would be enlisted to coach the rest of the class.

When I was a classroom teacher I encouraged peer to peer mentoring.  I'd explain for awhile then say:  "OK, now teach each other" -- something like that. 

Those who "get quadrays" raise your hand and work with anyone who didn't. 

Works well at a distance when the software supports it.
 

"Teams" would not receive credit until ALL members of the team could demonstrate competence.

That's one way to do it.  I think of board games (not bored games). 

So many ways to set them up!

That's when it comes down to making choices.  Some people won't push themselves to try new kinds of game or sport.  Others thrive on diversity / variety / novelty.

That's where encouraging pep talks and motivational presentations come in, as a genre. 

If I'm playing with the idea of enrolling in X, let me first go visit the recruiters for X and see if they persuade me.  I check into their PR.  How do they evangelize for their discipline?

I like education systems based on recruitment and persuasion more than "you must do this because the state mandates that I'm your boss, as your teacher". 

Sending kids to school so they get used to being bossed around is not the goal of every parent, either.  That's maybe what military academies are all about (learning to follow orders) but civilian schools are more about learning to be an autonomous agent who acts reliably (honors agreements).

The "stars" at first resisted [why should my grade get pulled down my somebody else?], 

but changed their mind when they discovered, as has every teacher, that the best way to learn something it to teach it.

Yes.  Sometimes it's the group result that matters, as in theater and dance.  If one actor or dancer is unable to sustain the action, then the whole performance suffers.

Other times "work" is a solo activity. 

Watching the Olympics or other sporting meetups featuring a wide variety of sports, reminds us of these "design patterns".

Indeed, many more "math games" could be devised to blend in various types of physical effort. 

I think of geocaching (using GPS to treasure hunt, triangulate). 

I think of "math labs" you might only reach by walking three miles and climbing 2000 feet.  Whole "math villages" could be relatively remote and require "camping skills" to sustain.  I call these Ecovillages and started Project Earthala to make them happen (name suggested by a Friend).
 

Sitting in rows, rank and file, for hours and hours, during the prime time for physical development, is unhealthy for the body as well as the mind. 

Math is an Outdoor Sport.

Of course, the traditional school system doesn't really encourage "cooperative learning" either,

because in reality the system is not structured so much to teach as to rank--to separate the classes.

Yeah, a lot used to depend on the teacher. 

I'm not sure how that's changing with #CCSS and the need to teach to the test or have the whole school suffer. 

Schools are being treated like those "teams" we were discussing, with "taking tests" the primary athletic event.  If the school does poorly on the test, the authorities start breathing down your neck.  Could these authorities pass these tests?  Do they eat their own dogfood?  Dogfooding is an important practice in my world (akin to "practice what you preach").

These tests are often multiple choice and do not involve "showing one's work" or "the steps" (like the SAT).

The problem is the tests themselves don't have to get tested by the people who use them, that's not built in as a feedback loop (poor design).

The presumption is every test "gets an A" meaning we're not supposed to judge or criticize the tests themselves, either their content or format.  Asking "where are the hex numbers?" is verboten because CCSS is "approved in advance" (by whom, when did we get to vote?).

From my point of view, as someone into hex numbers and tetravolumes, the ethnicity in control of the math tests is basically an oppressor / conqueror who refuses to teach what's important to my culture.  I feel my religious freedoms have been infringed, that I'm being railroaded. 

My subculture likes tetrahedrons OK?  We want a rhombic dodecahedron of volume six introduced by sixth grade, or we have no reason to respect the curriculum, is that a problem?  So yeah, lots of jokes at home about the "muggles".

I'm in the position of those native Americans for whom school was an institution designed to cleanse them of their ethnic traits so they could fit in to the dominant society.  When in Rome...

Fortunately, in America we have religious freedom and I'm gradually getting more Quakers, for example, interested in stuff like quadrays and "nonsense numbers" (natural numbers with infinity digits, just like the irrationals have). 

I gave a talk at Earlham College (Quaker) to their philosophy club about these things.
http://coffeeshopsnet.blogspot.com/2016/04/nonsense-numbers.html

[ Best wishes to the away team leaving for Iceland from there, today.  My daughter is on that team.  Soil science and biophysics among the field sciences they'll practice.  Math is an Outdoor Sport. ]

My focus is having enough of a home to be able to study at home, which includes Internet access.  I call that a personal workspace (PWS). 

"Fixing the schools" is not nearly as important as making homes livable.

