In fact, the very idea that in "algebra", the letters (or other expressions) should have *numeric* meanings is very much a SCHOLASTIC notion ... which still leads many educators to mistakenly think that arithmetic *should/must* come before algebra.
Professional algebraists invoke no such requirement. Neither do children ... who (age 5+) readily speak of "2 knives and 3 forks and 4 spoons" ... and are easily prompted to write it as 2K+3F+4S. Mathematically, Gattegno's rod-ventures actually were not actually about colored rods, as such ... but about the vector-algebra that is intrinsic to all civilized languages. Educators have yet to recognize that all civilized children ... if there are such things ((:- )) ... internally use (indeed, rely on) elements of vector algebra before they can make personal sense of "3H(undred), 4T(ee),5". ... 3H+4T+5S ... as in S(ingles).
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.
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.
From: Louis Talman Sent: Monday, October 29, 2012 11:58 PM To: Robert Hansen Cc: email@example.com 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 <firstname.lastname@example.org> wrote:
On Oct 29, 2012, at 1:26 PM, Joe Niederberger <email@example.com> 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?
-- --Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State College of Denver