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Fw: Why?
Posted:
Oct 30, 2012 4:41 PM


  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/~okarmaa/news/okarproceedings/OKAR2005/clgreenopart1.html > .... and > http://arapaho.nsuok.edu/~okarmaa/news/okarproceedings/OKAR2006/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: <mathteach@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 vectoralgebra 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 samekind *quantities* ... as with > 0R(eds), 1R(ed), 2R, 3R, .... > > Such (polynamial) combos can be scalaradded/subtracted and > multiplied/remainder divided by whole numbers. But yours seems to go the > further step ... of imposing equivalence classes (e.g. 2 of kinda ~ 3 of > kindb). Such "ratio" perceptions are crucial not only for rodfractions, > but also for childmeasurements 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 specialcontext 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 > graphtheory 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 Flike object, >> a crease where two walls adjoin is an Elike 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 writeup, with >> all that stuff about V + F == E + 2 and 10 * f * f + 2 >> (cuboctahedron).[1] >> >> In the Montessori preschool 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, opensource, NPO, MathematicsAsCommonSense^TM video > library about familylife 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: mathteach@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 algebraafter 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>