You're fighting our "getting ahead" culture!  The only way for some to "get ahead" is to insure "most children left behind".

Yes, it's a rigged system where a lot of people start out with so many societal handicaps that any concept of "fairness" is inapplicable.

But that's the Global U in general.  Many are penned in as detainees.  Huge numbers of would-be learners get to spend their time in camps, trying to escape the perps (i.e. perpetrators).

Definitely it's an emergency situation for so many on the planet. 

The Global U curriculum really needs to be about addressing the issue of low living standards, and should be judged on its ability to alleviate these conditions. 

If people are suffering unnecessarily, then curriculum is out of whack, by definition, a no-brainer.  GST shows us things *could be* much better.  The sun is generous (see below).

Of course by "curriculum" in this case I don't just mean what goes on in classrooms.  I mean all the role playing that goes on in the theater outside.  A theater in the round.  The stage is spherical.

 

We pay lip service to diversity until "one of them" gets ahead of my kid (or if my kid doesn't catch up,

depending on which side of the divide you start on.)

It can be like that, and again the sports analogy is apropos.

I'm always feeling protective of the few who've learned to think in tetravolumes a lot, and try to find them opportunities.  D. Koski for example.  He's going to Bridges in Finland at least (not my doing).

I feel the same about those learning GST and try to place its most serious students as CEOs of giant companies whenever possible.  :-D 

Or maybe GST students wind up in smaller still-dynamic companies, as the giants tend to be awkward and unwieldy unless internally modularized.

Google turned itself into Alphabet for that reason, though few people think in those business namespace terms. 

The driverless car stuff was diluting the search engine brand and will now be easier to spin off if need be, ditto the Deep Learning.

Driverless trucks in a convoy on a special road, with maybe one supervisor / engineer, is what we call a train by the way.  The engineer in charge could be replaced with an AI bot but we still think a trained human is safer.

With new software, that road might be paved (not rails) with even ordinary trucks adapted to form convoys without physically coupling. 

More likely a new kind of truck (with no cab) will be devised for such convoys and specific roads set aside or newly made for their exclusive use.  Stay tuned. 

On the physics listserv (PHYSLRNR) I called it the "zombie truck system" (ZTS) some years ago.  They're talking about maglev these days (I'm just lurking).

 

I believe the initiative for making homes livable must come from those living in it.

Depends what we mean by "initiative".  From my point of view it's all one big Global U and if the housing situation is poor, that means the curriculum is poor, either in content or delivery.  That's a tautology.

North America has many "square" miles of substandard dorms that need replacing.  The curriculum is obviously insufficient, but then upgrades are occurring daily, as more students stay home to learn about the whole new cities we might be building, such as Old Man River city (stadium shaped, but hugely bigger -- already on the books before the flooding of New Orleans, but never mentioned by CNN).

I know the Chinese have been good at building spanking new cities, many almost still empty, and if able to overcome xenophobia, might enjoy growth by accepting more detainees from the camps -- or from places with changing shorelines owing to ocean level deltas. 

The devil is in the details. 

There's great PR value in accepting refugees from any part of the Global U where physical disasters and/or virulent meme viruses (e.g. wars) have destroyed the mental health of most faculty and students.  Showing compassion still goes a long way towards winning hearts and minds.

People take notice when living standards greatly improve, for anyone.  "What math are they using?" people want to know.

 

As teachers, we may be able to open people's eyes to what is possible, but we can't make them want it,

or want it enough to give up something else (time, money, effort) to get it.

I think longing is inbuilt. 

Even the greatest billionaires today suffer low living standards because they're confined to a slum planet. 

You could get out of your ghetto with a curriculum custom tailored for our planetary concerns.  Billionaires take note.

I advertise and recruit for General Systems Theory, as far more brilliant than Economics, which feeds into it as a less science-informed discipline.  GST and Economics share an appreciation for Cybernetics.

According to Bucky, it was Einstein who said humans have two primary motivations:  fear and longing.

Fear leads to running from distopia whereas longing means moving towards relative utopia. 

A lot of the time, when just trying to escape bad circumstances (a distopia), we have no time to plan for or envision where we're trying to go (a relative utopia).  "Plan or panic" is a mantra for me. 

Planning means sharing blueprints for a more utopian tomorrow.  Notice how unskilled politicians rarely do that.  They're fear merchants more than planners.  'The Power of Nightmares' is a movie about how fear-mongering goes with climbing a political ladder in many cases.  I cite it often.

