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Topic: Non-Euclidean Arithmetic
Replies: 108   Last Post: Sep 13, 2012 3:39 PM

 Messages: [ Previous | Next ]
 Clyde Greeno @ MALEI Posts: 220 Registered: 9/13/10
Re: Non-Euclidean Arithmetic
Posted: Sep 10, 2012 4:13 AM

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

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).]

)]

Cordially,

Clyde

- --------------------------------------------------
From: "Joe Niederberger" <niederberger@comcast.net>
Sent: Sunday, September 09, 2012 11:01 AM
To: <math-teach@mathforum.org>
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

Date Subject Author
9/1/12 Jonathan J. Crabtree
9/1/12 Paul A. Tanner III
9/2/12 kirby urner
9/3/12 Paul A. Tanner III
9/3/12 kirby urner
9/3/12 Paul A. Tanner III
9/4/12 kirby urner
9/4/12 Paul A. Tanner III
9/4/12 kirby urner
9/5/12 Paul A. Tanner III
9/5/12 Robert Hansen
9/6/12 kirby urner
9/1/12 kirby urner
9/1/12 Joe Niederberger
9/1/12 Wayne Bishop
9/1/12 Joe Niederberger
9/2/12 Robert Hansen
9/3/12 Paul A. Tanner III
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Paul A. Tanner III
9/4/12 Joe Niederberger
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 kirby urner
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Joe Niederberger
9/6/12 Joe Niederberger
9/8/12 Robert Hansen
9/7/12 Jonathan J. Crabtree
9/8/12 kirby urner
9/8/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Joe Niederberger
9/8/12 Jonathan J. Crabtree
9/9/12 kirby urner
9/8/12 Clyde Greeno @ MALEI
9/8/12 Jonathan J. Crabtree
9/8/12 Jonathan J. Crabtree
9/8/12 Joe Niederberger
9/8/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/10/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/9/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Wayne Bishop
9/9/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/10/12 Paul A. Tanner III
9/10/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/13/12 Paul A. Tanner III
9/13/12 kirby urner
9/13/12 Paul A. Tanner III
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 kirby urner
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 israeliteknight
9/11/12 Joe Niederberger
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/11/12 Jonathan J. Crabtree