Date: Nov 11, 2012 8:01 PM
Author: kirby urner
Subject: Re: How teaching factors rather than multiplicand & multiplier<br> confuses kids!
On Sun, Nov 11, 2012 at 3:00 PM, Robert Hansen <firstname.lastname@example.org> wrote:
>> Is that what you're talking about? Rhetoric?
> No, not rhetoric. Even though some rhetoric is based on formal reasoning, formal reasoning is not a required element of rhetoric, persuasion is. Formal reasoning is the ability to understand and work with an abstract theory. It is pedantic (for lack of a better word). Persuasion is not an element of formal reasoning.
> Note: All theories are abstract so "abstract theory" is redundant.
So I'm still wondering if your touristic excursions into math teaching
pedantry is "formal reasoning". Apparently you're not equating it
with "formal logic" ala the logic of Frege - Russell - Wittgenstein
(the latter in fragments, as by Philosophical Investigations it's
mostly prose, though deeply worked to be grammatical in a certain
>>> I am good with the extra for experts inserts, but you have to get the
>>> algebra first before you can teach the student the why behind it, otherwise
I think saying you're not needing to give "the why behind it" up front
is what too many teachers are saying: we'll tell you later what this
is for, "just trust us" or "you need to know it because it's on the
test" (a smug tautology -- don't let your teacher get away with such
cheap and easy retorts, have some standards!).
There's a credibility problem among math teachers that's mostly
overcome in STEM, which supplies its own context. Add physics to
calculus and voila, a healthy brew. You can subtract the physics away
again, but at least you knew enough to stick with hybrids. STEM is
all about hybridizing the disciplines, not keeping them "pure" (that
fixation of "purity" was a horrible mistake that so many fell into,
bless their souls).
Speaking of which: a shout out to Dan Suttin of OCTA-TETRA Museum, in
San Antonio, TX:
STEM recruiting drive -- check the cool Einstein poster!).
>>> it is just pretend. Besides, Clyde will tell you that since they are doing
>>> algebra, and since algebra is controlled by group theory, they understand
>>> group theory. So, no reason to teach group theory twice.:)
>> This is reinforcing the stereotype that you need to be especially
>> gifted to get it in a different sequence.
> No, I am only saying that you have to get algebra before you attempt to get the theory of algebra. In the context of my statement, expert simply means that the kid got algebra fairly well.
That's very confusing to students, to say we can't give you theory
before we give you practice. Why not both together? What's so hard
about Closure, for example?
We're just saying if you have two of this type of something, like a
turtle or integer or donut or fruit, that if you combine them using an
operation we'll call "multiply" (why not), that you always get a
resulting output of the same type.
Two donuts make a donut; two fruits make a fruit. But then these
aren't intuitive examples. What's intuitive is "begetting".
My hypothesis is the reason we don't get into group theory that much
with kids has more to do with the schoolish bias against R-rated
topics, i.e. sex.
Fibonacci numbers had to do with rabbits mating, but how many text
books for pre-17 year olds dare mentioning such a thing? Censoring in
this way is a neo-Victorian reflex.
In the meantime kids are "trash talking" at recess and on the bus in
so many R- and X-rated ways, comparing household yammerings, getting
an ethnographic mix (good prep for later prez types). Welcome to
Spike Lee movies and life in the big city.
Speaking in general terms of "types begetting types" is what Bible
Camp might be about, since Genesis has lots of begets in it.
That's where the puritanical will tolerate maybe a tiny bit of sex
talk: in a Biblical context. Heaven forbid though, that secular
schools run by the gummint should ever talk about rabbits "doing it"
(gummint sex education was the reason many would home school). No
Fibonacci numbers until you're in college!
In other words, to use 60s talk: teachers are / were too "hung up" to
teach higher math, because higher math is metaphoric and metaphors are
dangerously day dreamy and promoting of imagination.
Staid arithmetic, on the other hand (like what Paul teaches), safely
kills or neutralizes the imagination, sweeps it clean of all objects
but an abacus (if you're Japanese), or a calculator if you're American
(just press the buttons, what's with all this finger twitching?).
