On Tue, Oct 30, 2012 at 8:07 AM, Clyde Greeno @ MALEI <email@example.com> 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).
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:
> 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: firstname.lastname@example.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 <email@example.com> wrote: >> >> >> On Oct 29, 2012, at 1:26 PM, Joe Niederberger <firstname.lastname@example.org> >> 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>