> 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.