Hitting two targets: OO + group theory (pedagogy)

[Kirby Urner]

ullrich at math.okstate.edu (David C. Ullrich) wrote:

>On Fri, 02 Mar 2001 18:07:16 -0800, Kirby Urner
><urner at alumni.princeton.edu> wrote:
>
>>
>>============== OREGON CURRICULUM NETWORK =====================
>
>Does the Oregon Curriculum Network have any official status, ie
>any actual connection with the state of Oregon?
>

Other than the fact that I live and work in Oregon, as do
some of the people I collaborate with, no.

On the home page it says, in italics at the bottom:

      The OCN is privately sponsored and does not claim 
      to represent the official views of any government 
      agency

I visit schools sometimes, public and/or private.  Just
this week I guest-taught two segments of 1st grade re 
polyhedra.  Teacher workshops too -- was at the Math Summit
in 1997, which was about mapping a state-wide strategy
(Sir Roger Penrose was there to help us out).

>One comment would be that if we're talking about pedagogy and
>curricula and things we probably want to get the terminology
>straight. There's no such thing as a "relative prime" - you
>meant "the positive integers less than n which are
>relatively prime to n."

Yes.  All integers k, 0 < k < n, such that gcd(k,n)=1.

Thanks for the pointer re no such animal as a "relative
prime".  I'll need to tighten up my syntax.

>And more important there's no reason why the modulus n
>should be prime! This gizmo is a group under multiplication
>whether n is prime or not. (That's the point to specifying
>we're talking about the integers relatively prime to n...)

Very true.  In the referenced math-learn draft
http://www.mathforum.com/epigone/math-learn/shehlimpmimp
I do most of my example with modulus n = 20, showing how 
integers coprime with 20 (< 20) form a group.  I shouldn't
have stipulated a prime modulus in the above post.

>Seems more interesting to me to start with a class
>Group that includes stuff common to all groups, so
>that a subclass can represent any group at all. If we're
>going to implement what's above as part of a
>"curriculum" then at least we should make it clear
>that our sandbox is not "groups", it's a tiny part of
>the category of all groups.

Yes, just an example of a group.  In another essay, I 
go with permutations, showing how these might be associated
with "clubhouse codes" (exchanging letters for other 
letters as in a Julius Caesar code).  I'd also like 
to build on my OO + Polyhedra approach to get into 
symmetry groups -- getting topics to connect around to
other topics as much as feasible is what I mean by a
"curriculum" (stuff connecting around in all 
circumferential directions, to form a system -- the
notion of circling-back is built right into the 
etymology).

I try to be fairly general in my approach to the 
group of relatively prime integers under multiplication
vis-a-vis modulus n in my little essay for 8th graders:

   When we study group theory, we have a mnemonic 
   that's helpful if you've studied the book of 
   Genesis at least enough to know the story of 
   Cain and Abel (use a search engine maybe).  Use 
   the letters CAIN to stand for "Closure, Associative, 
   Inverse, Neutral" -- and there you have the 
   properties you need to call your set & operation 
   a "group".

   These properties are defined with respect to a set 
   of elements, and an operation for relating them 
   (we generally call the operation "multiplication", 
   no matter what it actually does). If multiplication 
   is not just associative, but commutative as well, 
   then that's where Abel comes in -- we call it an 
   Abelian Group.

Probably there's some syntax that could be tightened
up here too.

Thanks for your useful feedback.  Much appreciated.

Kirby