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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 30, 2005 7:42 PM
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> Well, if we are going to split hairs, it's "phee",
> but who cares!

In English we say fie, as in fee FIE foe fum.

> The two roots (actually, halves of the two
> roots, if I recall correctly) generate the Fibonacci
> sequence.

The phi sequence is both geometric and recursively additive.

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.

Makes for a fun computer programming experience.

> I'll leave you with this point as an exercise.
> But back to Kirby's question, my answer would simple.
> We could just as well take another polynomial to
> establish some sort of a special or classical
> relationship. The question really should be, "How is
> the application of this polynomial important to the
> student's learning progress?" The issue should not be
> of the importance of the particular piece in the
> *history* of mathematics, but its importance in the
> learning process.

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).

If you've got other polynomials with deep histories to share, by all means do so. Let's add to our list.

> There are no irreplaceable parts in the curriculum, but
> many of the ideas we want students to learn may well be
> irreplaceable.

phi is an idea and is irreplacable.

> The trouble is, we cannot agree on which ideas these
> are.

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

> Kirby, if you want to use a particular polynomial as
> the cornerstone of some part of the curriculum, more
> power to you. But it does not mean that everyone (or
> anyone) should follow your example.
> VS-)

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.

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

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.

Remember, Computer Programming for Everybody (CP4E) was a DARPA-funded initiative. So *of course* we're interested in our Pentagon's geometry.

Ten-hut! March!


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