Actually, it goes rather well ... much better than does the prevailing curriculum (repeated addition). Of course, you have to use a language that the child already owns ... or can soon learn
Our video tapes are used for clinical research and are revealing, but anyone else get the same results.
One effective model is U.S. coins. On the personal (1.5'x2') whiteboard, the child makes a "cash drawer" chart .... one column for each denomination of coins. Column headings: $, H,Q,D,N,C. Grab a handful of coins, and write out the "combo" on the whiteboard ... say 2$ +7Q + 4N +9C. Grade-1 level: add and subtract such combos. [Yup, vector algebra.]
In Grade 2, build the "evaluation" language ... each coin is worth #? cents. Then count out the C-values (@ *rate* of exchange) for the nickels ...and write down the results. 1N=5, 2N=10, .... [The variable is not the N; the N is the rate/slope/constant-of-proportionality. The variable is the "multiplier."] Children soon learn that finger counting works ... but for the Q, H, and $ denominations, it is too slow. So they learn to use the calculator ... and usually are surprised by what they see along the D and $ scales. [For non-U.S. denominations ...2s, 3s, 4s, etc. ... game chips suffice.]
Except for their writings. there is nothing at all new. Almost every curriculum includes "the chants", and almost every child sooner of later uses those with coins. But in fact, the C-per-N rate (5-per-1) is the "slope" for the (5x) "counting by 5s" progression/series/function/line/proportion/chant. The fact that early-childhood education must "speak in childish languages" does not change the mathematical substance. Indeed, the technical rhetoric that is used by professional mathematicians commonly obscures the fact that the mathematical substance of the discourse actually is very "childlike." So, the phrases "slope" and "mx functions" superficially appear to refer to relatively advanced constructs ... yes, a bit too advanced for most first graders.
So YES: very young children do "run" x, at the rate of m-per-+1 ... and the resulting value is mx.
[Re, your earlier note: Children do need to own the integers, before they can "run" x+yi at the rate of m+ni - per- 1+0i ... and no, I have not yet tried to teach (image) "multiplication of complexes" at levels below beginning algebra. No mathematical need to have more than the integers, because any 4-quadrant coordinate plane will suffice. Since complex multiplication is a simplistic kind of "weather map" animation, the meanings of complex numbers could be handled in any up-to-par grade-6 program.] [Whether or not they "should" is quite another matter.]
Hm. When -b+ai is regarded as the "image" of a+bi, a complex rhyme pertains: I have a little image that goes in and out with me ... its always shares my origin, and fits me to a T. That?s because its truly right from me, in a leftish sort of way ... cause if I left-turned 90, I'd be where it is, today. It?s quite reciprocative, though negatively so. For, the angle in between us is as right as it can go. [Don't try to look up the rhyme, you won?t find it. Just groan and draw the picture ... ... and also draw 3 of (a+bi) + 7 of (-b+ai) ... which is (3+7i) *of* (a+bi).]
- -------------------------------------------------- From: "Joe Niederberger" <firstname.lastname@example.org> Sent: Sunday, September 09, 2012 11:01 AM To: <email@example.com> Subject: Re: Non-Euclidean Arithmetic
> Clyde Greeno says: >>Within the arithmetics of real numbers (and subordinate systems), the >>global (mx) meaning of "multiplications" is all about using using >>multipliers, m, as per-1 rates/slopes of the mx *proportions* through the >>origin ... "run" x, at the rate of m-per-+1 ... and the resulting value is >>mx. > >>That describes grade-2 "multiplication" of Arabic digits ... > > I see. You've taught second graders this way I presume. > How did it go? > > Joe N