 

  In his 'Critical Path' Bucky Fuller gives designs for whole cities, including Old Man River City which I know the folks of East St. Louis were excited about (I went to some meetups).

I grew up in St. Louis.

Great city.  Did you ever hear about OMR?  Probably not, given the poor curriculum.
 

Judging from how people are behaving today, education of the current generation has not been effective.  TV was probably a big part of it.  Too much fiction.  People grow up believing what they've viewed.  So sad.

I believe TV has precipitated a profound de-socialization.

When I was a kid, on a summer evening the parents would sit outside on the steps and the kids would play together up and down the block.

Then came TV.

Yeah, TV made a huge difference, as did the telephone before it.  Internet and smartphones... yet another revolution in human behavior.

 

As each family got one, they would stay inside and watch instead of going out and mingling.

Within a decade, the only people on the streets at night were up to no good.

We can't change the past, only learn from it.

TV also made people more fearful I think, as the news shows tend to capitalize on showing emergency situations and violence. 

The world became a much scarier place for a lot of people thanks to TV.

Advertising and motivational psychology had a lot to do with it too, as these work together to exploit fear and longing quite effectively.

Appreciating the power of advertising, a form of brainwashing / propaganda, is partly what inspired me to wanna be a spin doctor when I grew up.  Spin doctors made the world go around, I could see that as a kid. 

Learning about "precession" from Bucky Fuller helped a lot when it came to learning about spin.  He was a great teacher.

 

I believe another effect is the dis-incentivizing of "ordinary" talent development.

Via TV, (now internet) we are exposed to "the world's greatest" whatever.

Yes and no.  There's evidence that Youtube for example is inspiring people through peer group example.  People who talk act and look like me are doing these amazing things, maybe I could too.

http://bit.ly/28PBhHh

Case in point:  six year old body builder types.  Kids are seeing other kids put on muscle way before we thought that could happen in the old days.   https://youtu.be/gNhFUPBv1p4

I'm going to take the position that Youtube is having the net effect of inspiring us to higher levels of achievement, as we learn more about what's really possible.
 

The ordinary performances of ordinary people (unless it's your own kid) is passed over in favor of the mediated performance of a world-class star.

But to be on Youtube you don't need to be a world-class star.  Lots of ordinary people, doing some amazing things, plus sharing epic failures which we also learn from.

I think people are also learning that there are no superheros out there, i.e. no one with truly magical powers like in the comic books.  The same laws of physics apply to all. 

In the old days, people believed in all kinds of hidden powers, but thanks to Youtube, we're better able to cross check all that and out the hoaxers.

I'm not talking about "paranormal abilities" necessarily as some of those will likely turn out to be quite normal.  People do have natural capacities we don't really learn about until we have access to Big Data and see the patterns. 

For example, humans sometimes spontaneously self organize in new ways sociologists still can't quite explain.
 

The schools are perhaps the last venue of amateur talent.

We'll go see the school play if one of our kids is in it;

we may to to the community theater if one of our friends is performing,

but do we support home-grown sports or arts "for art's sake"?

I find the Internet to be like this too.  It's not either / or of course.
 

Not their fault really.  Lots of inertia.  Karma some call it.

 

At the same time I'm for upgrading the math curriculum, I'm in favor of not using it as a filter such as by making calculus mandatory for college, or anything like that.

Ah, but the "filter" aspect is what industry and prestige colleges want!  It doesn't matter that it's calculus so much as that it's "hard," that only a few are able or willing to do it.

So what though, right, who cares about them? 

Industry and prestige colleges that require calculus in some mindless way that make no sense won't seem all that important in light of their poor design and lazy admissions / employment policies. 

Companies with silly interviewing practices and criteria get noticed for their idiocy.  I hear lots of jokes about lame companies and their ridiculous screening practices -- worse than frat houses sometimes.

We can afford to just scoff at those dinosaur numbskulls who never got it.  Other opportunities await us.

 

If I have a job to fill, it may be than any HS grad could do it. But if there are 3 college grad applicants and 30 HS grads, my decision process (and EEO justification)  is a lot easier if I say "college required."

Yeah, the so-called "job market" is pretty tough.

I'll defer to other filing cabinets regarding GST might help us improve this vista. 

'Education Automation' is a classic title in the early literature.  Give everyone a scholarship-based lifestyle to stay home and not waste gas.  We do it in agriculture already (subsidize to prevent wasteful action).

The sun is our sponsor.  People always ask "where will the money come from".

 

What we see today is twice as many college grads as "white collar" jobs.  So I predict within a few years, it will become "masters required".

