Date: Oct 30, 2012 4:41 PM Author: Clyde Greeno Subject: Fw: Why?

- --------------------------------------------------

From: "Clyde Greeno @ MALEI" <greeno@malei.org>

Sent: Tuesday, October 30, 2012 3:35 PM

To: "kirby urner" <kirby.urner@gmail.com>

Subject: Re: Why?

> I believe (?) that you have already read "The Vector Algebraic Theory of

> Arithmetic" ...

> http://arapaho.nsuok.edu/~okar-maa/news/okarproceedings/OKAR-2005/clgreeno-part1.html

> .... and

> http://arapaho.nsuok.edu/~okar-maa/news/okarproceedings/OKAR-2006/greeno.htm

> ???

>

> But, see #, below

> --------------------------------------------------

> From: "kirby urner" <kirby.urner@gmail.com>

> Sent: Tuesday, October 30, 2012 12:44 PM

> To: "Clyde Greeno @ MALEI" <greeno@malei.org>

> Cc: <math-teach@mathforum.org>

> Subject: Re: Why?

>

>> On Tue, Oct 30, 2012 at 8:07 AM, Clyde Greeno @ MALEI <greeno@malei.org>

>> wrote:

>>

>>> Gattegno was not actually *proposing* that algebra "should" come before

>>> arithmetic. He was *observing* that vector-algebra necessarily *always

>>> does*

>>> 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

>> (cuboctahedron).[1]

>>

>> 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

>> prime:

>>

>> http://www.flickr.com/photos/kirbyurner/3725917904/ (after their

>> prime, seen in later life, largely retired from roadshow appearances

>> - -- back lot pose).

>>

>> Kirby

>> [1] 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?

>

> Cordially,

> Clyde

>

>

>

>

>

>>

>>> The mathematics of the colored rods does not come from the rods, as

>>> such,

>>> but from how the teacher uses them. The teacher who is unaware of

>>> children's

>>> use of vector algebra is unlikely to perceive the rods within a

>>> mathematical

>>> context.

>>>

>>> Cordially,

>>> Clyde

>>>

>>> From: Louis Talman

>>> Sent: Monday, October 29, 2012 11:58 PM

>>> To: Robert Hansen

>>> Cc: math-teach@mathforum.org

>>> Subject: Re: Why?

>>>

>>> Traditional algebra requires letters. But words are symbols, too. Use

>>> of

>>> 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 <bob@rsccore.com> wrote:

>>>>

>>>>

>>>> On Oct 29, 2012, at 1:26 PM, Joe Niederberger

>>>> <niederberger@comcast.net>

>>>> wrote:

>>>>

>>>> > 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

>>>

>>> <http://rowdy.mscd.edu/%7Etalmanl>