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RE: [xmca] Why Does N to the Power of Zero = 1?
Thanks to everybody who responded (of course)! It seems to me that Rod's answer is what Anna Sfard would call a PROCEDURAL one while David Ki's is more STRUCTURAL. That is, Rod is concerned with how the game is played and whether the answer makes sense in the context of the game rules (as he points out, it doesn't). David is concerned with how the different powers are related to to each other in a framework that is consistent.
Anna's point is that sociogenetically (and probably ontogenetically) structural knowledge is late emerging, just as it is in language: form follows function. But in the classroom, that's not always true; we sometimes start with the system and then show what we can do with it (for example, in foreign language teaching).
What interests me are the holes in the structure. For example, when we explicitly teach children that an English sentence is SVO, we can see that "subject" and "object" are at a different level of abstraction than "verb" (the former are structurally defined, while a verb is a functional category).
It seems to me that the same thing can be said about the number "zero", which is a number but not a quantity, and thus represents a kind of "hole". Even in David Ki's structural system, he has to represent 3 to the zeroeth power as "?" rather than as a known quantity.
A very short comment on the TITLE of the article for discussion. I don't think Vygotsky would have liked the term “sociocultural” because:
a) it seems redundant. I never heard of a society without a culture or a culture without a society. It seems to me that society and culture are like body and mind. Like Andy, I dislike terms like “embodied cognition”, which imply that somehow you can have cognition that is not embodied.
b) “sociocultural” has no obvious role for history, and I never heard of any change (learning OR development) that does not involve TIME.
I think Vygotsky would not have liked the term “cultural-historical” either, because:
a) it seems to me that culture is the product of history, and not the other way around, so it should really be historical-cultural. Society and culture are the way they are because they do what they have to do, but they do what they have to do because they have been what they have been
b) neither culture nor history (nor biology) is what we want to study here in Seoul. We want to study development and learning! So I guess we should call it "onto-microgenetic" or something.
To tell you the truth, I don’t really care what we call Vygotsky’s theory. I just call it “Vygotsky’s theory". But Vygotsky probably would have hated that most of all; it sounds like the name of a Shakespearian tragedy.
David Kellogg
Seoul National University of Education
--- On Mon, 2/28/11, Rod Parker-Rees <R.Parker-Rees@plymouth.ac.uk> wrote:
From: Rod Parker-Rees <R.Parker-Rees@plymouth.ac.uk>
Subject: RE: [xmca] Why Does N to the Power of Zero = 1?
To: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>
Date: Monday, February 28, 2011, 12:24 PM
Does raising to the power of x count as one of the four arithmetic operations, though?
Rod
-----Original Message-----
From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On Behalf Of ERIC.RAMBERG@spps.org
Sent: 28 February 2011 16:34
To: eXtended Mind, Culture, Activity
Subject: Re: [xmca] Why Does N to the Power of Zero = 1?
ooops typed after DKi's response. thank you DKi
From: ERIC.RAMBERG@spps.org
To: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>
Date: 02/28/2011 09:07 AM
Subject: Re: [xmca] Why Does N to the Power of Zero = 1?
Sent by: xmca-bounces@weber.ucsd.edu
proof is very long and complicated. had to do this for college level
class once. this is a relative late comer to the math game and has to do
with complex calculus. sure and xmcer out there has a copy of the proof.
From: David Kellogg <vaughndogblack@yahoo.com>
To: xmca <xmca@weber.ucsd.edu>
Date: 02/28/2011 12:42 AM
Subject: [xmca] Why Does N to the Power of Zero = 1?
Sent by: xmca-bounces@weber.ucsd.edu
I've been working on math games to play in the back seat of a car while
stuck in a Korean traffic jam (during the various lunar holidays when
everybody visits the ancestral graves).
So I've got a version of "Twenty Four" with cell phone numbers called
"Cell Phone Golf". In Korea, cell phone numbers usually have eight digits
once you remove the company code. Mine, for example is 3475 2505.
For the first hole of cell phone golf we split my number in two. My young
opponent takes 3475 and I take 2505. We both work to hit the number ONE
using the four arithmetic operations (and then two, and so on all the way
up to eighteen, or until the traffic clears up). My opponent tees off
first,
(3-4) + (7-5) = 1
Man, she's got a mean drive. Just look at those negative numbers. A hole
in one!
Now it's my turn. This is going to be hard to beat.
(2-5) + (0+5). Nah. that doesn't work....
I've got it!
(2 + 5 + 5) RAISED to the POWER of ZERO!
A hole in one! Ha! I explain to my young opponent that any number raised
to the power of zero is 1.
But she gives me a skeptical sidelong glance: I am suspected of
cheating. She demands to know WHY any number raised to the power of zero
is equal to one.
Can anyone help me out before the traffic improves?
David Kellogg
Seoul National University of Education
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