>> The segment I'm talking about involves distilling the totatives of a
>> number N, using the GCD algorithm and then showing how totatives
>> multiplied modulo N have
>> (a) Closure
>> (b) Associativity
>> (c) Inverse elements
>> (d) an Neutral element
>> (CAIN -- plus this group is also Abelian (Biblical pun)).
>> Explain what each of those properties of a group means. Play around
>> with more of them. It's simple stuff, easy peasy. Amenable to
Inverse and Neutral go together.
Every rabbit has a perfectly opposite rabbit such that, were these two
to find each other and have a child (unlikely), that child would be a
The Neutral One, when multiplied with another rabbit, gives a clone of
that other rabbit. A miracle. Like a black swan (these do exist by
the way -- some philosophers pretend otherwise).
In the integer world (domain) we call the rabbits "ints" and
((3 * 1) == 3) is True
(that's Python syntax), just as
((1 * 4) == 4) is True
i.e. each "coupling" with a Neuter gives a Same (clone). Lots of
games we could play, spiraling into it.
>> This is what's called "spiraling" by the way, where you go into
>> something a little, from one angle, and then get into it more later,
>> from another angle. John Saxon was emphatic about "spiraling".
> Yeah, well that isn't spiraling. Spiraling starts with a foundational treatment and the "spiraling" occurs after that, not before. What you wrote would be called enrichment or extra for experts.
There's nothing more foundational than the Bible and Genesis according
to many Americans, so that might be how we'll start at those summer
camps (when we're away from those public schools and their secular
Another beget story goes like this (the operation, instead of
"multiply" is called "dualing" -- not "dueling"), a unary operation:
Dual( Tetrahedron ) --> Tetrahedron
Dual( Octahedron ) --> Cube
Dual( Icosahedron ) --> Dodecahedron (pentagonal faced)
Yes, Dual( ) is a function, very good. It's also bijective. Note in
the domain above, they're are all members of the all-triangles set,
with Tetrahedron its own dual.
Yes, these are the Platonics, so very important to be called that
(check all USA-marketed 1980s - 2012 10th grade textbooks for mention
of "Platonic" -- a mixed bag, do a data visualization).
Now comes the Combine operation, which takes a poly and its dual and
combines them in a certain way, so that edges "criss cross":
Combine( Tetrahedron, Dual(Tetrahedron)) --> Cube
Combine( Octahedron, Dual(Octahedron)) --> Rhombic Dodecahedron
Combine( Icosahedron, Dual ( Icosahedron )) --> Rhombic Triacontahedron
- --> may be read as "begets".
Combine is a binary operation, a function of two arguments, or,
curried: Combine(Tetrahedron) is a function in search of
Dual(Tetrahedron) as its argument. This feeds in to Haskell and, more
generally (when we do some more turns) Lambda Calculus.
In Haskell, Tetrahedron `Combined` (Dual Tetrahedron) would be how to
use 'infix' notation instead.
That's probably what you mean by the difference between Algebra and
Algebra Theory, like the difference between Haskell and Lambda
In that sense, there are many algebras.
Anyway, none of the above is difficult and if that's the curriculum in
San Antonio, but not Sarasota, then so what?
Not everyone needs to be on the same page, in this great country of ours.
So lets *never* pretend we're in search of the one great National
Curriculum that all must adhere too.
That'd be anti-American both in spirit and in practice. I would never
want to be associated with such an unpatriotic endeavor.
>>> Junior is saddled with the same
>>> linear sequence is grandparents had. Is that a good thing? By
>>> Well, Junior is just human, like his grandparents. Naturally, the
>>> progression would be the same, right?
>> Is this what you call "formal reasoning" then?
>> You seem to consider yourself a good example of what your favored
>> curriculum would turn out.
> I would like more students to think rationally.
>> I assume we're being treated to an example of what "reasoning" means,
>> am I safe to assume that?
Different from just passing the Turing Test I assume. One doesn't
need to be especially rational to pass the Turing Test.
> While a forum like this would be classified as rhetoric, my rhetoric is based mostly on formal reasoning and yours mostly on ideology and word play.
Actually my thinking is quite self-consistent and disciplined
according to David Feinstein, Applied Mathematician. He likes that
I teach logical thinking for a living one could say. To hundreds. I
think my rhetoric is really top notch, one of the fastest horses on
the track (some days).
