>The convergence is quite easily derived. With Phi = (1+SQRT5)/2, as n goes >to infinity, |(1-SQRT5)/2|^n (<1) goes to zero so lim[f(n+1)/f(n)] for n -> >inf is Phi. > >It's indeed a very rich topic (multiple points of entry -> connections)
Thank you for doing the derivation. It's been long enough that I did not have the full recall and I have been swamped this week so didn't have the opportunity to redo the detailed analysis.
I want to point out that my objection to Kirby had nothing to do with the richness of the problem or the context. My concern is the same as in the case of any prescriptive curriculum. The fact that something is rich and interesting is not a justification for mandating, but merely a strong indication that it can be a recommended approach. I have noted in the past that neither the reform nor anti-reform crowd seems to get this. I know that this exactly what got Wayne & Co to arbitrarily place me deeply in the ultra-fuzzy camp. Of course, they chant the Bush mantra everyday: "Either you are with us or you are with the terrorists." So it's pointless to argue with them about relative merits of anything. Of course, left to their own devices, they will eventually encounter parts of curricula on which they disagree. At the point, they'll either resort to another (un)civil war or will lose interest in the subject. My concern is not with their eventual demise but the damage they would do in the meantime.
Many would-be reformers suffer from a similar Napoleonic complex. Sometimes, well-intentioned attempts to improve the standards also run into absolutists bureaucrats who don't get the difference between mandates and recommendations. This is how examples of good ideas (as I think Kirby's is) end up as required elements of state curricula. Unfortunately, both sides recognize only the extremes. The fact that the majority of practitioners is and should be in the middle, does not stop them from pulling the standards to the extremes. Although there is a reason to be concerned that being "centrist" is a wishy-washy, spineless position (just take a look at Lincoln Chafee), this need not be the default state for a "centrist". Being in the middle means carefully navigating the issues and making independent decisions--something that the zealots on either sides could never accept.