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Topic: Which Polynomial?
Replies: 28   Last Post: Jul 27, 2006 3:13 PM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Re: Which Polynomial?
Posted: Jul 31, 2005 11:10 AM
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> At 07:42 PM 7/30/2005, Kirby Urner wrote:
> >In English we say fie, as in fee FIE foe fum.
>
> My dictionary says otherwise.
>


American Heritage gives both fie and fee. At minimum, both are correct. To say fie is incorrect, is incorrect.

> > The usual thing we do with the Fibonacci sequence is
> > show how the ratio (F[n+1]/F[n]) --> phi as n -->
> > infinity, where F[n+1] and F[n] are successive terms
> > in this series.

>
> No, no, Kirby. Look at the sequence f1^n+f2^n. Every
> recursively-generated sequence has a generating
> polynomial. f1 and f2 are the two roots.


No really, the usual thing is to approach phi using two successive terms of the sequence. That's not to say your exercise is irrelevant though -- although the sequence I'm getting from your instructions isn't 0,1,1,2,3,5,8,13... but 1,3,4,7,11,18,29...

> >Math-through-storytelling: part of what we're doing
> >is passing on a history, a "where we've been" story.
> >It's not like history is irrelevant. On the other
> >hand, phi to this day has geometric importance
> >-- its value undimished with time (like gold).

>
> Weeellll... It's a stretch. We no longer build
> Parthenons.


So many K-12 mathematics curricula have pathologically refused to take any responsibility for imparting any kind of historical perspective, except maybe in a few wimpy sidebars. I consider this inexcusible. Per my earlier posts, all math is "ethno" and we should leverage this to positive advantage, by letting kids in on the story line.

> >phi is an idea and is irreplacable.
>
> In itself--yes. But not in the curriculum.


Phi very much needs to be in the K-12 curriculum. That's my judgment. You may disagree. Is it in the California Standard as of 2005? That'll be interesting to look up.

> > I'd think phi, pi and e we could at least agree on,
> > though we might take different approaches in
> > introducing them.

>
> These are not in the same category. One is essential
> at an elementary level, the other--a bit later. The
> third is a mere curiosity--a bump on a path to
> knowledge.


These are not especially helpful characterizations -- such a haughty and dismissive tone, snooty. Ridiculous.

Given we're doing polyhedra a lot more (maybe not in your preferred curriculum, but certainly in mine), we cannot afford to bleep over phi. Plus sequences in general are getting new life, as we invest in the figurate and polyhedral numbers (ala 'The Book of Numbers' by Conway and Guy). So of course we want the Fibonaccis.

> >It's implicit in writing curriculum that one is
> >saying something like "follow me" -- and then only
> >some do, maybe none. That's a given.

>
> Really? You obviously are not a member of
> Mathematically Erect. Also, see the comment about mx+b
> above--the notation became a convention because it
> was used in a popular book, not because it has any
> meaning or convenience.


That's true with *many* conventions in use today. Some popular book used them, and has been imitated ever since. And you're correct that I'm obviously not a member of whatever it was.

> >However, within the rhetorical conceit of offering
> >leadership, it behooves me to be persuasive, to
> >exercise whatever leadership experience and
> >qualities I possess.

>
> If at first you don't succeed, try and try again...
>


Exactly. And even when I *do* succeed (which is fairly often), that doesn't mean our work is done.

> So: NCLB people, listen up, we're going to push Phi
> (FIE) and we're going to do it in connection with
> solving a Polynomial. This will provide positive
> reinforcement for (a) the quadratic formula (b)
> completing the square, plus it'll lead to connected
> topics such as: Fibonacci numbers, sequences, and
> five-fold symmetry ala pentagons.
>
> The NCLB people don't give a rat's ass!
>
> VS-)


Speaking for yourself of course.

Kirby



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