The points I've been making, that the curriculum is far from fixed,
and that my New Math education was already different than yours, even
one generation apart... did those points register at all? Do you
admit that the sequence is far from fixed?
> I have this theory that there is a natural progression to all of this. When we lay down the ideas and theory of mathematics in a student, that process inevitably follows the same pattern in which it was discovered by humankind.
Ontogeny recapitulates phylogeny -- an over-used, over-applied cliche
that's often misleading when not plain wrong
>I am not saying that we must visit every success and failure of the past, but the progression of sophistication is the same. This isn't a unique theory, it is known fairly well in art and music. When you study art through history, the topics like form, shadow, foreground, perspective, appear in the same order as they do when we teach art. This is because children begin with the same primitive understanding of art as did primitive adults in the past. And they advance through the layers of sophistication in art in the same way that the generations of artists did, in the past. Whether it is one child learning art, or the whole human race discovering it, the progression is the same. It is quicker for the modern child (it doesn't take 100's or 1000's of years) because they do not have to discover it as did humankind, we are able to teach it to them. Otherwise we would stay forever at square one. Irregardless of the fact that we can teach, we are still bound by the same natural p!
These types of theories usually assume the present civilization is a
pinnacle i.e. it was all leading up to now.
Stephan J. Gould and Lynn Margulis on the other hand, were more into
seeing "evolution" as "transformation" i.e. it meanders, wanders,
adapts. It's not getting "more this" or "more that" overall i.e. it's
not a progression so much as a random exploration of a possibility
space, a continuous morphing.
Humanity is not a pinnacle of anything cosmically speaking and our
current civilization is not a pinnacle of anything either. We
stumbled into this niche and we'll stumble out of it. So it goes.
<< epic tale snipped >>
<< ad for Cloud Atlas, epic movie with Tom Hanks and Halle Berry, other stars >>
> When a child begins the journey of acquiring the theory of mathematics they are very much like our settlers on the new earth starting from scratch. In fact, as the child advances in mathematics aren't they advancing the state of the art of their own thinking? They are building a sort of technology, a technology of mind. Just like are settlers, they have to go through the progression and just like our settlers they have to linger a bit at each iteration to ensure that the infrastructure they are building will support the next iteration.
"The theory of mathematics" -- as if we all knew what that means, or
as if there were only one. Is that what's believed in the state of
> The settlers already had the blueprint for technology. They knew the journey. When teaching a chid however, it is the teacher that holds that information. They have been there and done that. It is their job to guide the child on that journey.
> Bob Hansen
Lets not forget the Library phase of life.
In many cultures, it's important for neophytes / noobs to wander in
search of a teacher or teachers. It's not like you just immediately
have the right teachers at hand, just because that's how they were
assigned by the union and/or district on the basis of seniority or
whatever politics. No, you must search in the Library.
In the old days, that might mean leaving Florida, as Florida was not
known to have wise people or great libraries, but only those seeking
the Fountain of Youth (i.e. misguided cretins over-due for rebirth).
Today, however, there's Youtube, and you can seek teachers from almost
anywhere. A man named Khan has been popular. A young woman named Vi
has a following (they've collaborated as each recognizes the other as
talented). Find a spelling teacher while you're at it.
Find people who speak your language, relate to your concerns.
In this model, the job of early teachers is to teach reading and
research skills, some alpha-numeracy, and then to encourage the use of
these skills to browse the Internet in search of their next teachers.
They find some locally as well.
Parents are a good start, but they're often not home in primitive
societies that aren't smart enough to arrange for parenting by parents
(the USA had fallen to this extreme low of cultural IQ in many zip
codes by the late 1900s, before telecommuting and village living
really got going again).
The Dual-Combine way of introducing Polyhedrons, based on the model of
Genesis, may lead to more R-rated segments than some schools allow.
Fortunately, as an andragog (a teacher of adults), I'm not obligated
to censor my language as much as the pedagogs.
The Friends (religious group) have also come to realize that their
"plain speech" testimony is consistent with R rated speech, i.e. the
more explicit vernacular of the playground and Hollywood films is
sometimes the fastest way to get into the mathematics at hand.