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Topic: Is this an exceptionally hard set of questions to answer?
Replies: 68   Last Post: Nov 11, 2002 7:54 PM

 Messages: [ Previous | Next ]
 Kevin Foltinek Posts: 680 Registered: 12/8/04
Re: Is this an exceptionally hard set of questions to answer?
Posted: Oct 29, 2002 12:38 PM

Alberto Moreira <junkmail@moreira.mv.com> writes:

> I would postpone the concept of
> associativity to senior high or so, when they first need to face it in
> its formal embodiment.

Associativity (and commutativity and distributivity) are needed and
used much sooner than senior high. For example, distributivity is
used extensively in the "long multiplication" algorithm.

> The fact that 6=5+1=1+2+3 doesn't need to be explained to a kid, it
> evolves naturally as the kid's number intuition develops

There's a little trap there: "1+2+3" does not even make sense until
you know that addition is associative.

The trouble with intuition is that it easily can be, and often is,
wrong. Indeed, it is intuitive to many students that (1/2) + (2/3) =
(1+2)/(2+3). The student who understands the properties of the
operators is (in my experience) less likely to make this sort of
error, while the student who merely develops an intuition that
operations "behave nicely", without understanding exactly what
"nicely" means, expects that the addition of fractions can be
accomplished using the "nice" and "intuitive" method.

I have heard comments that arithmetic and basic algebra appear as
"magic" to students, that there is some mysterious force involved that
makes things work. Things are only magic if you don't know what's
going on, if you don't know the rules.

> We have fingers - our hands are our abacuses, and couting with fingers
> is intuitive. An abacus is just an extension of our hands.

Nonsense. With two rows on an abacus, you can count to 100. With two
pairs of hands, you can count to 20 (assuming you use the usual simple
form of finger-counting). If you really think that an abacus is an
extension of our use of our fingers as counting tools, you don't
understand our digits representation of numbers (or you don't know how
to use an abacus).

> And as I pointed out before, the length of a rod is a
> false representation of a number: I can call the same length "1", "2"
> or "3.141592...", depending on how and where I use it. But a digit is
> a digit, and a finger is a finger.

If you refuse to recognize that the length of a rod is a perfectly
valid representation of a number once you have agreed on what has
length 1, fine. Make little cuts on the rods so that it looks like a
bunch of cubes glued together, then you can count the cubes. Many
children have no difficulty imagining these cuts (or similarly
imagining the rod made up of cubes). Apparently you do.

Kevin.

Date Subject Author
9/28/02 Karl M. Bunday
9/30/02 Alberto C Moreira
9/30/02 Shmuel (Seymour J.) Metz
10/5/02 Moufang Loop
10/7/02 Shmuel (Seymour J.) Metz
9/30/02 Stephen Herschkorn
9/30/02 Magi D. Shepley
10/1/02 Karl M. Bunday
10/2/02 Kevin Foltinek
10/2/02 Karl M. Bunday
10/3/02 Alberto C Moreira
10/3/02 Kevin Foltinek
10/3/02 Jim Hunter
10/4/02 Herman Rubin
10/4/02 Alberto C Moreira
10/5/02 Herman Rubin
10/4/02 Alberto C Moreira
10/4/02 Kevin Foltinek
10/5/02 Alberto C Moreira
10/6/02 Virgil
10/6/02 Herman Rubin
10/6/02 Jim Hunter
10/6/02 Virgil
10/7/02 Kevin Foltinek
10/8/02 Alberto C Moreira
10/8/02 Kevin Foltinek
10/9/02 Alberto C Moreira
10/10/02 Kevin Foltinek
10/11/02 Alberto C Moreira
10/14/02 Kevin Foltinek
10/15/02 Alberto C Moreira
10/15/02 Kevin Foltinek
10/16/02 Alberto C Moreira
10/16/02 Kevin Foltinek
10/14/02 Kevin Foltinek
10/16/02 Alberto C Moreira
10/16/02 Kevin Foltinek
10/12/02 Shmuel (Seymour J.) Metz
10/14/02 Kevin Foltinek
10/25/02 Van Bagnol
10/25/02 Alberto C Moreira
10/26/02 Van Bagnol
10/27/02 Alberto C Moreira
10/27/02 Herman Rubin
10/28/02 Kevin Foltinek
10/29/02 Alberto C Moreira
10/24/02 Van Bagnol
10/25/02 Van Bagnol
10/26/02 Alberto C Moreira
10/28/02 Kevin Foltinek
10/29/02 Alberto C Moreira
10/29/02 Kevin Foltinek
10/31/02 Alberto C Moreira
10/31/02 Kevin Foltinek
11/2/02 Alberto C Moreira
11/2/02 David Redmond
11/3/02 Alberto C Moreira
11/3/02 Alberto C Moreira
11/4/02 Kevin Foltinek
11/2/02 Virgil
11/4/02 Kevin Foltinek
11/5/02 Alberto C Moreira
11/5/02 Kevin Foltinek
11/6/02 Alberto C Moreira
11/7/02 Kevin Foltinek
11/9/02 Alberto C Moreira
11/11/02 Kevin Foltinek
10/3/02 Kevin Foltinek
10/5/02 Magi D. Shepley