So Robert intuitively relied on vectors and quantities ... without having that vocabulary.
But his question, below, is fair enough. Much of vector-arithmetic is explained in my 2005 paper written to mathematicians ... but that is a bit terse for persons having only traditional education in mathematics. More casual illustrations are in order. His passage from "apples" to number processing was by no means is a departure from quantities.
Pronounce "345" in normal English ... "3H(undred), 4T(ee), 5" ... a linguistically condensed form for "3H(undreds) + 4T(ees) + 5(Singles) ... which young children easily abbreviate as "3H+4T+5S". The English *phonics* for Arabic numerals ... and also for the "... illions" numerals ... totally relies on vector-combinations of quantities. When so written, it LOOKS like algebra, and it is ... vector algebra ... but it is not the kind of algebra that presently is taught in the American core-curriculum. http://en.wikipedia.org/wiki/Vector_algebra Hardly surprising that non-math majors would not want to call it "algebra."
Are the H, T, and S "variables?" In the English language, a "variable" is anything that can vary ... over time, over locations, over populations, over choices, etc. The state "color" on the maps of the presidential election varied from state to state.
Young children commonly learn to pronounce "345" in normal English ... at first knowing only that it follows 344, but without actually owning the concepts of 3(hundreds) or 4(tees) ... much less of "3H(hundreds) + 4T(ees) + 5(Singles)." For a while, "hundred" and "tee" are ambiguous "variables" whose specific or alternative meanings are not yet perceived.
Even "apple" is a variable ... whose meaning varies, in accord with what apple(s ) is/are being presently meant by the word. For sure, a variable might have alternative *numeric values* for each of its alternative states ... just as the *price* or "weight" of an apple might vary, from apple to apple ... or from day to day, or from store to store. [electoral votes?] But most variables are devoid of numeric values for their various alternative meanings ... as "Mister x" can readily verify. And the labels for variables [like GNP] very rarely are single letters.
The supposition that an algebraic variable *must* be as single letter that has alternative numeric values is perpetuated by a scholastic core-curriculum that is traditionally geared toward imparting skills for calculating. In truth, the variables are not the *letters* [GNP or "make"], but their (perhaps numerical) meanings.
The mathematics of vector algebra does not *depend* on its vectors having numeric values, although it does *allow* for such special-case possibilities. For example, if one does not read into it more than is shown, 3(x^2) + 4(x^1) + 5(x^0) is a vector. It uses 3, 4, and 5 as scalars ... and also uses denominations (x^2) , (x^1) , (x^0). Those three de-name-inations make the vector to be of the tri-nomen kind. Whether or not their variable meanings are numbers depends on who is talking about what.
In the core curriculum, those polynomial denominations are given numeric meanings. For the Arabic numerals, those three vector-denominations are given *constant* meanings: H, T, and S ... or if you prefer Roman numerals, C, V, and I. In base-number arithmetic, the numeric values of those denominations vary, depending on what base-numbers are chosen. When the numeric values are varied throughout the rational numbers, the vector-formula generates a parabolic curve. All of matrix algebra depends combos of "row vectors" and "column vectors" ... which use scalar entries, but do not have numerical values for the vectors, themselves.
In vector algebra, in general, the ("linear") combination, 3(x^2) + 4(x^1) + 5(x^0) ... or (3,4,5) ... does not rely on the denominations having any specific meaning ... numeric or otherwise ... as long as none of them is a combo of the others. Indeed, to minimize the confusion, the superscript exponents commonly are dropped into subscript exponents.
More directly pertinent to arithmetic: the vector, 3(4ths) +5(6ths), is a "combo" of two quantities having two differing denominations. The "measurement conversion tables" make that particular combo equivalent to the combo, 9(12ths)+10(12ths).
As for the decimal equivalence: the quantity, 3(7ths) , is quantitatively seen as being the result of dividing 3(wholes) by 7 ... perhaps by fractioning the wholes, one at a time. Decimate 3(wholes) into 30(10ths), and divide by 7(shares) ... yielding 4(10ths) per share, with a remainder of 2(10ths). Decimate that remainder into 20(100ths), and divvy for seven quotas, thereby yielding 4(10ths)+2(100ths) per share ... remainder 6(100ths). Decimate that remainder into ....
Because the remainders' scalars always are digits, the process always will "loop." But as the remainders get ever smaller, the successive values for the *decimal-point* quotients get ever closer to 3(7ths).
The quantity, 256(1-squares) can be organized as 128(1-squares)-by-2(1-squares) ... 128S x 2S ... or in other ways ... not because of their vector-ness, but of the kind of "multiplication" used for quantities of that particular kind.
For sure, one can deal with numbers without reverence to quantities and their vector-combos. But the commonsensibility of numeric procedures rests in quantitative conceptual under-standings ... which is why persons who wish to perceive or teach "number sense" normally revert to (vector-combos of) quantities.
Hope that helps
From: Robert Hansen Sent: Friday, November 09, 2012 12:52 PM To: firstname.lastname@example.org Subject: Re: How teaching factors rather than multiplicand & multiplier confuses kids!
On Nov 9, 2012, at 12:50 PM, Joe Niederberger <email@example.com> wrote:
I agree with Clyde - children learn to count, and even to count perhaps in pure numbers. Its a game, with its own rhythm, like skipping rope. But expressions like "how many" beg for an object: "how many apples". Beginning arithmetic such as addition has always been greatly aided by grounding in real world objects. That's how children begin to understand expressions like 3 + 5; they relate it to a real-world situation, like "3 apples + 5 apples".
One last thing Joe. If you agree with Clyde then you must understand Clyde, correct? Then explain to us, if you will, Clyde's theory of vector algebra for toddlers. Maybe you jumped into the middle of a long and continuing debate, but I never said that I didn't use apples to teach counting or to start addition. Who doesn't? The difference is that after that relatively short phase, I then went on to numbers, and we haven't looked back since. Clyde on the other hand is doing something with vector algebra. You agree with him, so tell us, WHAT IS IT?
The counting/adding/apple phase ends pretty quickly in the second grade. I shed a tear every time I remember my son adding on his fingers. They grow up so fast.