I'm looking to break the hold of the idea that people are required to work for a living.  Or rather, I accept the physics definition of work, which is to use energy.  We can't help but do that.  Pay people for breathing (i.e. working).

More important, pay people for learning.  The biggest mistake we make as a society is erecting barriers to self improvement. 

Every polynomial a kid meaninglessly yet correctly factors should result in 0.0001 bitcoin or something like that (less meaningless that way).  If you study hard, you can make ends meet.  Work / Study is what we do in the Global U.

The primary income to the Global U is of course the terawatts from the Sun.  We're like a kitchen appliance plugged in to a wall socket to get juice and that juice is from a fusion powered furnace called a "star" in the English language.

A big mistake is to think all that's of value is a result of humans and human labor.  Nothing could be further from the truth.  Marxism is a weak philosophy compared to GST's.

What humans do is channel energy i.e. insert their water wheels to make gears turn to mill wheat or sew garments or whatever.  They capitalize on what nature provides.

We exploit the angular momentum of wind, water combined with gravity and pressure differentials.  We also exploit rapid cell division, also sun powered.

Note that "money" has nothing to do with anything fundamental.  No biosystems use "money" until we get to the idea of debt and who owes whom. 

What money mirrors, is potential energy.  However people sometimes forget that "energy" by itself is not sufficient.  You need circuitry.  You need a design for the energy to flow into. 

Just "throwing money" at a problem is like kicking sand around on the beach and expecting a Tesla to magically materialize.

 

It's the basic logic fallacy: if people in good paying jobs have college degrees, then if I get a college degree, I'll get a good paying job.

I thought The Math Myth by Andrew Hacker was effective and will be interested if we get back to that.

I need to check that out!

Yes, it's a pretty important book I think.
 

The way math is abused today, it's no wonder that (A) so many hate it and (B) it's in the pits.

I think of Common Core Math a delay tactic, i.e. a political compromise aimed at keeping the CS side of things from being too disruptive.  Keep CS at arms length.  Keep math just like it used to be.

In NC they are doing that literally. The legislature is allowing schools to go back to Algebra 1 Geometry Algebra 2 instead of the integrated Math 1 2 3.

I call this "exporting future shock" i.e. "we can't cope with this brave new world and need braver people than us to deal with it". :-D

How did this conspiracy to teach only Base 10 arise?  Many interesting books could be written. 

In Oregon, with an economy built on hex numbers, there's no way we can follow Common Core except maybe politically we'll give it lip service, while meanwhile sneering and snickering amongst ourselves.

Portland is a boom city right now with a net of 1500 new people a week I think it was.  I should check my figures.  A lot of them are looking for technology jobs, many not finding them, and creating startups, a symptom of talent needing to monetize somehow. 

A lot of these people are also talented at teaching and know quite a bit about technology.

http://www.oregonlive.com/silicon-forest/index.ssf/2016/01/portlandia_no_longer_oregons_y.html

So I think it's inevitable, with lead in the schools, tutors looking for work, high tech, and CCSS being as terrible as it is, that we'll get more math teachers breaking away from base 10 and teaching 0-F in addition -- plus some other verboten topics not currently on any Pearson test. 

Portland's students are getting the message loud and clear that the math in their textbooks is ridiculously out of date.  We learn that from the BBC even (home of the Micro:bit).

Parents are complaining, because more and more of them are geeks (Linux is that old already). 

I wouldn't be surprised if Portland Public Schools breaks away from CCSS quite explicitly i.e. we might see a backlash here soon and a public disavowal.  "We can't afford to be that stupid" might be our motto.

"Base 10 only" smacks of xenophobia and an inability to cope, a need to export future shock, like North Carolina does (NC is already the butt of jokes around here, lots of bathroom humor).

 

The problem isn't that the students can't learn it so much as the teachers can't teach it.  And the parents can't help because the parents learned it the same way the teachers did.

Changing demographic:  more and more parents use Linux.  The first generation of open source is now mid 30s, with kids in school.  Windows is also bringing out a bash shell.  They have little choice in the matter.

"You may have a masters degree, but if you don't know bash, you don't know enough to teach my kids" is the growing attitude.

Math teachers who can't code is seeming more oxymoronic by the day, like English teachers who can't spell.

 

I think the major flaw in our education system is curriculum development. We academics discover stuff. We know stuff. But we don't know how to teach stuff. That's why we need the filter.  Students have to prove they can learn in spite of our "instruction" or we won't be able to continue to pretend that we are "teaching" them.  In grad school, at least, we get to go thru the motions of 'teaching" the elementary concepts.

