Date: Oct 30, 2012 1:44 PM Author: kirby urner 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. 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").

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). 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/

> 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>