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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 ]
 Alberto C Moreira Posts: 266 Registered: 12/6/04
Re: Is this an exceptionally hard set of questions to answer?
Posted: Oct 29, 2002 7:35 AM

Kevin Foltinek <foltinek@math.utexas.edu> said:

>So rather than talk about the properties of addition, you would rather
>define and understand addition in terms of the counting operation?
>This can be done but it requires a fairly sophisticated understanding
>of counting - a level of sophistication which most six-year-olds do
>not have.

I wouldn't bother to define anything, nor would I bother to tell them
that something like a "property" even exists. That's for much later.

>Indeed, I'd like to see you first write down the axioms or properties
>of counting, and then use those to explain associativity of addition.
>Or do an example: starting with your written down axioms of counting,
>explain why 6=5+1=1+2+3. (Make sure you do this in a way that a
>six-year-old will be able to follow.)

I would do nothing of the sort. I would postpone the concept of
associativity to senior high or so, when they first need to face it in
its formal embodiment. 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: count to 6, you get to 6; count to five then one more, you
get to 6; count one, then two more, then three more, you get to 6.
This is so intuitively obvious that there's no need to spend any time
on it ! I first faced modern math when I was in senior high, and I had
trouble taking it seriously, because everything that my teacher passed
down to me was basically, as I saw it then, reiterations of the
obvious, but done in a rather complicated way - and it took me a while
to see the point of it, and even now, years after I got over that
hump, I hardly see the point in complicating with formalism something
that's much easier acquired with intuition. There will be time later
for the formalism !

>An abacus involves a "digits" representation of a number, and this is
>a much more sophisticated concept than the "length of a rod"
>representation. Indeed, to understand digits, you first must
>understand both addition and multiplication.

We have fingers - our hands are our abacuses, and couting with fingers
is intuitive. An abacus is just an extension of our hands. Counting
is, in fact, well upstream from continuous realizations such as the
length of a rod. 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.

Alberto.

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