Good point, there's a lot to that.  Effective pedagogy / andragogy is scarce and therefore valuable.
 

My daughter majored in secondary ed.  She came home from practice teaching and said, Daddy, I love teaching.  But I never get to do it.

I have to spend all my time "reasoning" with parents and doing paperwork.

Lots of truth in this.

I liked teaching for O'Reilly and never having to hand out grades.  Instead I'd just ask for improvements and let them keep working on it. 

But in the regimented dead-line driven way they do it now, we can't afford this kind if "set your own pace" approach.  It's a machine that turns at an inexorable pace and "fails people" based on whether they work fast enough.  The old factory model.  A form of Taylorism.

In some games and sports, the clock does matter (timed chess), as it does in war.  As one of the Math Warriors (a non-violent pass time), I'm aware of time as an important factor. 

Making the clock dictate when it shouldn't is an exploitable weakness that helps my Invisible Army win its war against ignorance and stupidity.  Isn't that what Capitalism is all about (a competitive sport?).

Bucky writes about Capitalism's Invisible Army quite a lot (why he got so much help from Applewhite? :-D).

 

So we need to focus on the parents.  If we want better education, we need better homes. We need to read to our kids, take them to libraries and give them books. But we also need to prepare them for math and computing.  We need to do more counting, more quantitative comparison, more quantitative prediction.  E.g., if it took us 12 minutes to bake this one sheet of cookies (or mow this side of the lawn), what time will we be finished baking all three sheets?  [We do this stuff in our head all the time, but the kids can't see what we are thinking."]  Get them a stack of paper places or coasters and fold circles.  Get them straws and pipe cleaners and built polytopes. Get them a tablet and spend an "hour of code" with them and build some games.

I'm all for guardians being teachers!

We learn the most from peers and family members sometimes.

My 87 year old mom, who stays with me part of the year, is currently reading a book on the history of DARPA and sharing the juicy bits. 

Here I am, 58, and still learning new stuff from my own mother!
 

When I was a kids, we played board games with dice, and card games.  We were doing comparisons, ordering, at least adding, even multiplying (5 points for each this and 10 points for each that). We went to the store and had to count change and see how many 2 cent candies we could get for our nickel.  Now the game machine does all the calculating for them.

The <guild /> code school I've been writing about a lot (helps to have a brick and mortar example front and center) is all about teaching programming in conjunction with board games. 

Our lead instructor when I got here, fresh from O'Reilly Media, Tiffany, was a board games enthusiast. 

Geek culture is extremely into board games it seems. 

Our IT staff at O'Reilly repurposed an HDTV to become a Settlers of Catan board.  It lay on its back on the lunch room in Champaign-Urbana.  They played on lunch break.

 

This setup of making something named "math" mandatory with something named "CS" playing second fiddle, is a symptom of deeply flawed thinking, a dying civilization (good riddance).

The kids themselves will revolt--at least, the ones who have computers.

How ya gonna keep 'em learning their times tables after they've played Minecraft?

How ya gonna keep 'em using pencil and paper after they've done "hour of code"?

This revolt is already at least ten years behind us?  It's a fairly quiet revolt as revolts go.

The kids bending over backwards to humor the adults, still. 

Most kids want to go along to get along.  They're smaller, easier to cow.
 

I've seen it happen in other fields.

To the consternation of many a piano and violin teacher,

the kids of my generation all learned to play guitar instead.

They didn't need teachers; they taught each other.

I remember reading some "advice to young people" that said guitar was a poor choice of instrument because there was so little repertoire for it.

But the pattern of older and therefore possibly more experienced, teaching the younger, is not going to go away.  Younger teaches older too though.  The majority of my still-living teachers are by now younger than me.
 

Integrating numeracy and computation, as well as composition and calculation, is probably not something our present setup can accomplish. 

Not a problem.  We're setting up new systems as we speak.

We have the Internet, and more and more people staying home to really learn something, to study. 

Yes, they will survive us.

Joe

I'm planning on joining them, staying home and studying. 

I'll keep doing outdoor stuff too, and maybe teach in a classroom now and then. 

I've been teaching on-line to Californians until recently (why doesn't Oregon have a similar program?).

However I'm not planning to enslave myself to a "testing machine" that "fails people" just because they're more interested in learning something relevant, like hex numbers.  I'm too much of a capitalist for that.

Kirby