Date: Oct 30, 2012 4:41 PM
Author: Clyde Greeno
Subject: Fw: Why?
From: "Clyde Greeno @ MALEI" <firstname.lastname@example.org>
Sent: Tuesday, October 30, 2012 3:35 PM
To: "kirby urner" <email@example.com>
Subject: Re: Why?
> I believe (?) that you have already read "The Vector Algebraic Theory of
> Arithmetic" ...
> .... and
> But, see #, below
> From: "kirby urner" <firstname.lastname@example.org>
> Sent: Tuesday, October 30, 2012 12:44 PM
> To: "Clyde Greeno @ MALEI" <email@example.com>
> Cc: <firstname.lastname@example.org>
> Subject: Re: Why?
>> On Tue, Oct 30, 2012 at 8:07 AM, Clyde Greeno @ MALEI <email@example.com>
>>> Gattegno was not actually *proposing* that algebra "should" come before
>>> arithmetic. He was *observing* that vector-algebra necessarily *always
>>> come before arithmetic. What he proposed was that educators could/should
>>> capitalize on that aspect of human nature.
>> That's an interesting statement feel free to elaborate. If you mean a
>> colored rod with a ratio to other rods is a "vector" and laying them
>> side by side in colorful patterns, like weaving a rug of threads, is
>> arithmetical in nature, then yes, I agree with you.
> # Your scenario is interesting, but seemingly not what I meant. What I
> meant was that all children early learn to think and talk in "combos" ...
> "linear" combinations of things.
> Each "combination" of rods can be perceived as a combo --- using whole
> numbers as "scalars" for counting all rods of a single (length/color)
> kind. Each "scale" is the succession of same-kind *quantities* ... as with
> 0-R(eds), 1-R(ed), 2R, 3R, ....
> Such (poly-namial) combos can be scalar-added/subtracted and
> multiplied/remainder divided by whole numbers. But yours seems to go the
> further step ... of imposing equivalence classes (e.g. 2 of kind-a ~ 3 of
> kind-b). Such "ratio" perceptions are crucial not only for rod-fractions,
> but also for child-measurements and for Arabic arithmetic: in Roman, 345
> = 3C+4X+5I ... where 1C~10X and 1X~10I. Vector algebra does not
> *necessarily* invoke equivalence classes of combos, but it certainly does
> allow them.
> >A "vector" is a
>> kind of "edge" in primitive terminology, with directionality a
>> secondary characteristic. Notions of "ray" and "line" as well as
>> "line segment" come in from Greek metaphysics, where all terms are
>> infinite by default (points being infinitely "not sizable").
> # Most geometric meanings of "vectors" are special-context applications of
> algebraic vectors ... having lengths and/or directions or neither ... and
> enjoying all of the scalar operations. But I am not well enough versed on
> graph-theory to perceive any specific connections to edges.
>> In the few times I've scuba dived to 2nd grade and earlier, when my
>> daughter was in those years, I'd show up in the school and have them
>> categorize their surroundings in terms of V (corner), F (surface,
>> window, gap), E (edge, vector, boundary).
> # Categorizing things into "kinds" does set the stage for envisioning all
> possible quantities of each kind. The (additive) combos of such quantities
> then become subject to the vector operations.
>>A door is an F-like object,
>> a crease where two walls adjoin is an E-like object and so on. That
>> gets V, F and E anchored in experience, then we quick apply them to
>> polyhedrons, which we make, import, view on screen (project), hold in
>> hands etc. Here's a memory of me in Bhutan, doing just such a Lesson.
>> We called them "shapes". I left behind a high level write-up, with
>> all that stuff about V + F == E + 2 and 10 * f * f + 2
>> In the Montessori pre-school I visited, we talked about polyhedrons as
>> "measuring cups" as are found in the kitchen. Chances are that even
>> at that age there's been some home schooling in measurement. My
>> polyhedra had open lids and we poured beans or rice from one to
>> another. They were sized in a canonical way. I shared this in
>> Lesotho as well. Here's a picture of those polyhedrons, when in their
>> http://www.flickr.com/photos/kirbyurner/3725917904/ (after their
>> prime, seen in later life, largely retired from roadshow appearances
>> - -- back lot pose).
>>  http://www.flickr.com/photos/kirbyurner/3859886616/
> # COOL photo. Also a very interesting mode of distinguishing volumes from
> surface areas. Might also be used for "basic literacy" education in
> various aspects of geometric measurements ... including the vector theory:
> e.g. 3gal+ 5Qt + 37oz can be *reduced* to ????.
> Would you ... or yours ... or anybody ... be interested in contributing to
> an emerging, open-source, NPO, Mathematics-As-Common-Sense^TM video
> library about family-life measurements?
>>> The mathematics of the colored rods does not come from the rods, as
>>> but from how the teacher uses them. The teacher who is unaware of
>>> use of vector algebra is unlikely to perceive the rods within a
>>> From: Louis Talman
>>> Sent: Monday, October 29, 2012 11:58 PM
>>> To: Robert Hansen
>>> Cc: firstname.lastname@example.org
>>> Subject: Re: Why?
>>> Traditional algebra requires letters. But words are symbols, too. Use
>>> words is no reason to say a kid isn't doing algebra---after all, the
>>> beginning of algebra is the replacement of numbers with symbols for
>>> arbitrary numbers.
>>> On Mon, Oct 29, 2012 at 12:49 PM, Robert Hansen <email@example.com> wrote:
>>>> On Oct 29, 2012, at 1:26 PM, Joe Niederberger
>>>> > Clyde says:
>>>> >> The child who has already learned to calculate the area of a
>>>> >> rectangle
>>>> >> is ready to *abstract* such proceedings by creating and using a
>>>> >> FORMULA for
>>>> >> doing so ... perhaps LxW or BxA..
>>>> > Oops! I forgot (regarding above): How about "length x width"?
>>>> > Joe N
>>>> No, it can't be length x width, those are not letters. It has to be
>>>> letters. Don't you know algebra?
>>>> Bob Hansen
>>> --Louis A. Talman
>>> Department of Mathematical and Computer Sciences
>>> Metropolitan State College of Denver